[sc:ecwidpage id=”28191144″ price=”$47.97″ title=”Pi and the Lost Function Bundle”] 
Includes:

Introduction
[sc:bquote quote=”Pi and The Lost Function helps each individual student with their own knowledge of each strand. It builds; it gives them variety to choose how they learn. I feel that Pi is a unique and wonderful tool for them to learn.” cite=”Michelle Roan” title=”Middle School Math Teacher”] Pi and the Lost Function math educational series is aligned with the Common Core State Standards in Mathematics, Grades 68. Features: Alignment to state standards
 Online Tracking and Reporting Tools
 Content selection to fit your students’ needs
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Additional Information for Educators
[sc:standardswidget]Pi and the Lost Function aligns with the Common Core State Standards in Mathematics, Grades 68.
Topic Areas Covered:
 Whole Numbers
 Integers
 Fractions & Decimals
 Exponents & Roots
 Rates, Ratios, & Proportions
 Equations
 Inequalities
 Graphing
 Probability & Statistics
 Geometry
Common Core State Standards  2010
6
Whole Numbers
6.EE.A.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
6.EE.A.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.A.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
6.EE.A.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.A.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
6.EE.A.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
6.EE.A.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
6.EE.B.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
6.NS.B.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
6.NS.B.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
6.NS.C.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
6.NS.C.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.C.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
6.NS.C.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
6.NS.C.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.C.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
6.NS.C.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
6.NS.C.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
6.NS.C.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
6.NS.A.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
6.NS.B.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
6.NS.C.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
6.NS.C.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
6.EE.A.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
6.EE.A.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
6.EE.A.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
6.NS.C.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
6.NS.C.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
6.NS.C.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
6.NS.C.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.A.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
6.RP.A.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
6.RP.A.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
6.RP.A.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.A.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
6.RP.A.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
6.EE.A.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.A.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.A.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
6.EE.B.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
6.EE.B.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
6.NS.B.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
6.EE.B.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
6.EE.B.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
6.NS.C.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
6.NS.C.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
6.EE.C.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
6.NS.C.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.C.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
6.NS.C.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.C.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
6.RP.A.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
6.RP.A.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
6.SP.A.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
6.SP.A.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.A.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.B.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
6.SP.B.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.B.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.B.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.B.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.B.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
6.G.A.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
6.G.A.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
6.G.A.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
6.G.A.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
6.NS.C.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
7.EE.A.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
7.EE.A.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
7.EE.B.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.EE.B.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.NS.A.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
7.NS.A.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.A.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
7.NS.A.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
7.NS.A.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.A.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
7.NS.A.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
7.NS.A.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.A.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
7.NS.A.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
7.NS.A.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.A.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
7.NS.A.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.A.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
7.NS.A.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.A.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.A.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
7.NS.A.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.A.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
7.NS.A.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.A.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.A.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.A.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
7.NS.A.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.A.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
7.NS.A.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
7.NS.A.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
7.RP.A.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
7.RP.A.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
7.RP.A.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
7.RP.A.2c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
7.RP.A.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
7.EE.A.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.B.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.B.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.NS.A.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.A.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.A.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
7.EE.A.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
7.EE.B.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
7.EE.B.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
7.RP.A.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.A.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.A.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.A.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
7.SP.A.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
7.SP.A.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
7.SP.B.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
7.SP.B.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
7.SP.C.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
7.SP.C.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.C.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.C.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.C.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.C.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.C.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.C.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.C.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
7.G.A.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
7.G.A.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
7.G.A.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
7.G.B.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
7.G.B.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
7.G.B.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
8.EE.C.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
8.EE.C.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
8.EE.A.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
8.NS.A.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
8.EE.A.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
8.EE.A.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
8.EE.A.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
8.NS.A.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
8.EE.B.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
8.EE.C.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
8.EE.C.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
8.EE.B.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
8.EE.B.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.EE.C.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.C.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
8.EE.C.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.C.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
8.F.A.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
8.F.A.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
8.F.A.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
8.F.B.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.F.B.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
8.SP.A.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
8.SP.A.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
8.SP.A.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
8.SP.A.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
8.SP.A.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
8.G.A.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.A.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.A.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.A.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.A.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
8.G.A.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
8.G.A.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
8.G.A.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
8.G.B.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.B.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.B.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
8.G.C.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
Arkansas Common Core State Standards  2012
6
Whole Numbers
6.EE.A.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
6.EE.A.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.A.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
6.EE.A.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.A.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
6.EE.A.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
6.EE.A.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
6.EE.B.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
6.NS.B.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
6.NS.B.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
6.NS.C.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
6.NS.C.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.C.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
6.NS.C.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
6.NS.C.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.C.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
6.NS.C.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
6.NS.C.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
6.NS.C.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
6.NS.A.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
6.NS.B.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
6.NS.C.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
6.NS.C.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
6.EE.A.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
6.EE.A.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
6.EE.A.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
6.NS.C.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
6.NS.C.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
6.NS.C.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
6.NS.C.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.A.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
6.RP.A.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
6.RP.A.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
6.RP.A.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.A.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
6.RP.A.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
6.EE.A.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.A.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.A.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
6.EE.B.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
6.EE.B.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
6.NS.B.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
6.EE.B.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
6.EE.B.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
6.NS.C.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
6.NS.C.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
6.EE.C.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
6.NS.C.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.C.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
6.NS.C.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.C.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
6.RP.A.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
6.RP.A.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
6.SP.A.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
6.SP.A.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.A.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.B.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
6.SP.B.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.B.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.B.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.B.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.B.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
6.G.A.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
6.G.A.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
6.G.A.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
6.G.A.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
6.NS.C.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
7.EE.A.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
7.EE.A.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
7.EE.B.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.EE.B.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.NS.A.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
7.NS.A.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.A.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
7.NS.A.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
7.NS.A.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.A.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
7.NS.A.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
7.NS.A.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.A.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
7.NS.A.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
7.NS.A.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.A.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
7.NS.A.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.A.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
7.NS.A.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.A.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.A.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
7.NS.A.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.A.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
7.NS.A.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.A.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.A.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.A.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
7.NS.A.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.A.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
7.NS.A.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
7.NS.A.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
7.EE.B.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
7.RP.A.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
7.RP.A.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
7.RP.A.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
7.RP.A.2c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
7.RP.A.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
7.EE.A.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.B.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.B.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.NS.A.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.A.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.A.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
7.EE.A.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
7.EE.B.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
7.EE.B.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
7.RP.A.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.A.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.A.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.A.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
7.SP.A.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
7.SP.A.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
7.SP.B.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
7.SP.B.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
7.SP.C.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
7.SP.C.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.C.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.C.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.C.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.C.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.C.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.C.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.C.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
7.G.A.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
7.G.A.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
7.G.A.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
7.G.B.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
7.G.B.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
7.G.B.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
8.EE.C.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
8.EE.C.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
8.EE.A.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
8.NS.A.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
8.EE.A.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
8.EE.A.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
8.EE.A.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
8.NS.A.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
8.EE.B.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
8.EE.C.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
8.EE.C.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
8.EE.B.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
8.EE.B.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.EE.C.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.C.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
8.EE.C.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.C.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
8.F.A.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
8.F.A.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
8.F.A.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
8.F.B.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.F.B.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
8.SP.A.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
8.SP.A.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
8.SP.A.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
8.SP.A.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
8.SP.A.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
8.G.A.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.A.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.A.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.A.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.A.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
8.G.A.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
8.G.A.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
8.G.A.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
8.G.B.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.B.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.B.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
8.G.C.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
California Common Core State Standards  2013
6
Whole Numbers
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
6.EE.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
6.EE.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
6.NS.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
6.NS.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
6.NS.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
6.NS.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
6.NS.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
6.RP.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
6.RP.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
6.EE.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
6.EE.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
6.NS.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
6.EE.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
6.SP.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
6.SP.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
6.SP.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
6.G.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
6.G.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
6.G.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
6.G.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
7.EE.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
7.NS.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
7.RP.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
7.RP.2c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
7.RP.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
7.EE.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
7.RP.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
7.SP.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
7.SP.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
7.SP.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
7.SP.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
7.SP.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
7.SP.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
7.G.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
7.G.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
7.G.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
7.G.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
7.G.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
7.G.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
8.EE.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
8.NS.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
8.EE.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
8.EE.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
8.NS.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
8.EE.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
8.EE.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.EE.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
8.EE.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
8.F.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
8.F.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
8.F.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
8.F.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.F.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
8.SP.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
8.SP.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
8.SP.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
8.G.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
8.G.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
8.G.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
8.G.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
8.G.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
8.G.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
Florida Common Core State Standards  2010
6
Whole Numbers
MACC.6.EE.1.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
MACC.6.EE.1.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
MACC.6.EE.1.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
MACC.6.EE.1.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
MACC.6.EE.1.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
MACC.6.EE.1.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
MACC.6.EE.1.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
MACC.6.EE.2.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
MACC.6.NS.2.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
MACC.6.NS.2.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
MACC.6.NS.3.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
MACC.6.NS.3.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
MACC.6.NS.3.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
MACC.6.NS.3.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
MACC.6.NS.3.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
MACC.6.NS.3.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
MACC.6.NS.3.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
MACC.6.NS.3.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
MACC.6.NS.3.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
MACC.6.NS.1.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
MACC.6.NS.2.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
MACC.6.NS.3.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
MACC.6.NS.3.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
MACC.6.EE.1.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
MACC.6.EE.1.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
MACC.6.EE.1.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
MACC.6.NS.3.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
MACC.6.NS.3.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
MACC.6.NS.3.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
MACC.6.NS.3.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
MACC.6.RP.1.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
MACC.6.RP.1.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
MACC.6.RP.1.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
MACC.6.RP.1.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
MACC.6.RP.1.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
MACC.6.RP.1.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
MACC.6.EE.1.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
MACC.6.EE.1.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
MACC.6.EE.1.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
MACC.6.EE.2.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
MACC.6.EE.2.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
MACC.6.NS.2.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
MACC.6.EE.2.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
MACC.6.EE.2.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
MACC.6.NS.3.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
MACC.6.NS.3.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
MACC.6.EE.3.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
MACC.6.NS.3.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
MACC.6.NS.3.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
MACC.6.NS.3.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
MACC.6.NS.3.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
MACC.6.RP.1.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
MACC.6.RP.1.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
MACC.6.SP.1.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
MACC.6.SP.1.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
MACC.6.SP.1.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
MACC.6.SP.2.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
MACC.6.SP.2.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
MACC.6.SP.2.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
MACC.6.SP.2.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
MACC.6.SP.2.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
MACC.6.SP.2.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
MACC.6.G.1.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
MACC.6.G.1.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
MACC.6.G.1.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
MACC.6.G.1.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
MACC.6.NS.3.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
MACC.7.EE.1.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
MACC.7.EE.1.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
MACC.7.EE.2.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
MACC.7.EE.2.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
MACC.7.EE.2.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
MACC.7.NS.1.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
MACC.7.NS.1.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
MACC.7.NS.1.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
MACC.7.NS.1.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
MACC.7.NS.1.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
MACC.7.NS.1.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
MACC.7.NS.1.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
MACC.7.EE.2.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
MACC.7.NS.1.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
MACC.7.NS.1.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
MACC.7.NS.1.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
MACC.7.NS.1.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
MACC.7.NS.1.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
MACC.7.NS.1.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
MACC.7.NS.1.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
MACC.7.NS.1.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
MACC.7.NS.1.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
MACC.7.NS.1.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
MACC.7.EE.2.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
MACC.7.NS.1.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
MACC.7.NS.1.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
MACC.7.NS.1.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
MACC.7.NS.1.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
MACC.7.NS.1.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
MACC.7.NS.1.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
MACC.7.NS.1.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
MACC.7.NS.1.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
MACC.7.NS.1.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
MACC.7.NS.1.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
MACC.7.EE.2.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
MACC.7.EE.2.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
MACC.7.RP.1.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
MACC.7.RP.1.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
MACC.7.RP.1.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
MACC.7.RP.1.2c  Represent proportional relationships by equations.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
MACC.7.RP.1.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
MACC.7.EE.1.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
MACC.7.EE.2.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
MACC.7.EE.2.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
MACC.7.NS.1.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
MACC.7.NS.1.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
MACC.7.NS.1.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
MACC.7.EE.1.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
MACC.7.EE.2.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
MACC.7.EE.2.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
MACC.7.RP.1.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
MACC.7.RP.1.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
MACC.7.RP.1.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
MACC.7.RP.1.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
MACC.7.SP.1.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
MACC.7.SP.1.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
MACC.7.SP.2.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
MACC.7.SP.2.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
MACC.7.SP.3.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
MACC.7.SP.3.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
MACC.7.SP.3.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
MACC.7.SP.3.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
MACC.7.SP.3.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
MACC.7.SP.3.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
MACC.7.SP.3.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
MACC.7.SP.3.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
MACC.7.SP.3.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
MACC.7.G.1.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
MACC.7.G.1.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
MACC.7.G.1.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
MACC.7.G.2.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
MACC.7.G.2.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
MACC.7.G.2.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
MACC.8.EE.3.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
MACC.8.EE.3.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
MACC.8.EE.1.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
MACC.8.NS.1.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
MACC.8.EE.1.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
MACC.8.EE.1.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
MACC.8.EE.1.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
MACC.8.NS.1.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
MACC.8.EE.2.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
MACC.8.EE.3.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
MACC.8.EE.3.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
MACC.8.EE.2.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
MACC.8.EE.2.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
MACC.8.EE.3.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
MACC.8.EE.3.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
MACC.8.EE.3.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
MACC.8.EE.3.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
MACC.8.F.1.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
MACC.8.F.1.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
MACC.8.F.1.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
MACC.8.F.2.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
MACC.8.F.2.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
MACC.8.SP.1.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
MACC.8.SP.1.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
MACC.8.SP.1.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
MACC.8.SP.1.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
MACC.8.SP.1.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
MACC.8.G.1.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
MACC.8.G.1.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
MACC.8.G.1.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
MACC.8.G.1.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
MACC.8.G.1.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
MACC.8.G.1.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
MACC.8.G.1.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
MACC.8.G.1.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
MACC.8.G.2.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
MACC.8.G.2.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
MACC.8.G.2.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
MACC.8.G.3.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
Georgia Common Core Performance Standards  2011
6
Whole Numbers
MCC6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
MCC6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
MCC6.EE.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
MCC6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
MCC6.EE.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
MCC6.EE.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
MCC6.EE.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
MCC6.EE.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
MCC6.NS.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
MCC6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
MCC6.NS.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
MCC6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
MCC6.NS.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
MCC6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
MCC6.NS.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
MCC6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
MCC6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
MCC6.NS.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
MCC6.NS.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
MCC6.NS.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
MCC6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
MCC6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
MCC6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
MCC6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
MCC6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
MCC6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
MCC6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
MCC6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
MCC6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
MCC6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
MCC6.RP.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
MCC6.RP.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
MCC6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
MCC6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
MCC6.RP.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
MCC6.RP.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
MCC6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
MCC6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
MCC6.EE.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
MCC6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
MCC6.EE.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
MCC6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
MCC6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
MCC6.EE.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
MCC6.NS.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
MCC6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
MCC6.EE.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
MCC6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
MCC6.NS.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
MCC6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
MCC6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
MCC6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
MCC6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
MCC6.SP.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
MCC6.SP.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
MCC6.SP.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
MCC6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
MCC6.SP.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
MCC6.SP.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
MCC6.SP.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
MCC6.SP.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
MCC6.SP.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
MCC6.G.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
MCC6.G.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
MCC6.G.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
MCC6.G.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
MCC6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
MCC7.EE.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
MCC7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
MCC7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
MCC7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
MCC7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
MCC7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
MCC7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
MCC7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
MCC7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
MCC7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
MCC7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
MCC7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
MCC7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
MCC7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
MCC7.NS.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
MCC7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
MCC7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
MCC7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
MCC7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
MCC7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
MCC7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
MCC7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
MCC7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
MCC7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
MCC7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
MCC7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
MCC7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
MCC7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
MCC7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
MCC7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
MCC7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
MCC7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
MCC7.NS.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
MCC7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
MCC7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
MCC7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
MCC7.RP.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
MCC7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
MCC7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
MCC7.RP.2c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
MCC7.RP.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
MCC7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
MCC7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
MCC7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
MCC7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
MCC7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
MCC7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
MCC7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
MCC7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
MCC7.EE.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
MCC7.RP.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
MCC7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
MCC7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
MCC7.RP.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
MCC7.SP.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
MCC7.SP.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
MCC7.SP.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
MCC7.SP.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
MCC7.SP.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
MCC7.SP.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
MCC7.SP.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
MCC7.SP.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
MCC7.SP.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
MCC7.SP.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
MCC7.SP.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
MCC7.SP.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
MCC7.SP.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
MCC7.G.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
MCC7.G.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
MCC7.G.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
MCC7.G.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
MCC7.G.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
MCC7.G.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
MCC8.EE.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
MCC8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
MCC8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
MCC8.NS.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
MCC8.EE.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
MCC8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
MCC8.EE.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
MCC8.NS.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
MCC8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
MCC8.EE.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
MCC8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
MCC8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
MCC8.EE.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
MCC8.EE.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
MCC8.EE.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
MCC8.EE.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
MCC8.EE.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
MCC8.F.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
MCC8.F.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
MCC8.F.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
MCC8.F.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
MCC8.F.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
MCC8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
MCC8.SP.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
MCC8.SP.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
MCC8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
MCC8.SP.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
MCC8.G.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
MCC8.G.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
MCC8.G.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
MCC8.G.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
MCC8.G.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
MCC8.G.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
MCC8.G.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
MCC8.G.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
MCC8.G.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
MCC8.G.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
MCC8.G.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
MCC8.G.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
Michigan Common Core State Standards  2010
6
Whole Numbers
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
6.EE.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
6.EE.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
6.NS.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
6.NS.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
6.NS.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
6.NS.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
6.NS.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
6.RP.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
6.RP.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
6.EE.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
6.EE.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
6.NS.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
6.EE.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
6.SP.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
6.SP.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
6.SP.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
6.G.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
6.G.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
6.G.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
6.G.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
7.EE.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
7.NS.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
7.RP.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
7.RP.2c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
7.RP.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
7.EE.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
7.RP.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
7.SP.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
7.SP.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
7.SP.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
7.SP.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
7.SP.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
7.SP.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
7.G.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
7.G.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
7.G.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
7.G.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
7.G.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
7.G.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
8.EE.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
8.NS.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
8.EE.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
8.EE.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
8.NS.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
8.EE.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
8.EE.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.EE.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
8.EE.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
8.F.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
8.F.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
8.F.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
8.F.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.F.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
8.SP.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
8.SP.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
8.SP.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
8.G.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
8.G.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
8.G.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
8.G.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
8.G.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
8.G.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
New York Common Core State Standards  2012
6
Whole Numbers
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
6.EE.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
6.EE.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
6.NS.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
6.NS.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
6.NS.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
6.NS.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
6.NS.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
6.RP.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
6.RP.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
6.EE.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
6.EE.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
6.NS.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
6.EE.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
6.SP.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
6.SP.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
6.SP.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
6.G.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
6.G.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
6.G.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
6.G.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
7.EE.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
7.NS.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
7.RP.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
7.RP.2c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
7.RP.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
7.EE.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
7.RP.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
7.SP.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
7.SP.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
7.SP.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
7.SP.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
7.SP.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
7.SP.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
7.G.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
7.G.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
7.G.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
7.G.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
7.G.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
7.G.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
8.EE.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
8.NS.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
8.EE.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
8.EE.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
8.NS.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
8.EE.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
8.EE.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.EE.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
8.EE.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
8.F.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
8.F.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
8.F.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
8.F.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.F.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
8.SP.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
8.SP.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
8.SP.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
8.G.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
8.G.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
8.G.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
8.G.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
8.G.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
8.G.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
North Carolina Common Core State Standards  2012
6
Whole Numbers
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
6.EE.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
6.EE.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
6.NS.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
6.NS.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
6.NS.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
6.NS.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
6.NS.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
6.RP.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
6.RP.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
6.EE.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
6.EE.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
6.NS.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
6.EE.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
6.SP.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
6.SP.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
6.SP.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
6.G.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
6.G.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
6.G.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
6.G.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
7.EE.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
7.NS.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
7.RP.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
7.RP.2c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
7.RP.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
7.EE.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
7.RP.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
7.SP.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
7.SP.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
7.SP.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
7.SP.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
7.SP.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
7.SP.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
7.G.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
7.G.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
7.G.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
7.G.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
7.G.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
7.G.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
8.EE.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
8.NS.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
8.EE.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
8.EE.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
8.NS.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
8.EE.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
8.EE.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.EE.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
8.EE.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
8.F.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
8.F.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
8.F.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
8.F.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.F.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
8.SP.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
8.SP.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
8.SP.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
8.G.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
8.G.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
8.G.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
8.G.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
8.G.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
8.G.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
South Carolina Common Core State Standards  2010
6
Whole Numbers
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating Expressions Using the Order of OperationsWriting Algebraic Expressions 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2a  Write expressions that record operations with numbers and with letters standing for numbers.  Writing Algebraic Expressions 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Simplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
6.EE.2c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Identifying Properties of Addition and MultiplicationIdentifying the Greatest Common Factor (GCF)Simplifying Expressions by Combining Like Terms 
6.EE.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).  Simplifying Expressions by Combining Like Terms 
6.EE.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.  Writing Algebraic Expressions 
6.NS.2  Fluently divide multidigit numbers using the standard algorithm.  Using the Divisibility Rules 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Identifying the Greatest Common Factor (GCF)Identifying the Least Common Multiple (LCM)Listing Factors 
Integers
6.NS.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.  Identifying IntegersIdentifying Opposite Integers and Absolute ValueModeling Real Life Using Integers 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.6a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.  Identifying Opposite Integers and Absolute Value 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Graphing Integers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing IntegersGraphing IntegersIdentifying Opposite Integers and Absolute Value 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Integers 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Integers 
6.NS.7c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation.  Identifying Opposite Integers and Absolute Value 
6.NS.7d  Distinguish comparisons of absolute value from statements about order.  Comparing Integers 
Fractions & Decimals
6.NS.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.  Dividing FractionsIdentifying Parts of a FractionMultiplying Fractions 
6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.  Adding Decimal NumbersDividing Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal Numbers 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions and Decimals 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents.  Evaluating All PowersIdentifying Bases and Exponents 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Factoring Monomials 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Factoring Monomials 
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Numbers in Scientific Notation 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
6.NS.7  Understand ordering and absolute value of rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
6.NS.7b  Write, interpret, and explain statements of order for rational numbers in realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  Expressing Ratios in Simplified Form 
6.RP.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship.  Expressing Unit Rates 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Comparing Fractions, Decimals, Percents, and Ratios 
6.RP.3b  Solve unit rate problems including those involving unit pricing and constant speed.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Practical Problems Using Ratios 
6.RP.3c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.  Applying the Percent ProportionSolving Proportions to Find Missing Terms 
Equations
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.2b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.  Applying the Distributive Property to Write Equivalent Expressions with VariablesWriting OneStep Equations and Solving Word Problems 
6.EE.3  Apply the properties of operations to generate equivalent expressions.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
6.EE.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using Inequalities 
6.EE.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
6.NS.7  Understand ordering and absolute value of rational numbers  Graphing Inequalities on a Number Line 
6.NS.7a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.  Graphing Inequalities on a Number Line 
Graphing
6.EE.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.  Graphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent Variables 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.6b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed Point 
6.NS.6c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValues 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
6.RP.3a  Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Identifying Quadrants and Graphing an Ordered PairPredicting Values in Tables Using Numerical Patterns 
Probability & Statistics
6.SP.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.  Selecting a Representative Sampling Method for a Population 
6.SP.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
6.SP.5  Summarize numerical data sets in relation to their context, such as by:  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5a  Reporting the number of observations.  Analyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
6.SP.5b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data DisplaySelecting a Representative Sampling Method for a Population 
6.SP.5c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
6.SP.5d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
6.G.1  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.  Finding Area of Polygons Using Composing and Decomposing Techniques 
6.G.2  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.  Solving Problems Involving Volume 
6.G.3  Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.  Solving Problems Involving Polygons on the Coordinate Plane 
6.G.4  Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.  Investigating Surface Area and CrossSections of Solids 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.  Solving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
7.EE.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.  Identifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Writing Algebraic Expressions 
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Evaluating Expressions Using the Order of Operations 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Identifying Properties of Addition and Multiplication 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Identifying Properties of Addition and Multiplication 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationUsing the Divisibility Rules 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Identifying Properties of Addition and Multiplication 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and Multiplication 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Evaluating Expressions Using the Order of Operations 
Integers
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Combining Integer Multiplication and DivisionComparing Integers 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1a  Describe situations in which opposite quantities combine to make 0.  Adding Integers with Unlike SignsIdentifying Opposite Integers and Absolute Value 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute Value 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Combining Integer Subtraction and AdditionIdentifying Opposite Integers and Absolute ValueSubtracting Integers 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Subtraction and AdditionSubtracting Integers 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying and Dividing Integers 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Adding Decimal NumbersComparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsDividing Decimal NumbersDividing FractionsIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike Denominators 
7.NS.1c  Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.  Subtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.1d  Apply properties of operations as strategies to add and subtract rational numbers.  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Converting Fractions to DecimalsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Multiplying Decimal NumbersMultiplying Fractions 
7.NS.2b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.  Dividing Decimal NumbersDividing FractionsMultiplying FractionsWriting an Integer as a Fraction or Decimal 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
7.NS.2d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.  Converting Fractions to DecimalsWriting an Integer as a Fraction or Decimal 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers.1  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
Exponents & Roots
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
Rates, Ratios, & Proportions
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate assess the reasonableness of answers using mental computation and estimation strategies.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
7.RP.2  Recognize and represent proportional relationships between quantities.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Determining Proportional Relationships Between Quantities 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Solving Practical Problems Using Ratios 
7.RP.2c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Applying the Percent ProportionIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
7.RP.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
Equations
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.EE.4a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
7.NS.2c  Apply properties of operations as strategies to multiply and divide rational numbers.  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
7.EE.4b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsGraphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionSolving Word Problems Using InequalitiesWriting Inequalities Using Graphs 
Graphing
7.RP.2  Recognize and represent proportional relationships between quantities.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
7.RP.2d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.  Finding the Rate of Change 
Probability & Statistics
7.SP.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.  Selecting a Representative Sampling Method for a Population 
7.SP.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.  Drawing Inferences About a Population 
7.SP.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
7.SP.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
7.SP.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.  Determining the Likelihood of Events 
7.SP.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.  Determining the Likelihood of EventsIdentifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
7.SP.7b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.  Identifying and Comparing ProbabilitiesMaking Predictions 
7.SP.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.SP.8c  Design and use a simulation to generate frequencies for compound events.  Finding the Probabilities of Compound Events 
Geometry
7.G.1  Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.  Solving Problems Involving Scale Drawings of Geometric Figures 
7.G.2  Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.  Analyzing TrianglesIdentifying and Classifying Polygons 
7.G.3  Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.  Investigating Surface Area and CrossSections of Solids 
7.G.4  Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.  Solving Problems Involving Area and Circumference of a Circle 
7.G.5  Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.  Solving Equations for an Unknown Angle in a Figure 
7.G.6  Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8
Whole Numbers
8.EE.7  Solve linear equations in one variable.  Simplifying Expressions by Combining Like Terms 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Simplifying Expressions by Combining Like Terms 
Fractions & Decimals
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Classifying Numbers 
8.NS.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.  Classifying NumbersConverting Fractions to DecimalsDefining Rational and IrrationalWriting an Integer as a Fraction or Decimal 
Exponents & Roots
8.EE.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions.  Evaluating All PowersFactoring MonomialsFinding the Prime Factorization of a NumberIdentifying Bases and ExponentsIdentifying Prime and Composite NumbersLaws of ExponentsRepresenting Expressions with Only Positive ExponentsSimplifying Monomial Expressions (Division)Simplifying Monomial Expressions (Multiplication)Simplifying Monomial Expressions (Powers of Powers) 
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.  Identifying the Cube Root of a Perfect CubeIdentifying the Square Root of a Perfect Square 
8.EE.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
8.NS.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.  Estimating Square Roots 
Rates, Ratios, & Proportions
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining Proportional Relationships Between QuantitiesExpressing Unit RatesSolving Proportions to Find Missing Terms 
Equations
8.EE.7  Solve linear equations in one variable.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
8.EE.7b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two SidesWriting OneStep Equations and Solving Word Problems 
Graphing
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.  Determining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting an Equation for Direct Variation 
8.EE.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.EE.8  Analyze and solve pairs of simultaneous linear equations.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations Graphically 
8.EE.8b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.  Determining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
8.EE.8c  Solve realworld and mathematical problems leading to two linear equations in two variables.  Solving Systems of Two Linear Equations Algebraically 
8.F.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Determining if a Relation is a FunctionIdentifying Domain and RangeIdentifying Independent and Dependent VariablesIdentifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointRepresenting Relations and Functions in Different Forms 
8.F.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).  Calculating the Slope of a Graphed LineFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
8.F.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.  Classifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsIdentifying a Graph of a Direct Variation 
8.F.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.F.5  Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using Tables 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Finding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Predicting Values Using a Direct VariationWriting an Equation for Direct Variation 
Probability & Statistics
8.SP.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.  Interpreting Scatter Plots to Investigate Patterns of Association 
8.SP.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.  Solving Problems Involving Bivariate Measurement Data 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.  Solving Problems Involving Bivariate Measurement Data 
8.SP.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.  Constructing and Interpreting a TwoWay Table 
Geometry
8.G.1  Verify experimentally the properties of rotations, reflections, and translations:  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1a  Lines are taken to lines, and line segments to line segments of the same length.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1b  Angles are taken to angles of the same measure.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.1c  Parallel lines are taken to parallel lines.  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.G.2  Understand that a twodimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.  Using Congruency Statements to Identify Corresponding Parts of a Polygon 
8.G.3  Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates.  Using the Coordinate Plane to Demonstrate Transformations 
8.G.4  Understand that a twodimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar twodimensional figures, describe a sequence that exhibits the similarity between them.  Describing the Sequence of Transformations of Two Similar Figures 
8.G.5  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles.  Identifying Angle Relationships Formed by Parallel Lines and a Transversal 
8.G.6  Explain a proof of the Pythagorean Theorem and its converse.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.7  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions.  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
8.G.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.  Finding Distance Between Two Points on the Coordinate Plane 
8.G.9  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve realworld and mathematical problems.  Solving Problems Involving Volume 
Texas Essential Knowledge and Skills for Mathematics  2012
6
Whole Numbers
111.26.b.3  The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. (Number and operations)  Evaluating Expressions Using the Order of Operations 
111.26.b.3.E  Multiply and divide positive rational numbers fluently.  Evaluating Expressions Using the Order of Operations 
111.26.b.4  The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality)  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationWriting Algebraic Expressions 
111.26.b.4.A  Compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships;  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationWriting Algebraic Expressions 
111.26.b.7  The student applies mathematical process standards to develop concepts of expressions and equations. (Expressions, equations, and relationships)  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
111.26.b.7.A  Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization;  Evaluating Expressions Using the Order of Operations 
111.26.b.7.B  Distinguish between expressions and equations verbally, numerically, and algebraically;  Evaluating ExpressionsSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
111.26.b.7.D  Generate equivalent expressions using the properties of operations : such as the inverse, identity, commutative, associative, and distributive properties.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like Terms 
Integers
111.26.b.2  The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations)  Comparing IntegersGraphing IntegersIdentifying Integers 
111.26.b.2.B  Identify a number, its opposite, and its absolute value;  Identifying Opposite Integers and Absolute Value 
111.26.b.2.C  Locate, compare, and order integers and rational numbers  Comparing IntegersGraphing IntegersIdentifying Integers 
111.26.b.3  The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. (Number and operations)  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionModeling Real Life Using IntegersMultiplying and Dividing IntegersSubtracting Integers 
111.26.b.3.C  Represent integer operations with concrete models and connect the actions with the models to standardized algorithms;  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionModeling Real Life Using IntegersMultiplying and Dividing Integers 
111.26.b.3.D  Add, subtract, multiply, and divide integers fluently;  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
111.26.b.3.E  Multiply and divide positive rational numbers fluently.  Combining Integer Multiplication and DivisionMultiplying and Dividing Integers 
111.26.b.9  The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships)  Comparing IntegersGraphing Integers 
111.26.b.9.B  Represent solutions for onevariable, onestep equations and inequalities on number lines;  Comparing IntegersGraphing Integers 
Fractions & Decimals
111.26.b.2  The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations)  Classifying NumbersComparing Fractions and DecimalsDefining Rational and IrrationalIdentifying Parts of a Fraction 
111.26.b.2.A  Classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers;  Classifying NumbersDefining Rational and Irrational 
111.26.b.2.C  Locate, compare, and order integers and rational numbers  Comparing Fractions and Decimals 
111.26.b.2.E  Extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.  Identifying Parts of a Fraction 
111.26.b.3  The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. (Number and operations)  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
111.26.b.3.A  Recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values;  Dividing Fractions 
111.26.b.3.E  Multiply and divide positive rational numbers fluently.  Dividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying Fractions 
111.26.b.4  The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality)  Converting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
111.26.b.4.C  Give examples of ratios as multiplicative comparisons of two quantities describing the same attribute;  Writing Fractions in Simplest Form 
111.26.b.4.F  Represent benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers;  Converting Fractions to Decimals 
111.26.b.4.G  Generate equivalent forms of fractions, decimals, and percents using realworld problems, including problems that involve money;  Converting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
111.26.b.5  The student applies mathematical process standards to solve problems involving proportional relationships. (Proportionality)  Converting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
111.26.b.5.C  Use equivalent fractions, decimals, and percents to show equal parts of the same whole.  Converting Fractions to DecimalsConverting Mixed Numbers and Improper FractionsWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
Exponents & Roots
111.26.b.2  The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations)  Comparing Numbers in Scientific Notation 
111.26.b.2.C  Locate, compare, and order integers and rational numbers  Comparing Numbers in Scientific Notation 
111.26.b.7  The student applies mathematical process standards to develop concepts of expressions and equations. (Expressions, equations, and relationships)  Evaluating All PowersFinding the Prime Factorization of a NumberIdentifying Prime and Composite Numbers 
111.26.b.7.A  Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization;  Evaluating All PowersFinding the Prime Factorization of a NumberIdentifying Prime and Composite Numbers 
Rates, Ratios, & Proportions
111.26.b.2  The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations)  Comparing Fractions, Decimals, Percents, and RatiosExpressing Ratios in Simplified Form 
111.26.b.2.C  Locate, compare, and order integers and rational numbers  Comparing Fractions, Decimals, Percents, and Ratios 
111.26.b.2.D  Order a set of rational numbers arising from mathematical and realworld contexts;  Comparing Fractions, Decimals, Percents, and Ratios 
111.26.b.2.E  Extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.  Expressing Ratios in Simplified Form 
111.26.b.4  The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality)  Comparing Fractions, Decimals, Percents, and RatiosExpressing Ratios in Simplified FormExpressing Unit RatesExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a FractionIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using Ratios 
111.26.b.4.B  Apply qualitative and quantitative reasoning to solve prediction and comparison of realworld problems involving ratios and rates;  Expressing Ratios in Simplified FormExpressing Unit RatesSolving Practical Problems Using Ratios 
111.26.b.4.C  Give examples of ratios as multiplicative comparisons of two quantities describing the same attribute;  Expressing Ratios in Simplified Form 
111.26.b.4.D  Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients;  Expressing Ratios in Simplified FormExpressing Unit RatesSolving Practical Problems Using Ratios 
111.26.b.4.E  Represent ratios and percents with concrete models, fractions, and decimals;  Expressing Ratios in Simplified FormExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
111.26.b.4.F  Represent benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers;  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
111.26.b.4.G  Generate equivalent forms of fractions, decimals, and percents using realworld problems, including problems that involve money;  Expressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
111.26.b.4.H  Convert units within a measurement system, including the use of proportions and unit rates.  Identify Actual Measurements and Scale Factors 
111.26.b.5  The student applies mathematical process standards to solve problems involving proportional relationships. (Proportionality)  Applying the Percent ProportionComputing Simple InterestDetermining Proportional Relationships Between QuantitiesExpressing Ratios in Simplified FormExpressing Unit RatesExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a FractionIdentify Actual Measurements and Scale FactorsIdentifying the Percent of ChangeSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
111.26.b.5.A  Represent mathematical and realworld problems involving ratios and rates using scale factors, tables, graphs, and proportions;  Determining Proportional Relationships Between QuantitiesExpressing Ratios in Simplified FormExpressing Unit RatesIdentify Actual Measurements and Scale FactorsSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
111.26.b.5.B  Solve realworld problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole , including the use of concrete and pictorial models ;  Applying the Percent ProportionComputing Simple InterestIdentifying the Percent of Change 
111.26.b.5.C  Use equivalent fractions, decimals, and percents to show equal parts of the same whole.  Expressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
Equations
111.26.b.10  The student applies mathematical process standards to use equations and inequalities to solve problems. (Expressions, equations, and relationships)  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
111.26.b.10.A  Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts;  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
111.26.b.10.B  Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
111.26.b.4  The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality)  Applying Knowledge of TwoStep Equations to Solve Word ProblemsWriting OneStep Equations and Solving Word Problems 
111.26.b.4.A  Compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships;  Applying Knowledge of TwoStep Equations to Solve Word ProblemsWriting OneStep Equations and Solving Word Problems 
111.26.b.6  The student applies mathematical process standards to use multiple representations to describe algebraic relationships. (Expressions, equations, and relationships)  Writing OneStep Equations and Solving Word Problems 
111.26.b.6.C  Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.  Writing OneStep Equations and Solving Word Problems 
111.26.b.7  The student applies mathematical process standards to develop concepts of expressions and equations. (Expressions, equations, and relationships)  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
111.26.b.7.B  Distinguish between expressions and equations verbally, numerically, and algebraically;  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Write Equivalent Expressions with VariablesSolving OneStep Equations Involving Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
111.26.b.7.C  Determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations;  Applying the Distributive Property to Solve EquationsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionWriting OneStep Equations and Solving Word Problems 
111.26.b.7.D  Generate equivalent expressions using the properties of operations : such as the inverse, identity, commutative, associative, and distributive properties.  Applying the Distributive Property to Write Equivalent Expressions with Variables 
111.26.b.9  The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships)  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
111.26.b.9.A  Write onevariable, onestep equations and inequalities to represent constraints or conditions within problems;  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
111.26.b.9.C  Write corresponding realworld problems given onevariable, onestep equations or inequalities.  Writing OneStep Equations and Solving Word Problems 
Inequalities
111.26.b.10  The student applies mathematical process standards to use equations and inequalities to solve problems. (Expressions, equations, and relationships)  Solving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving Word Problems Using Inequalities 
111.26.b.10.A  Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts;  Solving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving Word Problems Using Inequalities 
111.26.b.10.B  Determine if the given value(s) make(s) onevariable, onestep equations or inequalities true.  Solving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or Division 
111.26.b.9  The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships)  Graphing Inequalities on a Number LineSolving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
111.26.b.9.A  Write onevariable, onestep equations and inequalities to represent constraints or conditions within problems;  Solving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving Word Problems Using InequalitiesWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
111.26.b.9.B  Represent solutions for onevariable, onestep equations and inequalities on number lines;  Graphing Inequalities on a Number LineWriting Inequalities Using Graphs 
111.26.b.9.C  Write corresponding realworld problems given onevariable, onestep equations or inequalities.  Solving Word Problems Using InequalitiesWriting Mathematical Sentences Using Inequalities 
Graphing
111.26.b.11  The student applies mathematical process standards to use coordinate geometry to identify locations on a plane. The student is expected to graph points in all four quadrants using ordered pairs of rational numbers. (Measurement and data)  Identifying Quadrants and Graphing an Ordered PairIdentifying an Ordered Pair Given a Graphed PointIdentifying the x and yIntercepts 
111.26.b.6  The student applies mathematical process standards to use multiple representations to describe algebraic relationships. (Expressions, equations, and relationships)  Graphing a Linear Function in Two Variables Using TablesIdentifying Domain and RangeIdentifying Independent and Dependent VariablesRepresenting Relations and Functions in Different Forms 
111.26.b.6.A  Identify independent and dependent quantities from tables and graphs;  Identifying Domain and RangeIdentifying Independent and Dependent Variables 
111.26.b.6.C  Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.  Graphing a Linear Function in Two Variables Using TablesRepresenting Relations and Functions in Different Forms 
Probability & Statistics
111.26.b.12  The student applies mathematical process standards to use numerical or graphical representations to analyze problems. (Measurement and data)  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
111.26.b.12.A  Represent numeric data graphically, including dot plots, stemandleaf plots, histograms, and box plots;  Representing a Set of Data Using a Data Display 
111.26.b.12.B  Use the graphical representation of numeric data to describe the center, spread, and shape of the data distribution;  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
111.26.b.12.C  Summarize numeric data with numerical summaries, including the mean and median (measures of center) and the range and interquartile range (IQR) (measures of spread), and use these summaries to describe the center, spread, and shape of the data distribution;  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetDescribing a Set of Data (Mean, Median, Mode, Range) 
111.26.b.12.D  Summarize categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and use these summaries to describe the data distribution.  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Describing a Set of Data (Mean, Median, Mode, Range)Representing a Set of Data Using a Data Display 
111.26.b.13  The student applies mathematical process standards to use numerical or graphical representations to solve problems. (Measurement and data)  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetRepresenting a Set of Data Using a Data Display 
111.26.b.13.A  Interpret numeric data summarized in dot plots, stemandleaf plots, histograms, and box plots;  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data SetRepresenting a Set of Data Using a Data Display 
111.26.b.13.B  Distinguish between situations that yield data with and without variability.  Analyzing and Comparing Data SetsComputing the Interquartile Range and Mean Absolute Deviation of a Data Set 
Geometry
111.26.b.10  The student applies mathematical process standards to use equations and inequalities to solve problems. (Expressions, equations, and relationships)  Analyzing TrianglesFinding the Missing Dimension of a Rectangle or TriangleIdentifying Angle Relationships Formed by Parallel Lines and a TransversalSolving Equations for an Unknown Angle in a FigureSolving Problems Involving Area and Circumference of a CircleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
111.26.b.10.A  Model and solve onevariable, onestep equations and inequalities that represent problems, including geometric concepts;  Analyzing TrianglesFinding the Missing Dimension of a Rectangle or TriangleIdentifying Angle Relationships Formed by Parallel Lines and a TransversalSolving Equations for an Unknown Angle in a FigureSolving Problems Involving Area and Circumference of a CircleSolving Problems Involving Surface AreaSolving Problems Involving Volume 
111.26.b.8  The student applies mathematical process standards to use geometry to represent relationships and solve problems. (Expressions, equations, and relationships)  Analyzing TrianglesClassifying Triangles Using Angles and SidesFinding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleIdentifying and Classifying PolygonsSolving Equations for an Unknown Angle in a FigureSolving Problems Involving Polygons on the Coordinate Plane 
111.26.b.8.A  Extend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle;  Analyzing TrianglesClassifying Triangles Using Angles and SidesIdentifying and Classifying PolygonsSolving Equations for an Unknown Angle in a Figure 
111.26.b.8.B  Model area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes;  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Polygons on the Coordinate Plane 
111.26.b.8.C  Write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers;  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Polygons on the Coordinate Plane 
111.26.b.8.D  Determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Polygons on the Coordinate Plane 
7
Whole Numbers
111.27.b.3  The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. (Number and operations)  Evaluating Expressions Using the Order of Operations 
111.27.b.3.A  Add, subtract, multiply, and divide rational numbers fluently;  Evaluating Expressions Using the Order of Operations 
111.27.b.3.B  Apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.  Evaluating Expressions Using the Order of Operations 
111.27.b.4  The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)  Evaluating Expressions 
111.27.b.4.A  Represent constant rates of change in mathematical and realworld problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt;  Evaluating Expressions 
Integers
111.27.b.3  The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. (Number and operations)  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
111.27.b.3.A  Add, subtract, multiply, and divide rational numbers fluently;  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
111.27.b.3.B  Apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and Addition 
Fractions & Decimals
111.27.b.2  The student applies mathematical process standards to represent and use rational numbers in a variety of forms. The student is expected to extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers. (Number and operations)  Classifying NumbersDefining Rational and Irrational 
111.27.b.3  The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. (Number and operations)  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
111.27.b.3.A  Add, subtract, multiply, and divide rational numbers fluently;  Adding Decimal NumbersAdding Like DenominatorsAdding Unlike DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal NumbersSubtracting Like DenominatorsSubtracting Unlike Denominators 
111.27.b.3.B  Apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.  Adding Like DenominatorsDividing Decimal NumbersDividing FractionsMultiplying Decimal NumbersMultiplying FractionsSubtracting Decimal Numbers 
111.27.b.4  The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)  Converting Fractions to Decimals 
111.27.b.4.D  Solve problems involving ratios, rates, and percents, including multistep problems involving percent increase and percent decrease, and financial literacy problems;  Converting Fractions to Decimals 
Rates, Ratios, & Proportions
111.27.b.13.A  Calculate the sales tax for a given purchase and calculate income tax for earned wages ;  Applying the Percent Proportion 
111.27.b.13.E  Calculate and compare simple interest and compound interest earnings;  Computing Simple Interest 
111.27.b.4  The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosComputing Simple InterestDetermining Proportional Relationships Between QuantitiesExpressing Ratios in Simplified FormExpressing Unit RatesExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a FractionIdentify Actual Measurements and Scale FactorsIdentifying the Percent of ChangeSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
111.27.b.4.B  Calculate unit rates from rates in mathematical and realworld problems;  Expressing Unit RatesSolving Practical Problems Using Ratios 
111.27.b.4.C  Determine the constant of proportionality (k = y/x) within mathematical and realworld problems;  Determining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
111.27.b.4.D  Solve problems involving ratios, rates, and percents, including multistep problems involving percent increase and percent decrease, and financial literacy problems;  Applying the Percent ProportionComparing Fractions, Decimals, Percents, and RatiosComputing Simple InterestExpressing Ratios in Simplified FormExpressing Unit RatesExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a FractionIdentifying the Percent of ChangeSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
111.27.b.5  The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. (Proportionality)  Determining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale Factors 
111.27.b.5.A  Generalize the critical attributes of similarity, including ratios within and between similar shapes;  Determining Proportional Relationships Between QuantitiesIdentify Actual Measurements and Scale Factors 
111.27.b.5.C  Solve mathematical and realworld problems involving similar shape and scale drawings.  Identify Actual Measurements and Scale Factors 
Equations
111.27.b.10.A  Write onevariable, twostep equations and inequalities to represent constraints or conditions within problems;  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve Equations 
111.27.b.10.C  Write a corresponding realworld problem given a onevariable, twostep equation or inequality.  Applying Knowledge of TwoStep Equations to Solve Word Problems 
111.27.b.11.A  Model and solve onevariable, twostep equations and inequalities;  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or Subtraction 
111.27.b.11.B  Determine if the given value(s) make(s) onevariable, twostep equations and inequalities true;  Applying Knowledge of TwoStep Equations to Solve Word ProblemsSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or Subtraction 
Inequalities
111.27.b.10.A  Write onevariable, twostep equations and inequalities to represent constraints or conditions within problems;  Applying Knowledge of TwoStep Inequalities to Solve Word Problems 
111.27.b.10.B  Represent solutions for onevariable, twostep equations and inequalities on number lines;  Graphing Inequalities on a Number LineSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or SubtractionWriting Inequalities Using Graphs 
111.27.b.10.C  Write a corresponding realworld problem given a onevariable, twostep equation or inequality.  Applying Knowledge of TwoStep Inequalities to Solve Word Problems 
111.27.b.11.A  Model and solve onevariable, twostep equations and inequalities;  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or Subtraction 
111.27.b.11.B  Determine if the given value(s) make(s) onevariable, twostep equations and inequalities true;  Solving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or Subtraction 
Graphing
111.27.b.4  The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)  Determining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationFinding the Rate of ChangeGraphing a Linear Function in Two Variables Using TablesIdentifying a Table of a Direct VariationPredicting Values Using a Direct VariationPredicting Values in Tables Using Numerical Patterns 
111.27.b.4.A  Represent constant rates of change in mathematical and realworld problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt;  Determining Whether Functions are Linear or Nonlinear Using PatternsFinding the Rate of ChangeGraphing a Linear Function in Two Variables Using TablesPredicting Values in Tables Using Numerical Patterns 
111.27.b.4.C  Determine the constant of proportionality (k = y/x) within mathematical and realworld problems;  Determining the Constant of VariationIdentifying a Table of a Direct VariationPredicting Values Using a Direct Variation 
111.27.b.7  The student applies mathematical process standards to represent linear relationships using multiple representations. The student is expected to represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b. (Expressions, equations, and relationships)  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying Independent and Dependent VariablesIdentifying the Slope of a Line Using Two PointsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points) 
Probability & Statistics
111.27.b.12.A  Compare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads;  Analyzing and Comparing Data SetsAnalyzing and Describing the Distribution of a Data Set (Data Display)Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
111.27.b.12.B  Use data from a random sample to make inferences about a population;  Drawing Inferences About a PopulationSelecting a Representative Sampling Method for a Population 
111.27.b.12.C  Compare two populations based on data in random samples from these populations, including informal comparative inferences about differences between the two populations.  Analyzing and Comparing Data Sets 
111.27.b.6  The student applies mathematical process standards to use probability and statistics to describe or solve problems involving proportional relationships. (Proportionality)  Finding the Probabilities of Compound EventsIdentifying and Comparing ProbabilitiesIdentifying the Outcomes in a Sample SpaceIdentifying the Theoretical Probability of an EventMaking Predictions 
111.27.b.6.A  Represent sample spaces for simple and compound events using lists and tree diagrams;  Identifying the Outcomes in a Sample SpaceIdentifying the Theoretical Probability of an Event 
111.27.b.6.B  Select and use different simulations to represent simple and compound events with and without technology  Finding the Probabilities of Compound EventsIdentifying and Comparing ProbabilitiesIdentifying the Outcomes in a Sample SpaceIdentifying the Theoretical Probability of an EventMaking Predictions 
111.27.b.6.C  Make predictions and determine solutions using experimental data for simple and compound events;  Finding the Probabilities of Compound EventsIdentifying and Comparing ProbabilitiesMaking Predictions 
111.27.b.6.D  Make predictions and determine solutions using theoretical probability for simple and compound events;  Finding the Probabilities of Compound EventsIdentifying and Comparing ProbabilitiesIdentifying the Outcomes in a Sample SpaceIdentifying the Theoretical Probability of an EventMaking Predictions 
111.27.b.6.E  Find the probabilities of a simple event and its complement and describe the relationship between the two;  Identifying the Theoretical Probability of an EventMaking Predictions 
111.27.b.6.F  Use data from a random sample to make inferences about a population;  Drawing Inferences About a PopulationSelecting a Representative Sampling Method for a Population 
111.27.b.6.G  Solve problems using data represented in bar graphs, dot plots, and circle graphs, including parttowhole and parttopart comparisons and equivalents;  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
111.27.b.6.H  Solve problems using qualitative and quantitative predictions and comparisons from simple experiments;  Identifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
111.27.b.6.I  Determine experimental and theoretical probabilities related to simple and compound events using data and sample spaces.  Finding the Probabilities of Compound EventsIdentifying and Comparing ProbabilitiesIdentifying the Outcomes in a Sample SpaceIdentifying the Theoretical Probability of an EventMaking Predictions 
Geometry
111.27.b.11.C  Write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.  Analyzing TrianglesIdentifying Angle Relationships Formed by Parallel Lines and a TransversalSolving Equations for an Unknown Angle in a Figure 
111.27.b.5  The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. (Proportionality)  Describing the Sequence of Transformations of Two Similar FiguresSolving Problems Involving Area and Circumference of a CircleSolving Problems Involving Scale Drawings of Geometric FiguresUsing the Coordinate Plane to Demonstrate Transformations 
111.27.b.5.A  Generalize the critical attributes of similarity, including ratios within and between similar shapes;  Describing the Sequence of Transformations of Two Similar FiguresSolving Problems Involving Scale Drawings of Geometric FiguresUsing the Coordinate Plane to Demonstrate Transformations 
111.27.b.5.B  Describe pi () as the ratio of the circumference of a circle to its diameter;  Solving Problems Involving Area and Circumference of a Circle 
111.27.b.5.C  Solve mathematical and realworld problems involving similar shape and scale drawings.  Describing the Sequence of Transformations of Two Similar FiguresSolving Problems Involving Scale Drawings of Geometric FiguresUsing the Coordinate Plane to Demonstrate Transformations 
111.27.b.8  Use models to determine the approximate formulas for the circumference and area of a circle and connect the models to the actual formulas.  Solving Problems Involving Area and Circumference of a Circle 
111.27.b.8.A  The student applies mathematical process standards to develop geometric relationships with volume. (Expressions, equations, and relationships)  Solving Problems Involving Volume 
111.27.b.8.B  Model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas;  Solving Problems Involving Volume 
111.27.b.8.C  Explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connect that relationship to the formulas;  Solving Problems Involving Volume 
111.27.b.9.A  Solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids;  Solving Problems Involving Volume 
111.27.b.9.B  Determine the circumference and area of circles;  Solving Problems Involving Area and Circumference of a Circle 
111.27.b.9.C  Determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles;  Finding Area of Polygons Using Composing and Decomposing TechniquesSolving Problems Involving Area and Circumference of a CircleSolving Problems Involving Polygons on the Coordinate Plane 
111.27.b.9.D  Solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net.  Investigating Surface Area and CrossSections of SolidsSolving Problems Involving Surface Area 
8
Integers
111.28.b.2  The student applies mathematical process standards to represent and use real numbers in a variety of forms.  Comparing Integers 
111.28.b.2.D  Order a set of real numbers arising from mathematical and realworld contexts.  Comparing Integers 
Fractions & Decimals
111.28.b.2  The student applies mathematical process standards to represent and use real numbers in a variety of forms.  Classifying NumbersComparing Fractions and DecimalsDefining Rational and Irrational 
111.28.b.2.A  Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers;  Classifying NumbersDefining Rational and Irrational 
111.28.b.2.B  Approximate the value of an irrational number, including pi () and square roots of numbers less than 225, and locate that rational number approximation on a number line;  Defining Rational and Irrational 
111.28.b.2.D  Order a set of real numbers arising from mathematical and realworld contexts.  Comparing Fractions and Decimals 
Exponents & Roots
111.28.b.2  The student applies mathematical process standards to represent and use real numbers in a variety of forms.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormEstimating Square RootsIdentifying the Square Root of a Perfect SquareWriting in Scientific Notation 
111.28.b.2.B  Approximate the value of an irrational number, including pi () and square roots of numbers less than 225, and locate that rational number approximation on a number line;  Estimating Square RootsIdentifying the Square Root of a Perfect Square 
111.28.b.2.C  Convert between standard decimal notation and scientific notation  Converting from Scientific Notation to Standard FormWriting in Scientific Notation 
111.28.b.2.D  Order a set of real numbers arising from mathematical and realworld contexts.  Comparing Numbers in Scientific Notation 
Rates, Ratios, & Proportions
111.28.b.12  The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. (Personal financial literacy)  Computing Simple Interest 
111.28.b.12.A  Solve realworld problems comparing how interest rate and loan length affect the cost of credit;  Computing Simple Interest 
111.28.b.12.D  Calculate and compare simple interest and compound interest earnings;  Computing Simple Interest 
111.28.b.2  The student applies mathematical process standards to represent and use real numbers in a variety of forms.  Comparing Fractions, Decimals, Percents, and Ratios 
111.28.b.2.D  Order a set of real numbers arising from mathematical and realworld contexts.  Comparing Fractions, Decimals, Percents, and Ratios 
Equations
111.28.b.8  The student applies mathematical process standards to use onevariable equations or inequalities in problem situations. (Expressions, equations, and relationships)  Applying the Distributive Property to Solve EquationsSolving an Equation with Variables on Two Sides 
111.28.b.8.C  Model and solve onevariable equations with variables on both sides of the equal sign that represent mathematical and realworld problems using rational number coefficients and constants;  Applying the Distributive Property to Solve EquationsSolving an Equation with Variables on Two Sides 
Graphing
111.28.b.4  The student applies mathematical process standards to explain proportional and nonproportional relationships involving slope. (Proportionality)  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining the Constant of VariationFinding the Rate of ChangeGraphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying a Graph of a Direct VariationIdentifying the Slope of a Line Using Two PointsIdentifying the x and yInterceptsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points) 
111.28.b.4.A  Use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in yvalues to the change in xvalues, (y2  y1) / (x2  x1), is the same for any two points (x1, y1) and (x2, y2) on the same line;  Calculating the Slope of a Graphed LineIdentifying the Slope of a Line Using Two Points 
111.28.b.4.B  Graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship  Calculating the Slope of a Graphed LineDetermining the Constant of VariationFinding the Rate of ChangeIdentifying a Graph of a Direct Variation 
111.28.b.4.C  Use data from a table or graph to determine the rate of change or slope and yintercept in mathematical and realworld problems.  Calculating the Slope of a Graphed LineDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsFinding the Rate of ChangeGraphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying the Slope of a Line Using Two PointsIdentifying the x and yInterceptsPredicting Values in Tables Using Numerical PatternsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points) 
111.28.b.5  The student applies mathematical process standards to use proportional and nonproportional relationships to develop foundational concepts of functions. (Proportionality)  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining if a Relation is a FunctionDetermining the Constant of VariationGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying Domain and RangeIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct VariationPredicting Values Using a Direct VariationRepresenting Relations and Functions in Different FormsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
111.28.b.5.A  Represent linear proportional situations with tables, graphs, and equations in the form of y = kx;  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct VariationPredicting Values Using a Direct VariationWriting an Equation for Direct Variation 
111.28.b.5.B  Represent linear nonproportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0;  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points) 
111.28.b.5.E  Solve problems involving direct variation;  Determining the Constant of VariationPredicting Values Using a Direct VariationWriting an Equation for Direct Variation 
111.28.b.5.F  Distinguish between proportional and nonproportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0;  Determining the Constant of VariationIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
111.28.b.5.G  Identify functions using sets of ordered pairs, tables, mappings, and graphs;  Determining if a Relation is a FunctionIdentifying Domain and RangeRepresenting Relations and Functions in Different Forms 
111.28.b.5.H  Identify examples of proportional and nonproportional functions that arise from mathematical and realworld problems  Classifying Functions as Linear or NonlinearDetermining if a Relation is a FunctionIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct Variation 
111.28.b.5.I  Write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.  Determining Slope and yIntercept of Linear Equations in SlopeIntercept FormGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Writing Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points) 
111.28.b.9  The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations. (Expressions, equations, and relationships)  Determining if an Ordered Pair is a Solution to the SystemDetermining the Solution(s) of the System of Linear EquationsSolving Systems of Two Linear Equations AlgebraicallySolving Systems of Two Linear Equations Graphically 
Probability & Statistics
111.28.b.11  The student applies mathematical process standards to use statistical procedures to describe data. (Measurement and data)  Computing the Interquartile Range and Mean Absolute Deviation of a Data SetInterpreting Scatter Plots to Investigate Patterns of AssociationSelecting a Representative Sampling Method for a Population 
111.28.b.11.A  Construct a scatterplot and describe the observed data to address questions of association such as linear, nonlinear, and no association between bivariate data;  Interpreting Scatter Plots to Investigate Patterns of Association 
111.28.b.11.B  Determine the mean absolute deviation and use this quantity as a measure of the average distance data are from the mean using a data set of no more than 10 data points.  Computing the Interquartile Range and Mean Absolute Deviation of a Data Set 
111.28.b.11.C  Simulate generating random samples of the same size from a population with known characteristics to develop the notion of a random sample being representative of the population from which it was selected.  Selecting a Representative Sampling Method for a Population 
111.28.b.5  The student applies mathematical process standards to use proportional and nonproportional relationships to develop foundational concepts of functions. (Proportionality)  Interpreting Scatter Plots to Investigate Patterns of AssociationSolving Problems Involving Bivariate Measurement Data 
111.28.b.5.C  Contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation;  Interpreting Scatter Plots to Investigate Patterns of AssociationSolving Problems Involving Bivariate Measurement Data 
111.28.b.5.D  Use a trend line that approximates the linear relationship between bivariate sets of data to make predictions;  Solving Problems Involving Bivariate Measurement Data 
Geometry
111.28.b.10  The student applies mathematical process standards to develop transformational geometry concepts. (Twodimensional shapes)  Describing the Sequence of Transformations of Two Similar FiguresIdentifying Properties of Rotations, Reflections, and TranslationsSolving Problems Involving Area and Circumference of a CircleSolving Problems Involving Polygons on the Coordinate PlaneSolving Problems Involving Scale Drawings of Geometric FiguresUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.10.A  Generalize the properties of orientation and congruence of rotations, reflections, translations, and dilations of twodimensional shapes on a coordinate plane;  Describing the Sequence of Transformations of Two Similar FiguresIdentifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.10.B  Differentiate between transformations that preserve congruence and those that do not;  Describing the Sequence of Transformations of Two Similar FiguresIdentifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.10.C  Explain the effect of translations, reflections over the x or yaxis, and rotations limited to 90, 180, 270, and 360 as applied to twodimensional shapes on a coordinate plane using an algebraic representation;  Identifying Properties of Rotations, Reflections, and TranslationsUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.10.D  Model the effect on linear and area measurements of dilated twodimensional shapes.  Solving Problems Involving Area and Circumference of a CircleSolving Problems Involving Polygons on the Coordinate PlaneSolving Problems Involving Scale Drawings of Geometric FiguresUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.3  The student applies mathematical process standards to use proportional relationships to describe dilations. (Proportionality)  Describing the Sequence of Transformations of Two Similar FiguresSolving Problems Involving Scale Drawings of Geometric FiguresUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.3.A  Generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation;  Describing the Sequence of Transformations of Two Similar FiguresSolving Problems Involving Scale Drawings of Geometric FiguresUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.3.B  Compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane  Describing the Sequence of Transformations of Two Similar FiguresUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.3.C  Use an algebraic representation to explain the effect of a given positive rational scale factor applied to twodimensional figures on a coordinate plane with the origin as the center of dilation.  Describing the Sequence of Transformations of Two Similar FiguresSolving Problems Involving Scale Drawings of Geometric FiguresUsing the Coordinate Plane to Demonstrate Transformations 
111.28.b.6  The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. (Expressions, equations, and relationships)  Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsFinding Distance Between Two Points on the Coordinate PlaneSolving Problems Involving Volume 
111.28.b.6.A  Describe the volume formula V = Bh of a cylinder in terms of its base area and its height;  Solving Problems Involving Volume 
111.28.b.6.C  Use models and diagrams to explain the Pythagorean theorem.  Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsFinding Distance Between Two Points on the Coordinate Plane 
111.28.b.7  The student applies mathematical process standards to use geometry to solve problems. (Expressions, equations, and relationships)  Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsFinding Distance Between Two Points on the Coordinate PlaneInvestigating Surface Area and CrossSections of SolidsSolving Problems Involving Surface AreaSolving Problems Involving Volume 
111.28.b.7.A  Solve problems involving the volume of cylinders, cones, and spheres;  Solving Problems Involving Volume 
111.28.b.7.B  Use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders;  Investigating Surface Area and CrossSections of SolidsSolving Problems Involving Surface Area 
111.28.b.7.C  Use the Pythagorean Theorem and its converse to solve problems  Determining Unknown Side Lengths in Right Triangles of Polygons and Solids 
111.28.b.7.D  Determine the distance between two points on a coordinate plane using the Pythagorean Theorem.  Finding Distance Between Two Points on the Coordinate Plane 
111.28.b.8  The student applies mathematical process standards to use onevariable equations or inequalities in problem situations. (Expressions, equations, and relationships)  Analyzing TrianglesIdentifying Angle Relationships Formed by Parallel Lines and a Transversal 
111.28.b.8.D  Use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angleangle criterion for similarity of triangles. and  Analyzing TrianglesIdentifying Angle Relationships Formed by Parallel Lines and a Transversal 
Virginia Mathematics Standards of Learning  2009
6
Whole Numbers
6.18  Solve onestep linear equations in one variable involving whole number coefficients and positive rational solutions.  Simplifying Expressions by Combining Like Terms 
6.19a  Investigate and recognize the identity properties for addition and multiplication;  Identifying Properties of Addition and Multiplication 
6.19b  Investigate and recognize the multiplicative property of zero;  Identifying Properties of Addition and Multiplication 
6.8  Evaluate whole number numerical expressions, using the order of operations.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
Integers
6.2d  Compare and order fractions, decimals, and percents.  Comparing Integers 
6.3a  Identify and represent integers;  Identifying IntegersModeling Real Life Using Integers 
6.3b  Order and compare integers;  Comparing IntegersGraphing Integers 
6.3c  Identify and describe absolute value of integers.  Graphing IntegersIdentifying Opposite Integers and Absolute Value 
Fractions & Decimals
6.19c  Investigate and recognize the inverse property for multiplication.  Dividing Fractions 
6.2b  Identify a given fraction, decimal or percent from a representation;  Adding Like DenominatorsAdding Unlike DenominatorsConverting Mixed Numbers and Improper FractionsDefining Rational and IrrationalIdentifying Place Value and Rounding Decimal NumbersWriting Fractions in Simplest Form 
6.2c  Demonstrate equivalent relationships among fractions, decimals, and percents;  Converting Fractions to DecimalsConverting Mixed Numbers and Improper Fractions 
6.2d  Compare and order fractions, decimals, and percents.  Comparing Fractions and Decimals 
6.4  Demonstrate multiple representations of multiplication and division of fractions.  Dividing FractionsMultiplying Fractions 
6.6a  Multiply and divide fractions and mixed numbers;  Converting Mixed Numbers and Improper FractionsDividing FractionsMultiplying FractionsWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
6.6b  Estimate solutions and then solve singlestep and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.  Adding Like DenominatorsAdding Unlike DenominatorsConverting Mixed Numbers and Improper FractionsDividing FractionsMultiplying FractionsSubtracting Like DenominatorsSubtracting Unlike DenominatorsWriting Fractions in Simplest FormWriting an Integer as a Fraction or Decimal 
6.7  Solve singlestep and multistep practical problems involving addition, subtraction, multiplication, and division of decimals  Adding Decimal NumbersDividing Decimal NumbersIdentifying Place Value and Rounding Decimal NumbersMultiplying Decimal NumbersSubtracting Decimal NumbersWriting an Integer as a Fraction or Decimal 
Exponents & Roots
6.2c  Demonstrate equivalent relationships among fractions, decimals, and percents;  Converting from Scientific Notation to Standard Form 
6.2d  Compare and order fractions, decimals, and percents.  Comparing Numbers in Scientific Notation 
6.5  Investigate and describe concepts of positive exponents and perfect squares.  Evaluating All PowersIdentifying Bases and ExponentsIdentifying the Square Root of a Perfect Square 
6.8  Evaluate whole number numerical expressions, using the order of operations.  Evaluating All Powers 
Rates, Ratios, & Proportions
6.1  Describe and compare data, using ratios, and use appropriate notations, such as a/b , a to b, and a:b.  Expressing Ratios in Simplified FormExpressing Unit RatesSolving Practical Problems Using Ratios 
6.2a  Investigate and describe fractions, decimals and percents as ratios;  Expressing Ratios in Simplified Form 
6.2b  Identify a given fraction, decimal or percent from a representation;  Expressing a Fraction as a Percent 
6.2c  Demonstrate equivalent relationships among fractions, decimals, and percents;  Expressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
6.2d  Compare and order fractions, decimals, and percents.  Comparing Fractions, Decimals, Percents, and Ratios 
Equations
6.18  Solve onestep linear equations in one variable involving whole number coefficients and positive rational solutions.  Solving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionWriting OneStep Equations and Solving Word Problems 
Inequalities
6.20  Graph inequalities on a number line.  Graphing Inequalities on a Number LineWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
Graphing
6.11a  Identify the coordinates of a point in a coordinate plane;  Identifying an Ordered Pair Given a Graphed Point 
6.11b  Graph ordered pairs in a coordinate plane.  Identifying Quadrants and Graphing an Ordered PairIdentifying the Distance Between Two Points with Identical x or yValues 
6.12  Determine congruence of segments, angles, and polygons.  Identifying the Distance Between Two Points with Identical x or yValues 
6.17  Identify and extend geometric and arithmetic sequences.  Predicting Values in Tables Using Numerical Patterns 
Probability & Statistics
6.14a  Construct circle graphs;  Representing a Set of Data Using a Data Display 
6.14b  Draw conclusions and make predictions, using circle graphs;  Analyzing and Describing the Distribution of a Data Set (Data Display) 
6.14c  Compare and contrast graphs that present information from the same data set.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
6.15a  Describe mean as balance point;  Describing a Set of Data (Mean, Median, Mode, Range) 
6.15b  Decide which measure of center is appropriate for a given purpose.  Analyzing and Comparing Data Sets 
6.16a  Compare and contrast dependent and independent events;  Identifying the Outcomes in a Sample Space 
6.16b  Determine probabilities for dependent and independent events.  Determining the Likelihood of EventsFinding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample SpaceIdentifying the Theoretical Probability of an Event 
6.2b  Identify a given fraction, decimal or percent from a representation;  Representing a Set of Data Using a Data Display 
Geometry
6.10a  Define pi (π) as the ratio of the circumference of a circle to its diameter;  Solving Problems Involving Area and Circumference of a Circle 
6.10b  Solve practical problems involving circumference and area of a circle, given the diameter or radius;  Solving Problems Involving Area and Circumference of a Circle 
6.10c  Solve practical problems involving area and perimeter;  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Area and Circumference of a CircleSolving Problems Involving Polygons on the Coordinate Plane 
6.10d  Describe and determine the volume and surface area of a rectangular prism.  Investigating Surface Area and CrossSections of SolidsSolving Problems Involving Surface AreaSolving Problems Involving Volume 
6.12  Determine congruence of segments, angles, and polygons.  Analyzing TrianglesIdentifying and Classifying PolygonsUsing Congruency Statements to Identify Corresponding Parts of a Polygon 
6.13  Describe and identify properties of quadrilaterals.  Identifying PolygonsIdentifying QuadrilateralsIdentifying and Classifying Polygons 
7
Whole Numbers
7.13a  Write verbal expressions as algebraic expressions and sentences as equations and vice versa;  Writing Algebraic Expressions 
7.13b  Evaluate algebraic expressions for given replacement values of the variables.  Evaluating ExpressionsEvaluating Expressions Using the Order of OperationsSimplifying Expressions by Combining Like Terms 
7.16a  Apply the following properties of operations with real numbers: the commutative and associative properties for addition and multiplication;  Identifying Properties of Addition and Multiplication 
7.16c  Apply the following properties of operations with real numbers: the additive and multiplicative identity properties;  Identifying Properties of Addition and Multiplication 
7.16e  Apply the following properties of operations with real numbers: the multiplicative property of zero.  Identifying Properties of Addition and Multiplication 
Integers
7.16d  Apply the following properties of operations with real numbers: the additive and multiplicative inverse properties;  Adding Integers with Unlike Signs 
7.1e  Identify and describe absolute value for rational numbers.  Comparing IntegersIdentifying Opposite Integers and Absolute Value 
7.3a  Model addition, subtraction, multiplication and division of integers;  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionIdentifying IntegersModeling Real Life Using IntegersMultiplying and Dividing IntegersSubtracting Integers 
7.3b  Add, subtract, multiply, and divide integers.  Adding Integers with Like SignsAdding Integers with Unlike SignsCombining Integer Multiplication and DivisionCombining Integer Subtraction and AdditionMultiplying and Dividing IntegersSubtracting Integers 
Fractions & Decimals
7.16d  Apply the following properties of operations with real numbers: the additive and multiplicative inverse properties;  Dividing Fractions 
Exponents & Roots
7.1a  Investigate and describe the concept of negative exponents for powers of ten;  Representing Expressions with Only Positive Exponents 
7.1b  Determine scientific notation for numbers greater than zero;  Converting from Scientific Notation to Standard FormWriting in Scientific Notation 
7.1c  Compare and order fractions, decimals, percents and numbers written in scientific notation;  Comparing Numbers in Scientific Notation 
7.1d  Determine square roots;  Identifying the Square Root of a Perfect Square 
Rates, Ratios, & Proportions
7.4  Solve singlestep and multistep practical problems, using proportional reasoning.  Applying the Percent ProportionDetermining Proportional Relationships Between QuantitiesExpressing Unit RatesIdentify Actual Measurements and Scale FactorsSolving Proportions to Find Missing Terms 
Equations
7.13a  Write verbal expressions as algebraic expressions and sentences as equations and vice versa;  Writing OneStep Equations and Solving Word Problems 
7.14a  Solve one and twostep linear equations in one variable;  Applying the Distributive Property to Solve EquationsSolving OneStep Equations Involving Addition or SubtractionSolving OneStep Equations Involving Multiplication or DivisionSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two Sides 
7.14b  Solve practical problems requiring the solution of one and twostep linear equations.  Applying Knowledge of TwoStep Equations to Solve Word ProblemsWriting OneStep Equations and Solving Word Problems 
7.16b  Apply the following properties of operations with real numbers: the distributive property;  Applying the Distributive Property to Solve EquationsApplying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
7.15a  Solve onestep inequalities in one variable;  Solving OneStep Inequalities Involving Addition or SubtractionSolving OneStep Inequalities Involving Multiplication or DivisionSolving Word Problems Using Inequalities 
7.15b  Graph solutions to inequalities on the number line.  Graphing Inequalities on a Number LineWriting Inequalities Using GraphsWriting Mathematical Sentences Using Inequalities 
Graphing
7.12  Represent relationships with tables, graphs, rules, and words.  Determining if a Relation is a FunctionIdentifying Domain and RangeRepresenting Relations and Functions in Different Forms 
7.2  Describe and represent arithmetic and geometric sequences using variable expressions.  Predicting Values in Tables Using Numerical Patterns 
Probability & Statistics
7.10  Determine the probability of compound events, using the Fundamental (Basic) Counting Principle.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
7.11a  Given data in a practical situation, construct and analyze histograms;  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
7.11b  Compare and contrast histograms with other types of graphs presenting information from the same data set.  Analyzing and Describing the Distribution of a Data Set (Data Display)Representing a Set of Data Using a Data Display 
7.9  Investigate and describe the difference between the experimental probability and theoretical probability of an event.  Identifying and Comparing ProbabilitiesIdentifying the Theoretical Probability of an EventMaking Predictions 
Geometry
7.5a  Describe volume and surface area of cylinders;  Solving Problems Involving Volume 
7.5b  Solve practical problems involving the volume and surface area of rectangular prisms and cylinders;  Investigating Surface Area and CrossSections of SolidsSolving Problems Involving Surface AreaSolving Problems Involving Volume 
7.5c  Describe how changing one measured attribute of a rectangular prism affects its volume and surface area.  Solving Problems Involving Surface AreaSolving Problems Involving Volume 
7.6  Determine whether plane figures quadrilaterals and triangles are similar and write proportions to express the relationships between corresponding sides of similar figures.  Describing the Sequence of Transformations of Two Similar FiguresSolving Problems Involving Scale Drawings of Geometric Figures 
7.7  Compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid.  Identifying PolygonsIdentifying QuadrilateralsIdentifying and Classifying Polygons 
7.8  Given a polygon in the coordinate plane,represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane.  Describing the Sequence of Transformations of Two Similar FiguresIdentifying Properties of Rotations, Reflections, and TranslationsSolving Problems Involving Scale Drawings of Geometric FiguresUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8
Whole Numbers
8.15a  Solve multistep linear equations in one variable on one and two sides of the equation;  Simplifying Expressions by Combining Like Terms 
8.15b  Solve twostep linear inequalities and graph the results on a number line;  Simplifying Expressions by Combining Like Terms 
8.15c  Identify properties of operations used to solve an equation.  Identifying Properties of Addition and Multiplication 
8.1a  Simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;  Evaluating Expressions Using the Order of OperationsIdentifying Properties of Addition and MultiplicationSimplifying Expressions by Combining Like TermsWriting Algebraic Expressions 
8.4  Apply the order of operations to evaluate algebraic expressions for given replacement values of the variables.  Evaluating ExpressionsEvaluating Expressions Using the Order of Operations 
Integers
8.15c  Identify properties of operations used to solve an equation.  Adding Integers with Unlike Signs 
8.1b  Compare and order decimals, fractions, percents, and numbers written in scientific notation.  Comparing Integers 
8.2  Describe orally and in writing the relationships between the subsets of the real number system.  Identifying Integers 
Fractions & Decimals
8.15c  Identify properties of operations used to solve an equation.  Dividing Fractions 
8.1a  Simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;  Classifying NumbersDefining Rational and Irrational 
8.1b  Compare and order decimals, fractions, percents, and numbers written in scientific notation.  Comparing Fractions and DecimalsConverting Fractions to DecimalsConverting Mixed Numbers and Improper Fractions 
8.2  Describe orally and in writing the relationships between the subsets of the real number system.  Classifying NumbersDefining Rational and Irrational 
Exponents & Roots
8.1a  Simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;  Evaluating All Powers 
8.1b  Compare and order decimals, fractions, percents, and numbers written in scientific notation.  Comparing Numbers in Scientific NotationConverting from Scientific Notation to Standard FormWriting in Scientific Notation 
8.4  Apply the order of operations to evaluate algebraic expressions for given replacement values of the variables.  Identifying the Square Root of a Perfect Square 
8.5a  Determine whether a given number is a perfect square  Identifying the Square Root of a Perfect Square 
8.5b  Find the two consecutive whole numbers between which a square root lies.  Estimating Square Roots 
Rates, Ratios, & Proportions
8.1b  Compare and order decimals, fractions, percents, and numbers written in scientific notation.  Comparing Fractions, Decimals, Percents, and RatiosExpressing a Decimal Number as a PercentExpressing a Fraction as a PercentExpressing a Percent as a Decimal NumberExpressing a Percent as a Fraction 
8.3a  Solve practical problems involving rational numbers, percents, ratios, and proportions;  Applying the Percent ProportionComputing Simple InterestExpressing a Percent as a FractionSolving Practical Problems Using RatiosSolving Proportions to Find Missing Terms 
8.3b  Determine the percent increase or decrease for a given situation.  Identifying the Percent of Change 
Equations
8.15a  Solve multistep linear equations in one variable on one and two sides of the equation;  Applying Knowledge of TwoStep Equations to Solve Word ProblemsApplying the Distributive Property to Solve EquationsSolving TwoStep Equations Involving Division with Addition or SubtractionSolving TwoStep Equations Involving Multiplication with Addition or SubtractionSolving an Equation with Variables on Two Sides 
8.15c  Identify properties of operations used to solve an equation.  Applying the Distributive Property to Solve Equations 
8.1a  Simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;  Applying the Distributive Property to Write Equivalent Expressions with Variables 
Inequalities
8.15b  Solve twostep linear inequalities and graph the results on a number line;  Applying Knowledge of TwoStep Inequalities to Solve Word ProblemsSolving TwoStep Inequalities Involving Division with Addition or SubtractionSolving TwoStep Inequalities Involving Multiplication with Addition or Subtraction 
Graphing
8.14  Make connections between any two representations (tables, graphs, words, and rules) of a given relationship.  Determining if a Relation is a FunctionIdentifying Domain and RangeRepresenting Relations and Functions in Different Forms 
8.16  Graph a linear equation in two variables.  Calculating the Slope of a Graphed LineClassifying Functions as Linear or NonlinearDetermining Slope and yIntercept of Linear Equations in SlopeIntercept FormDetermining Whether Functions are Linear or Nonlinear Using PatternsDetermining if a Relation is a FunctionDetermining the Constant of VariationFinding the Rate of ChangeGraphing Linear Equations in SlopeIntercept Form Using the Slope and yIntercept Graphing a Linear Function in Two Variables Using InterceptsGraphing a Linear Function in Two Variables Using TablesIdentifying Quadrants and Graphing an Ordered PairIdentifying a Graph of a Direct VariationIdentifying a Table of a Direct VariationIdentifying an Ordered Pair Given a Graphed PointIdentifying the Distance Between Two Points with Identical x or yValuesIdentifying the Slope of a Line Using Two PointsIdentifying the x and yInterceptsPredicting Values Using a Direct VariationPredicting Values in Tables Using Numerical PatternsRepresenting Relations and Functions in Different FormsWriting Equations in SlopeIntercept Form (Slope and a Point)Writing Equations in SlopeIntercept Form (Slope and yIntercept)Writing Equations in SlopeIntercept Form (Two Points)Writing an Equation for Direct Variation 
8.17  Identify the domain, range, independent variable or dependent variable in a given situation.  Identifying Domain and RangeIdentifying Independent and Dependent Variables 
Probability & Statistics
8.12  Determine the probability of independent and dependent events with and without replacement.  Finding the Probabilities of Compound EventsIdentifying the Outcomes in a Sample Space 
8.13a  Make comparisons, predictions, and inferences, using information displayed in graphs;  Interpreting Scatter Plots to Investigate Patterns of AssociationSolving Problems Involving Bivariate Measurement Data 
Geometry
8.10a  Verify the Pythagorean Theorem  Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsFinding Distance Between Two Points on the Coordinate Plane 
8.10b  Apply the Pythagorean Theorem.  Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsFinding Distance Between Two Points on the Coordinate Plane 
8.11  Solve practical area and perimeter problems involving composite plane figures.  Finding Area of Polygons Using Composing and Decomposing TechniquesFinding the Missing Dimension of a Rectangle or TriangleSolving Problems Involving Area and Circumference of a CircleSolving Problems Involving Polygons on the Coordinate Plane 
8.6a  Verify by measuring and describe the relationships among vertical angles, adjacent angles, supplementary angles, and complementary angles  Identifying Angle Relationships Formed by Parallel Lines and a TransversalSolving Equations for an Unknown Angle in a Figure 
8.6b  Measure angles of less than 360°.  Classifying Angles 
8.7a  Investigate and solve practical problems involving volume and surface area of prisms, cylinders, cones, and pyramids;  Investigating Surface Area and CrossSections of SolidsSolving Problems Involving Surface AreaSolving Problems Involving Volume 
8.7b  Describe how changing one measured attribute of the figure affects the volume and surface area.  Solving Problems Involving Surface AreaSolving Problems Involving Volume 
8.8a  Apply transformations to plane figures  Identifying Properties of Rotations, Reflections, and TranslationsUsing Congruency Statements to Identify Corresponding Parts of a PolygonUsing the Coordinate Plane to Demonstrate Transformations 
8.8b  Identify applications of transformations.  Describing the Sequence of Transformations of Two Similar FiguresSolving Problems Involving Scale Drawings of Geometric FiguresUsing the Coordinate Plane to Demonstrate Transformations 
8.9  Construct a threedimensional model, given the top or bottom, side, and front views.  Investigating Surface Area and CrossSections of Solids 
Story
[sc:lfstory]Features
[sc:lffeatures]Documents
Document  Description 

Teachers Manual that discusses the game and its features in the context of classroom use.
General topics covered include:


Developmental Education Option: Alternative Math Remediation for High School Seniors and State College and Community College Students is a white paper which discusses the nationwide problems that entering college freshman are experiencing when taking college math placement exams. The paper highlights the benefits of using the AT< math game, Pi and the Lost Function, as an effective and efficient preparation tool for the different placement exams. 