Pi and the Lost Function Bundle
 \$47.97
Includes:
• Pi and the Lost Function Game (for Windows or Mac)
• Individual and Group Performance Reports

#### Introduction

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.

Michelle Roan

Middle School Math Teacher
Pi and the Lost Function math educational series is aligned with the Common Core State Standards in Mathematics, Grades 6-8. Features:
• Alignment to state standards
• Online Tracking and Reporting Tools
• Content selection to fit your students’ needs
Pi and the Lost Function comprises a comprehensive Pre-Algebra curriculum that can be used either as a tutoring tool (independent of classroom instruction) or as a classroom supplement. The game’s story keeps students engaged and provides a constructive learning context in which to explore math concepts. The in-game virtual math tutor provides differentiated instruction to students, so they are spending time where they need it most.

Educator Overview Video

For an in-depth look at how students learn while using The Lost Function products, visit:

##### Search Aligned Standards
 Standard: Grade: Any Filter: Refine Search
Pi and the Lost Function aligns with the Common Core State Standards in Mathematics, Grades 6-8.
###### Topic Areas Covered:
1. Whole Numbers
2. Integers
3. Fractions & Decimals
4. Exponents & Roots
5. Rates, Ratios, & Proportions
6. Equations
7. Inequalities
8. Graphing
9. Probability & Statistics
10. Geometry

`Common Core State Standards - 20106Whole Numbers6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions6.EE.A.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions6.EE.A.2bIdentify 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 Terms|Writing Algebraic Expressions6.EE.A.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of Operations6.EE.A.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like Terms6.EE.A.4Identify 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 Terms6.EE.B.6Use variables to represent numbers and write expressions when solving a real-world 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 Expressions6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules6.NS.B.4Find 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 FactorsIntegers6.NS.C.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using Integers6.NS.C.6Understand 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 Integers|Identifying Opposite Integers and Absolute Value6.NS.C.6aRecognize 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 Value6.NS.C.6cFind 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 Integers6.NS.C.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value6.NS.C.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers6.NS.C.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute Value6.NS.C.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & Decimals6.NS.A.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying Fractions6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal Numbers6.NS.C.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & Roots6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials6.EE.A.2bIdentify 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 Monomials6.NS.C.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & Proportions6.NS.C.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form6.RP.A.2Understand 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 Rates6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms6.RP.A.3aMake tables of equivalent ratios relating quantities with whole-number 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 Ratios6.RP.A.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios6.RP.A.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquations6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems6.EE.A.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word Problems6.EE.A.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables6.EE.B.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems6.EE.B.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems6.NS.B.4Find 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 VariablesInequalities6.EE.B.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities6.EE.B.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities6.NS.C.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line6.NS.C.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphing6.EE.C.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent Variables6.NS.C.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.C.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed Point6.NS.C.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.C.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-Values6.RP.A.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical Patterns6.RP.A.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & Statistics6.SP.A.1Recognize 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 Population6.SP.A.2Understand 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 Display6.SP.A.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.B.4Display 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 Display6.SP.B.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.B.5aReporting 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 Display6.SP.B.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.B.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.B.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometry6.G.A.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing Techniques6.G.A.2Find 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 real-world and mathematical problems.Solving Problems Involving Volume6.G.A.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate Plane6.G.A.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of Solids6.NS.C.8Solve real-world 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 Plane7Whole Numbers7.EE.A.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms7.EE.A.2Understand 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 Expressions7.EE.B.3Solve multi-step real-life 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 Operations7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of Operations7.EE.B.4aSolve 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 Expressions|Evaluating Expressions Using the Order of Operations7.NS.A.1Apply 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 Multiplication7.NS.A.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.A.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules7.NS.A.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegers7.EE.B.3Solve multi-step real-life 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 Division|Comparing Integers7.NS.A.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.A.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value7.NS.A.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value7.NS.A.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.A.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting Integers7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.A.2aUnderstand 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 real-world contexts.Multiplying and Dividing Integers7.NS.A.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & Decimals7.EE.B.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal7.NS.A.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.A.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators7.NS.A.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.A.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.A.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying Fractions7.NS.A.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions7.NS.A.2dConvert 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 Decimals|Writing an Integer as a Fraction or Decimal7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & Roots7.EE.B.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & Proportions7.EE.B.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction7.RP.A.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms7.RP.A.2aDecide 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 Quantities7.RP.A.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios7.RP.A.2cRepresent 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 Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms7.RP.A.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquations7.EE.A.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.B.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.A.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalities7.EE.A.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities7.EE.B.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphing7.RP.A.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.A.2aDecide 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 Variation|Identifying a Table of a Direct Variation7.RP.A.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.A.2dExplain 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 ChangeProbability & Statistics7.SP.A.1Understand 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 Population7.SP.A.2Use 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 Population7.SP.B.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set7.SP.B.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)7.SP.C.5Understand 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 Events7.SP.C.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making Predictions7.SP.C.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.C.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.C.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making Predictions7.SP.C.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space7.SP.C.8aUnderstand 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 Events|Identifying the Outcomes in a Sample Space7.SP.C.8bRepresent 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 Events|Identifying the Outcomes in a Sample Space7.SP.C.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometry7.G.A.1Solve 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 Figures7.G.A.2Draw (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 Triangles|Identifying and Classifying Polygons7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of Solids7.G.B.4Know 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 Circle7.G.B.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a Figure7.G.B.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole Numbers8.EE.C.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms8.EE.C.7bSolve 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 TermsFractions & Decimals8.EE.A.2Use 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 Numbers8.NS.A.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & Roots8.EE.A.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)8.EE.A.2Use 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 Cube|Identifying the Square Root of a Perfect Square8.EE.A.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific Notation8.NS.A.2Use 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 RootsRates, Ratios, & Proportions8.EE.B.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquations8.EE.C.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems8.EE.C.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphing8.EE.B.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct Variation8.EE.B.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.EE.C.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.C.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically8.EE.C.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.C.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically8.F.A.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different Forms8.F.A.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different Forms8.F.A.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct Variation8.F.B.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.F.B.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Tables8.SP.A.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & Statistics8.SP.A.1Construct 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 Association8.SP.A.2Know 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 Data8.SP.A.3Use 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 Data8.SP.A.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometry8.G.A.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.A.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.A.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.A.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.A.2Understand that a two-dimensional 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 Polygon8.G.A.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations8.G.A.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar Figures8.G.A.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a Transversal8.G.B.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane8.G.C.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeArkansas Common Core State Standards - 20126Whole Numbers6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions6.EE.A.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions6.EE.A.2bIdentify 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 Terms|Writing Algebraic Expressions6.EE.A.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of Operations6.EE.A.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like Terms6.EE.A.4Identify 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 Terms6.EE.B.6Use variables to represent numbers and write expressions when solving a real-world 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 Expressions6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules6.NS.B.4Find 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 FactorsIntegers6.NS.C.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using Integers6.NS.C.6Understand 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 Integers|Identifying Opposite Integers and Absolute Value6.NS.C.6aRecognize 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 Value6.NS.C.6cFind 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 Integers6.NS.C.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value6.NS.C.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers6.NS.C.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute Value6.NS.C.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & Decimals6.NS.A.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying Fractions6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal Numbers6.NS.C.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & Roots6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials6.EE.A.2bIdentify 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 Monomials6.NS.C.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & Proportions6.NS.C.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form6.RP.A.2Understand 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 Rates6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms6.RP.A.3aMake tables of equivalent ratios relating quantities with whole-number 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 Ratios6.RP.A.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios6.RP.A.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquations6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems6.EE.A.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word Problems6.EE.A.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables6.EE.B.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems6.EE.B.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems6.NS.B.4Find 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 VariablesInequalities6.EE.B.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities6.EE.B.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities6.NS.C.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line6.NS.C.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphing6.EE.C.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent Variables6.NS.C.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.C.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed Point6.NS.C.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.C.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-Values6.RP.A.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical Patterns6.RP.A.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & Statistics6.SP.A.1Recognize 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 Population6.SP.A.2Understand 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 Display6.SP.A.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.B.4Display 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 Display6.SP.B.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.B.5aReporting 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 Display6.SP.B.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.B.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.B.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometry6.G.A.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing Techniques6.G.A.2Find 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 real-world and mathematical problems.Solving Problems Involving Volume6.G.A.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate Plane6.G.A.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of Solids6.NS.C.8Solve real-world 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 Plane7Whole Numbers7.EE.A.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms7.EE.A.2Understand 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 Expressions7.EE.B.3Solve multi-step real-life 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 Operations7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of Operations7.EE.B.4aSolve 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 Expressions|Evaluating Expressions Using the Order of Operations7.NS.A.1Apply 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 Multiplication7.NS.A.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.A.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules7.NS.A.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegers7.EE.B.3Solve multi-step real-life 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 Division|Comparing Integers7.NS.A.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.A.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value7.NS.A.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value7.NS.A.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.A.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting Integers7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.A.2aUnderstand 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 real-world contexts.Multiplying and Dividing Integers7.NS.A.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & Decimals7.EE.B.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal7.NS.A.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.A.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators7.NS.A.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.A.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.A.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying Fractions7.NS.A.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions7.NS.A.2dConvert 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 Decimals|Writing an Integer as a Fraction or Decimal7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & Roots7.EE.B.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & Proportions7.EE.B.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction7.RP.A.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms7.RP.A.2aDecide 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 Quantities7.RP.A.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios7.RP.A.2cRepresent 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 Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms7.RP.A.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquations7.EE.A.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.B.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.NS.A.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.A.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalities7.EE.A.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities7.EE.B.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphing7.RP.A.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.A.2aDecide 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 Variation|Identifying a Table of a Direct Variation7.RP.A.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.A.2dExplain 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 ChangeProbability & Statistics7.SP.A.1Understand 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 Population7.SP.A.2Use 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 Population7.SP.B.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set7.SP.B.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)7.SP.C.5Understand 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 Events7.SP.C.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making Predictions7.SP.C.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.C.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.C.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making Predictions7.SP.C.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space7.SP.C.8aUnderstand 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 Events|Identifying the Outcomes in a Sample Space7.SP.C.8bRepresent 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 Events|Identifying the Outcomes in a Sample Space7.SP.C.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometry7.G.A.1Solve 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 Figures7.G.A.2Draw (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 Triangles|Identifying and Classifying Polygons7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of Solids7.G.B.4Know 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 Circle7.G.B.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a Figure7.G.B.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole Numbers8.EE.C.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms8.EE.C.7bSolve 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 TermsFractions & Decimals8.EE.A.2Use 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 Numbers8.NS.A.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & Roots8.EE.A.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)8.EE.A.2Use 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 Cube|Identifying the Square Root of a Perfect Square8.EE.A.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific Notation8.NS.A.2Use 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 RootsRates, Ratios, & Proportions8.EE.B.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquations8.EE.C.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems8.EE.C.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphing8.EE.B.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct Variation8.EE.B.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.EE.C.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.C.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically8.EE.C.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.C.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically8.F.A.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different Forms8.F.A.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different Forms8.F.A.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct Variation8.F.B.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.F.B.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Tables8.SP.A.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & Statistics8.SP.A.1Construct 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 Association8.SP.A.2Know 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 Data8.SP.A.3Use 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 Data8.SP.A.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometry8.G.A.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.A.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.A.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.A.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.A.2Understand that a two-dimensional 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 Polygon8.G.A.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations8.G.A.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar Figures8.G.A.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a Transversal8.G.B.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane8.G.C.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeCalifornia Common Core State Standards - 20136Whole Numbers6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions6.EE.2bIdentify 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 Terms|Writing Algebraic Expressions6.EE.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of Operations6.EE.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like Terms6.EE.4Identify 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 Terms6.EE.6Use variables to represent numbers and write expressions when solving a real-world 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 Expressions6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules6.NS.4Find 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 FactorsIntegers6.NS.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using Integers6.NS.6Understand 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 Integers|Identifying Opposite Integers and Absolute Value6.NS.6aRecognize 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 Value6.NS.6cFind 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 Integers6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers6.NS.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute Value6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & Decimals6.NS.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying Fractions6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal Numbers6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & Roots6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials6.EE.2bIdentify 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 Monomials6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & Proportions6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form6.RP.2Understand 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 Rates6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Ratios6.RP.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios6.RP.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquations6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems6.EE.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word Problems6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables6.EE.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems6.EE.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems6.NS.4Find 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 VariablesInequalities6.EE.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities6.EE.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphing6.EE.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent Variables6.NS.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed Point6.NS.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-Values6.RP.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical Patterns6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & Statistics6.SP.1Recognize 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 Population6.SP.2Understand 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 Display6.SP.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.4Display 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 Display6.SP.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5aReporting 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 Display6.SP.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometry6.G.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing Techniques6.G.2Find 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 real-world and mathematical problems.Solving Problems Involving Volume6.G.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate Plane6.G.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of Solids6.NS.8Solve real-world 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 Plane7Whole Numbers7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms7.EE.2Understand 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 Expressions7.EE.3Solve multi-step real-life 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 Operations7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of Operations7.EE.4aSolve 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 Expressions|Evaluating Expressions Using the Order of Operations7.NS.1Apply 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 Multiplication7.NS.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules7.NS.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegers7.EE.3Solve multi-step real-life 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 Division|Comparing Integers7.NS.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value7.NS.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value7.NS.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting Integers7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2aUnderstand 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 real-world contexts.Multiplying and Dividing Integers7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & Decimals7.EE.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal7.NS.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators7.NS.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying Fractions7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions7.NS.2dConvert 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 Decimals|Writing an Integer as a Fraction or Decimal7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & Roots7.EE.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & Proportions7.EE.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction7.RP.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms7.RP.2aDecide 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 Quantities7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios7.RP.2cRepresent 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 Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms7.RP.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquations7.EE.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalities7.EE.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities7.EE.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphing7.RP.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2aDecide 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 Variation|Identifying a Table of a Direct Variation7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2dExplain 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 ChangeProbability & Statistics7.SP.1Understand 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 Population7.SP.2Use 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 Population7.SP.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set7.SP.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)7.SP.5Understand 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 Events7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making Predictions7.SP.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making Predictions7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space7.SP.8aUnderstand 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 Events|Identifying the Outcomes in a Sample Space7.SP.8bRepresent 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 Events|Identifying the Outcomes in a Sample Space7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometry7.G.1Solve 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 Figures7.G.2Draw (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 Triangles|Identifying and Classifying Polygons7.G.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of Solids7.G.4Know 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 Circle7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a Figure7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole Numbers8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms8.EE.7bSolve 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 TermsFractions & Decimals8.EE.2Use 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 Numbers8.NS.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & Roots8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)8.EE.2Use 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 Cube|Identifying the Square Root of a Perfect Square8.EE.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific Notation8.NS.2Use 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 RootsRates, Ratios, & Proportions8.EE.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquations8.EE.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems8.EE.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphing8.EE.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct Variation8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.EE.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically8.EE.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically8.F.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different Forms8.F.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different Forms8.F.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct Variation8.F.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.F.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Tables8.SP.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & Statistics8.SP.1Construct 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 Association8.SP.2Know 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 Data8.SP.3Use 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 Data8.SP.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometry8.G.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.2Understand that a two-dimensional 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 Polygon8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations8.G.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar Figures8.G.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a Transversal8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeFlorida Common Core State Standards - 20106Whole NumbersMACC.6.EE.1.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic ExpressionsMACC.6.EE.1.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic ExpressionsMACC.6.EE.1.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic ExpressionsMACC.6.EE.1.2bIdentify 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 Terms|Writing Algebraic ExpressionsMACC.6.EE.1.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of OperationsMACC.6.EE.1.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like TermsMACC.6.EE.1.4Identify 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 TermsMACC.6.EE.2.6Use variables to represent numbers and write expressions when solving a real-world 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 ExpressionsMACC.6.NS.2.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility RulesMACC.6.NS.2.4Find 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 FactorsIntegersMACC.6.NS.3.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using IntegersMACC.6.NS.3.6Understand 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 Integers|Identifying Opposite Integers and Absolute ValueMACC.6.NS.3.6aRecognize 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 ValueMACC.6.NS.3.6cFind 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 IntegersMACC.6.NS.3.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute ValueMACC.6.NS.3.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing IntegersMACC.6.NS.3.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing IntegersMACC.6.NS.3.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute ValueMACC.6.NS.3.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & DecimalsMACC.6.NS.1.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying FractionsMACC.6.NS.2.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal NumbersMACC.6.NS.3.7Understand ordering and absolute value of rational numbersComparing Fractions and DecimalsMACC.6.NS.3.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & RootsMACC.6.EE.1.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and ExponentsMACC.6.EE.1.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring MonomialsMACC.6.EE.1.2bIdentify 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 MonomialsMACC.6.NS.3.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific NotationMACC.6.NS.3.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & ProportionsMACC.6.NS.3.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and RatiosMACC.6.NS.3.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and RatiosMACC.6.RP.1.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified FormMACC.6.RP.1.2Understand 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 RatesMACC.6.RP.1.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing TermsMACC.6.RP.1.3aMake tables of equivalent ratios relating quantities with whole-number 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 RatiosMACC.6.RP.1.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using RatiosMACC.6.RP.1.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquationsMACC.6.EE.1.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word ProblemsMACC.6.EE.1.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word ProblemsMACC.6.EE.1.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with VariablesMACC.6.EE.2.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word ProblemsMACC.6.EE.2.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word ProblemsMACC.6.NS.2.4Find 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 VariablesInequalitiesMACC.6.EE.2.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using InequalitiesMACC.6.EE.2.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using InequalitiesMACC.6.NS.3.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number LineMACC.6.NS.3.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphingMACC.6.EE.3.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent VariablesMACC.6.NS.3.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-InterceptsMACC.6.NS.3.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed PointMACC.6.NS.3.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-InterceptsMACC.6.NS.3.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-ValuesMACC.6.RP.1.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical PatternsMACC.6.RP.1.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & StatisticsMACC.6.SP.1.1Recognize 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 PopulationMACC.6.SP.1.2Understand 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 DisplayMACC.6.SP.1.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetMACC.6.SP.2.4Display 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 DisplayMACC.6.SP.2.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a PopulationMACC.6.SP.2.5aReporting 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 DisplayMACC.6.SP.2.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a PopulationMACC.6.SP.2.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetMACC.6.SP.2.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometryMACC.6.G.1.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing TechniquesMACC.6.G.1.2Find 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 real-world and mathematical problems.Solving Problems Involving VolumeMACC.6.G.1.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate PlaneMACC.6.G.1.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of SolidsMACC.6.NS.3.8Solve real-world 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 Plane7Whole NumbersMACC.7.EE.1.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like TermsMACC.7.EE.1.2Understand 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 ExpressionsMACC.7.EE.2.3Solve multi-step real-life 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 OperationsMACC.7.EE.2.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of OperationsMACC.7.EE.2.4aSolve 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 Expressions|Evaluating Expressions Using the Order of OperationsMACC.7.NS.1.1Apply 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 MultiplicationMACC.7.NS.1.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and MultiplicationMACC.7.NS.1.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and MultiplicationMACC.7.NS.1.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility RulesMACC.7.NS.1.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and MultiplicationMACC.7.NS.1.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and MultiplicationMACC.7.NS.1.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegersMACC.7.EE.2.3Solve multi-step real-life 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 Division|Comparing IntegersMACC.7.NS.1.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting IntegersMACC.7.NS.1.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute ValueMACC.7.NS.1.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute ValueMACC.7.NS.1.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting IntegersMACC.7.NS.1.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting IntegersMACC.7.NS.1.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing IntegersMACC.7.NS.1.2aUnderstand 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 real-world contexts.Multiplying and Dividing IntegersMACC.7.NS.1.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing IntegersMACC.7.NS.1.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing IntegersMACC.7.NS.1.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & DecimalsMACC.7.EE.2.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or DecimalMACC.7.NS.1.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsMACC.7.NS.1.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike DenominatorsMACC.7.NS.1.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsMACC.7.NS.1.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsMACC.7.NS.1.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or DecimalMACC.7.NS.1.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying FractionsMACC.7.NS.1.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or DecimalMACC.7.NS.1.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying FractionsMACC.7.NS.1.2dConvert 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 Decimals|Writing an Integer as a Fraction or DecimalMACC.7.NS.1.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & RootsMACC.7.EE.2.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & ProportionsMACC.7.EE.2.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a FractionMACC.7.RP.1.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing TermsMACC.7.RP.1.2aDecide 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 QuantitiesMACC.7.RP.1.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using RatiosMACC.7.RP.1.2cRepresent proportional relationships by equations.Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing TermsMACC.7.RP.1.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquationsMACC.7.EE.1.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsMACC.7.EE.2.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsMACC.7.EE.2.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsMACC.7.NS.1.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesMACC.7.NS.1.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesMACC.7.NS.1.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalitiesMACC.7.EE.1.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using InequalitiesMACC.7.EE.2.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using InequalitiesMACC.7.EE.2.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphingMACC.7.RP.1.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct VariationMACC.7.RP.1.2aDecide 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 Variation|Identifying a Table of a Direct VariationMACC.7.RP.1.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct VariationMACC.7.RP.1.2dExplain 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 ChangeProbability & StatisticsMACC.7.SP.1.1Understand 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 PopulationMACC.7.SP.1.2Use 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 PopulationMACC.7.SP.2.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetMACC.7.SP.2.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)MACC.7.SP.3.5Understand 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 EventsMACC.7.SP.3.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making PredictionsMACC.7.SP.3.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making PredictionsMACC.7.SP.3.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making PredictionsMACC.7.SP.3.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making PredictionsMACC.7.SP.3.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample SpaceMACC.7.SP.3.8aUnderstand 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 Events|Identifying the Outcomes in a Sample SpaceMACC.7.SP.3.8bRepresent 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 Events|Identifying the Outcomes in a Sample SpaceMACC.7.SP.3.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometryMACC.7.G.1.1Solve 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 FiguresMACC.7.G.1.2Draw (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 Triangles|Identifying and Classifying PolygonsMACC.7.G.1.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of SolidsMACC.7.G.2.4Know 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 CircleMACC.7.G.2.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a FigureMACC.7.G.2.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole NumbersMACC.8.EE.3.7Solve linear equations in one variable.Simplifying Expressions by Combining Like TermsMACC.8.EE.3.7bSolve 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 TermsFractions & DecimalsMACC.8.EE.1.2Use 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 NumbersMACC.8.NS.1.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & RootsMACC.8.EE.1.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)MACC.8.EE.1.2Use 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 Cube|Identifying the Square Root of a Perfect SquareMACC.8.EE.1.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationMACC.8.NS.1.2Use 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 RootsRates, Ratios, & ProportionsMACC.8.EE.2.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquationsMACC.8.EE.3.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsMACC.8.EE.3.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphingMACC.8.EE.2.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct VariationMACC.8.EE.2.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct VariationMACC.8.EE.3.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations GraphicallyMACC.8.EE.3.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations GraphicallyMACC.8.EE.3.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations GraphicallyMACC.8.EE.3.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations AlgebraicallyMACC.8.F.1.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different FormsMACC.8.F.1.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different FormsMACC.8.F.1.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct VariationMACC.8.F.2.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct VariationMACC.8.F.2.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using TablesMACC.8.SP.1.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & StatisticsMACC.8.SP.1.1Construct 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 AssociationMACC.8.SP.1.2Know 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 DataMACC.8.SP.1.3Use 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 DataMACC.8.SP.1.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometryMACC.8.G.1.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate TransformationsMACC.8.G.1.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate TransformationsMACC.8.G.1.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate TransformationsMACC.8.G.1.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate TransformationsMACC.8.G.1.2Understand that a two-dimensional 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 PolygonMACC.8.G.1.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate TransformationsMACC.8.G.1.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar FiguresMACC.8.G.1.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a TransversalMACC.8.G.2.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsMACC.8.G.2.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsMACC.8.G.2.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate PlaneMACC.8.G.3.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeGeorgia Common Core Performance Standards - 20116Whole NumbersMCC6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic ExpressionsMCC6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic ExpressionsMCC6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic ExpressionsMCC6.EE.2bIdentify 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 Terms|Writing Algebraic ExpressionsMCC6.EE.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of OperationsMCC6.EE.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like TermsMCC6.EE.4Identify 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 TermsMCC6.EE.6Use variables to represent numbers and write expressions when solving a real-world 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 ExpressionsMCC6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility RulesMCC6.NS.4Find 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 FactorsIntegersMCC6.NS.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using IntegersMCC6.NS.6Understand 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 Integers|Identifying Opposite Integers and Absolute ValueMCC6.NS.6aRecognize 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 ValueMCC6.NS.6cFind 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 IntegersMCC6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute ValueMCC6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing IntegersMCC6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing IntegersMCC6.NS.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute ValueMCC6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & DecimalsMCC6.NS.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying FractionsMCC6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal NumbersMCC6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and DecimalsMCC6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & RootsMCC6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and ExponentsMCC6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring MonomialsMCC6.EE.2bIdentify 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 MonomialsMCC6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific NotationMCC6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & ProportionsMCC6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and RatiosMCC6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and RatiosMCC6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified FormMCC6.RP.2Understand 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 RatesMCC6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing TermsMCC6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 RatiosMCC6.RP.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using RatiosMCC6.RP.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquationsMCC6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word ProblemsMCC6.EE.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word ProblemsMCC6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with VariablesMCC6.EE.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word ProblemsMCC6.EE.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word ProblemsMCC6.NS.4Find 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 VariablesInequalitiesMCC6.EE.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using InequalitiesMCC6.EE.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using InequalitiesMCC6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number LineMCC6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphingMCC6.EE.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent VariablesMCC6.NS.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-InterceptsMCC6.NS.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed PointMCC6.NS.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-InterceptsMCC6.NS.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-ValuesMCC6.RP.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical PatternsMCC6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & StatisticsMCC6.SP.1Recognize 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 PopulationMCC6.SP.2Understand 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 DisplayMCC6.SP.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetMCC6.SP.4Display 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 DisplayMCC6.SP.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a PopulationMCC6.SP.5aReporting 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 DisplayMCC6.SP.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a PopulationMCC6.SP.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetMCC6.SP.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometryMCC6.G.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing TechniquesMCC6.G.2Find 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 real-world and mathematical problems.Solving Problems Involving VolumeMCC6.G.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate PlaneMCC6.G.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of SolidsMCC6.NS.8Solve real-world 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 Plane7Whole NumbersMCC7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like TermsMCC7.EE.2Understand 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 ExpressionsMCC7.EE.3Solve multi-step real-life 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 OperationsMCC7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of OperationsMCC7.EE.4aSolve 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 Expressions|Evaluating Expressions Using the Order of OperationsMCC7.NS.1Apply 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 MultiplicationMCC7.NS.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and MultiplicationMCC7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and MultiplicationMCC7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility RulesMCC7.NS.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and MultiplicationMCC7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and MultiplicationMCC7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegersMCC7.EE.3Solve multi-step real-life 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 Division|Comparing IntegersMCC7.NS.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting IntegersMCC7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute ValueMCC7.NS.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute ValueMCC7.NS.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting IntegersMCC7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting IntegersMCC7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing IntegersMCC7.NS.2aUnderstand 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 real-world contexts.Multiplying and Dividing IntegersMCC7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing IntegersMCC7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing IntegersMCC7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & DecimalsMCC7.EE.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or DecimalMCC7.NS.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsMCC7.NS.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike DenominatorsMCC7.NS.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsMCC7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsMCC7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or DecimalMCC7.NS.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying FractionsMCC7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or DecimalMCC7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying FractionsMCC7.NS.2dConvert 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 Decimals|Writing an Integer as a Fraction or DecimalMCC7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & RootsMCC7.EE.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & ProportionsMCC7.EE.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a FractionMCC7.RP.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing TermsMCC7.RP.2aDecide 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 QuantitiesMCC7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using RatiosMCC7.RP.2cRepresent 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 Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing TermsMCC7.RP.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquationsMCC7.EE.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsMCC7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsMCC7.EE.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsMCC7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesMCC7.NS.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesMCC7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalitiesMCC7.EE.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using InequalitiesMCC7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using InequalitiesMCC7.EE.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphingMCC7.RP.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct VariationMCC7.RP.2aDecide 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 Variation|Identifying a Table of a Direct VariationMCC7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct VariationMCC7.RP.2dExplain 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 ChangeProbability & StatisticsMCC7.SP.1Understand 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 PopulationMCC7.SP.2Use 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 PopulationMCC7.SP.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetMCC7.SP.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)MCC7.SP.5Understand 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 EventsMCC7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making PredictionsMCC7.SP.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making PredictionsMCC7.SP.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making PredictionsMCC7.SP.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making PredictionsMCC7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample SpaceMCC7.SP.8aUnderstand 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 Events|Identifying the Outcomes in a Sample SpaceMCC7.SP.8bRepresent 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 Events|Identifying the Outcomes in a Sample SpaceMCC7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometryMCC7.G.1Solve 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 FiguresMCC7.G.2Draw (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 Triangles|Identifying and Classifying PolygonsMCC7.G.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of SolidsMCC7.G.4Know 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 CircleMCC7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a FigureMCC7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole NumbersMCC8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like TermsMCC8.EE.7bSolve 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 TermsFractions & DecimalsMCC8.EE.2Use 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 NumbersMCC8.NS.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & RootsMCC8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)MCC8.EE.2Use 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 Cube|Identifying the Square Root of a Perfect SquareMCC8.EE.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationMCC8.NS.2Use 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 RootsRates, Ratios, & ProportionsMCC8.EE.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquationsMCC8.EE.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsMCC8.EE.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphingMCC8.EE.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct VariationMCC8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct VariationMCC8.EE.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations GraphicallyMCC8.EE.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations GraphicallyMCC8.EE.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations GraphicallyMCC8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations AlgebraicallyMCC8.F.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different FormsMCC8.F.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different FormsMCC8.F.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct VariationMCC8.F.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct VariationMCC8.F.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using TablesMCC8.SP.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & StatisticsMCC8.SP.1Construct 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 AssociationMCC8.SP.2Know 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 DataMCC8.SP.3Use 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 DataMCC8.SP.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometryMCC8.G.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate TransformationsMCC8.G.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate TransformationsMCC8.G.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate TransformationsMCC8.G.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate TransformationsMCC8.G.2Understand that a two-dimensional 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 PolygonMCC8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate TransformationsMCC8.G.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar FiguresMCC8.G.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a TransversalMCC8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsMCC8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and SolidsMCC8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate PlaneMCC8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeMichigan Common Core State Standards - 20106Whole Numbers6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions6.EE.2bIdentify 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 Terms|Writing Algebraic Expressions6.EE.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of Operations6.EE.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like Terms6.EE.4Identify 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 Terms6.EE.6Use variables to represent numbers and write expressions when solving a real-world 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 Expressions6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules6.NS.4Find 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 FactorsIntegers6.NS.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using Integers6.NS.6Understand 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 Integers|Identifying Opposite Integers and Absolute Value6.NS.6aRecognize 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 Value6.NS.6cFind 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 Integers6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers6.NS.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute Value6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & Decimals6.NS.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying Fractions6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal Numbers6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & Roots6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials6.EE.2bIdentify 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 Monomials6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & Proportions6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form6.RP.2Understand 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 Rates6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Ratios6.RP.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios6.RP.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquations6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems6.EE.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word Problems6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables6.EE.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems6.EE.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems6.NS.4Find 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 VariablesInequalities6.EE.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities6.EE.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphing6.EE.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent Variables6.NS.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed Point6.NS.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-Values6.RP.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical Patterns6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & Statistics6.SP.1Recognize 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 Population6.SP.2Understand 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 Display6.SP.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.4Display 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 Display6.SP.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5aReporting 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 Display6.SP.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometry6.G.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing Techniques6.G.2Find 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 real-world and mathematical problems.Solving Problems Involving Volume6.G.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate Plane6.G.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of Solids6.NS.8Solve real-world 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 Plane7Whole Numbers7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms7.EE.2Understand 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 Expressions7.EE.3Solve multi-step real-life 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 Operations7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of Operations7.EE.4aSolve 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 Expressions|Evaluating Expressions Using the Order of Operations7.NS.1Apply 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 Multiplication7.NS.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules7.NS.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegers7.EE.3Solve multi-step real-life 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 Division|Comparing Integers7.NS.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value7.NS.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value7.NS.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting Integers7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2aUnderstand 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 real-world contexts.Multiplying and Dividing Integers7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & Decimals7.EE.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal7.NS.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators7.NS.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying Fractions7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions7.NS.2dConvert 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 Decimals|Writing an Integer as a Fraction or Decimal7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & Roots7.EE.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & Proportions7.EE.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction7.RP.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms7.RP.2aDecide 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 Quantities7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios7.RP.2cRepresent 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 Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms7.RP.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquations7.EE.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalities7.EE.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities7.EE.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphing7.RP.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2aDecide 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 Variation|Identifying a Table of a Direct Variation7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2dExplain 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 ChangeProbability & Statistics7.SP.1Understand 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 Population7.SP.2Use 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 Population7.SP.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set7.SP.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)7.SP.5Understand 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 Events7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making Predictions7.SP.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making Predictions7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space7.SP.8aUnderstand 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 Events|Identifying the Outcomes in a Sample Space7.SP.8bRepresent 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 Events|Identifying the Outcomes in a Sample Space7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometry7.G.1Solve 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 Figures7.G.2Draw (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 Triangles|Identifying and Classifying Polygons7.G.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of Solids7.G.4Know 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 Circle7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a Figure7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole Numbers8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms8.EE.7bSolve 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 TermsFractions & Decimals8.EE.2Use 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 Numbers8.NS.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & Roots8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)8.EE.2Use 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 Cube|Identifying the Square Root of a Perfect Square8.EE.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific Notation8.NS.2Use 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 RootsRates, Ratios, & Proportions8.EE.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquations8.EE.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems8.EE.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphing8.EE.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct Variation8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.EE.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically8.EE.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically8.F.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different Forms8.F.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different Forms8.F.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct Variation8.F.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.F.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Tables8.SP.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & Statistics8.SP.1Construct 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 Association8.SP.2Know 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 Data8.SP.3Use 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 Data8.SP.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometry8.G.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.2Understand that a two-dimensional 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 Polygon8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations8.G.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar Figures8.G.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a Transversal8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeNew York Common Core State Standards - 20126Whole Numbers6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions6.EE.2bIdentify 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 Terms|Writing Algebraic Expressions6.EE.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of Operations6.EE.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like Terms6.EE.4Identify 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 Terms6.EE.6Use variables to represent numbers and write expressions when solving a real-world 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 Expressions6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules6.NS.4Find 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 FactorsIntegers6.NS.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using Integers6.NS.6Understand 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 Integers|Identifying Opposite Integers and Absolute Value6.NS.6aRecognize 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 Value6.NS.6cFind 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 Integers6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers6.NS.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute Value6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & Decimals6.NS.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying Fractions6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal Numbers6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & Roots6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials6.EE.2bIdentify 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 Monomials6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & Proportions6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form6.RP.2Understand 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 Rates6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Ratios6.RP.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios6.RP.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquations6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems6.EE.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word Problems6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables6.EE.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems6.EE.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems6.NS.4Find 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 VariablesInequalities6.EE.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities6.EE.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphing6.EE.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent Variables6.NS.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed Point6.NS.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-Values6.RP.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical Patterns6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & Statistics6.SP.1Recognize 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 Population6.SP.2Understand 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 Display6.SP.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.4Display 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 Display6.SP.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5aReporting 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 Display6.SP.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometry6.G.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing Techniques6.G.2Find 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 real-world and mathematical problems.Solving Problems Involving Volume6.G.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate Plane6.G.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of Solids6.NS.8Solve real-world 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 Plane7Whole Numbers7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms7.EE.2Understand 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 Expressions7.EE.3Solve multi-step real-life 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 Operations7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of Operations7.EE.4aSolve 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 Expressions|Evaluating Expressions Using the Order of Operations7.NS.1Apply 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 Multiplication7.NS.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules7.NS.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegers7.EE.3Solve multi-step real-life 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 Division|Comparing Integers7.NS.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value7.NS.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value7.NS.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting Integers7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2aUnderstand 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 real-world contexts.Multiplying and Dividing Integers7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & Decimals7.EE.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal7.NS.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators7.NS.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying Fractions7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions7.NS.2dConvert 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 Decimals|Writing an Integer as a Fraction or Decimal7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & Roots7.EE.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & Proportions7.EE.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction7.RP.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms7.RP.2aDecide 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 Quantities7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios7.RP.2cRepresent 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 Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms7.RP.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquations7.EE.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalities7.EE.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities7.EE.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphing7.RP.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2aDecide 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 Variation|Identifying a Table of a Direct Variation7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2dExplain 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 ChangeProbability & Statistics7.SP.1Understand 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 Population7.SP.2Use 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 Population7.SP.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set7.SP.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)7.SP.5Understand 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 Events7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making Predictions7.SP.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making Predictions7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space7.SP.8aUnderstand 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 Events|Identifying the Outcomes in a Sample Space7.SP.8bRepresent 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 Events|Identifying the Outcomes in a Sample Space7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometry7.G.1Solve 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 Figures7.G.2Draw (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 Triangles|Identifying and Classifying Polygons7.G.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of Solids7.G.4Know 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 Circle7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a Figure7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole Numbers8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms8.EE.7bSolve 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 TermsFractions & Decimals8.EE.2Use 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 Numbers8.NS.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & Roots8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)8.EE.2Use 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 Cube|Identifying the Square Root of a Perfect Square8.EE.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific Notation8.NS.2Use 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 RootsRates, Ratios, & Proportions8.EE.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquations8.EE.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems8.EE.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphing8.EE.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct Variation8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.EE.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically8.EE.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically8.F.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different Forms8.F.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different Forms8.F.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct Variation8.F.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.F.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Tables8.SP.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & Statistics8.SP.1Construct 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 Association8.SP.2Know 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 Data8.SP.3Use 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 Data8.SP.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometry8.G.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.2Understand that a two-dimensional 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 Polygon8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations8.G.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar Figures8.G.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a Transversal8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeNorth Carolina Common Core State Standards - 20126Whole Numbers6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions6.EE.2bIdentify 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 Terms|Writing Algebraic Expressions6.EE.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of Operations6.EE.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like Terms6.EE.4Identify 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 Terms6.EE.6Use variables to represent numbers and write expressions when solving a real-world 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 Expressions6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules6.NS.4Find 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 FactorsIntegers6.NS.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using Integers6.NS.6Understand 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 Integers|Identifying Opposite Integers and Absolute Value6.NS.6aRecognize 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 Value6.NS.6cFind 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 Integers6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers6.NS.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute Value6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & Decimals6.NS.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying Fractions6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal Numbers6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & Roots6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials6.EE.2bIdentify 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 Monomials6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & Proportions6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form6.RP.2Understand 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 Rates6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Ratios6.RP.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios6.RP.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquations6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems6.EE.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word Problems6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables6.EE.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems6.EE.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems6.NS.4Find 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 VariablesInequalities6.EE.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities6.EE.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphing6.EE.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent Variables6.NS.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed Point6.NS.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-Values6.RP.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical Patterns6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & Statistics6.SP.1Recognize 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 Population6.SP.2Understand 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 Display6.SP.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.4Display 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 Display6.SP.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5aReporting 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 Display6.SP.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometry6.G.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing Techniques6.G.2Find 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 real-world and mathematical problems.Solving Problems Involving Volume6.G.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate Plane6.G.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of Solids6.NS.8Solve real-world 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 Plane7Whole Numbers7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms7.EE.2Understand 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 Expressions7.EE.3Solve multi-step real-life 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 Operations7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of Operations7.EE.4aSolve 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 Expressions|Evaluating Expressions Using the Order of Operations7.NS.1Apply 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 Multiplication7.NS.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules7.NS.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegers7.EE.3Solve multi-step real-life 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 Division|Comparing Integers7.NS.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value7.NS.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value7.NS.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting Integers7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2aUnderstand 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 real-world contexts.Multiplying and Dividing Integers7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & Decimals7.EE.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal7.NS.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators7.NS.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying Fractions7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions7.NS.2dConvert 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 Decimals|Writing an Integer as a Fraction or Decimal7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & Roots7.EE.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & Proportions7.EE.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction7.RP.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms7.RP.2aDecide 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 Quantities7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios7.RP.2cRepresent 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 Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms7.RP.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquations7.EE.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalities7.EE.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities7.EE.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphing7.RP.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2aDecide 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 Variation|Identifying a Table of a Direct Variation7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2dExplain 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 ChangeProbability & Statistics7.SP.1Understand 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 Population7.SP.2Use 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 Population7.SP.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set7.SP.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)7.SP.5Understand 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 Events7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making Predictions7.SP.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making Predictions7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space7.SP.8aUnderstand 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 Events|Identifying the Outcomes in a Sample Space7.SP.8bRepresent 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 Events|Identifying the Outcomes in a Sample Space7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometry7.G.1Solve 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 Figures7.G.2Draw (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 Triangles|Identifying and Classifying Polygons7.G.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of Solids7.G.4Know 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 Circle7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a Figure7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole Numbers8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms8.EE.7bSolve 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 TermsFractions & Decimals8.EE.2Use 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 Numbers8.NS.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & Roots8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)8.EE.2Use 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 Cube|Identifying the Square Root of a Perfect Square8.EE.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific Notation8.NS.2Use 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 RootsRates, Ratios, & Proportions8.EE.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquations8.EE.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems8.EE.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphing8.EE.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct Variation8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.EE.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically8.EE.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically8.F.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different Forms8.F.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different Forms8.F.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct Variation8.F.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.F.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Tables8.SP.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & Statistics8.SP.1Construct 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 Association8.SP.2Know 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 Data8.SP.3Use 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 Data8.SP.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometry8.G.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.2Understand that a two-dimensional 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 Polygon8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations8.G.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar Figures8.G.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a Transversal8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeSouth Carolina Common Core State Standards - 20106Whole Numbers6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions6.EE.2bIdentify 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 Terms|Writing Algebraic Expressions6.EE.2cEvaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).Evaluating Expressions|Evaluating Expressions Using the Order of Operations6.EE.3Apply the properties of operations to generate equivalent expressions.Identifying Properties of Addition and Multiplication|Identifying the Greatest Common Factor (GCF)|Simplifying Expressions by Combining Like Terms6.EE.4Identify 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 Terms6.EE.6Use variables to represent numbers and write expressions when solving a real-world 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 Expressions6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules6.NS.4Find 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 FactorsIntegers6.NS.5Understand 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 real-world contexts, explaining the meaning of 0 in each situation.Identifying Integers|Identifying Opposite Integers and Absolute Value|Modeling Real Life Using Integers6.NS.6Understand 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 Integers|Identifying Opposite Integers and Absolute Value6.NS.6aRecognize 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 Value6.NS.6cFind 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 Integers6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers6.NS.7cUnderstand 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 real-world situation.Identifying Opposite Integers and Absolute Value6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing IntegersFractions & Decimals6.NS.1Interpret 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 Fractions|Identifying Parts of a Fraction|Multiplying Fractions6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Adding Decimal Numbers|Dividing Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal Numbers6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and DecimalsExponents & Roots6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials6.EE.2bIdentify 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 Monomials6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & Proportions6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form6.RP.2Understand 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 Rates6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Ratios6.RP.3bSolve unit rate problems including those involving unit pricing and constant speed.Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Solving Practical Problems Using Ratios6.RP.3cFind 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 Proportion|Solving Proportions to Find Missing TermsEquations6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems6.EE.2bIdentify 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 Variables|Writing One-Step Equations and Solving Word Problems6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables6.EE.5Understand 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 Two-Step Equations to Solve Word Problems|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems6.EE.7Solve real-world 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 One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems6.NS.4Find 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 VariablesInequalities6.EE.5Understand 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 Two-Step Inequalities to Solve Word Problems|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities6.EE.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number LineGraphing6.EE.9Use variables to represent two quantities in a real-world 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 Tables|Identifying Independent and Dependent Variables6.NS.6Understand 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.6bUnderstand 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 Pair|Identifying an Ordered Pair Given a Graphed Point6.NS.6cFind 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts6.NS.8Solve real-world 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-Values6.RP.3Use ratio and rate reasoning to solve real-world 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 Pair|Predicting Values in Tables Using Numerical Patterns6.RP.3aMake tables of equivalent ratios relating quantities with whole-number 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 Pair|Predicting Values in Tables Using Numerical PatternsProbability & Statistics6.SP.1Recognize 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 Population6.SP.2Understand 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 Display6.SP.3Recognize 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.4Display 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 Display6.SP.5Summarize numerical data sets in relation to their context, such as by:Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5aReporting 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 Display6.SP.5bDescribing the nature of the attribute under investigation, including how it was measured and its units of measurement.Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display|Selecting a Representative Sampling Method for a Population6.SP.5cGiving 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set6.SP.5dRelating 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometry6.G.1Find 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 real-world and mathematical problems.Finding Area of Polygons Using Composing and Decomposing Techniques6.G.2Find 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 real-world and mathematical problems.Solving Problems Involving Volume6.G.3Draw 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 real-world and mathematical problems.Solving Problems Involving Polygons on the Coordinate Plane6.G.4Represent three-dimensional 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 real-world and mathematical problems.Investigating Surface Area and Cross-Sections of Solids6.NS.8Solve real-world 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 Plane7Whole Numbers7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms7.EE.2Understand 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 Expressions7.EE.3Solve multi-step real-life 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 Operations7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Evaluating Expressions|Evaluating Expressions Using the Order of Operations7.EE.4aSolve 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 Expressions|Evaluating Expressions Using the Order of Operations7.NS.1Apply 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 Multiplication7.NS.1cUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules7.NS.2aUnderstand 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 real-world contexts.Identifying Properties of Addition and Multiplication7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of OperationsIntegers7.EE.3Solve multi-step real-life 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 Division|Comparing Integers7.NS.1Apply 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 Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value7.NS.1bUnderstand 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 real-world contexts.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value7.NS.1cUnderstand 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 real-world contexts.Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Subtracting Integers7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2aUnderstand 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 real-world contexts.Multiplying and Dividing Integers7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & Decimals7.EE.3Solve multi-step real-life 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 Numbers|Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Dividing Decimal Numbers|Dividing Fractions|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal7.NS.1Apply 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 Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1bUnderstand 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 real-world contexts.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators7.NS.1cUnderstand 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 real-world contexts.Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Converting Fractions to Decimals|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2aUnderstand 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 real-world contexts.Multiplying Decimal Numbers|Multiplying Fractions7.NS.2bUnderstand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world contexts.Dividing Decimal Numbers|Dividing Fractions|Multiplying Fractions|Writing an Integer as a Fraction or Decimal7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions7.NS.2dConvert 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 Decimals|Writing an Integer as a Fraction or Decimal7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike DenominatorsExponents & Roots7.EE.3Solve multi-step real-life 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific NotationRates, Ratios, & Proportions7.EE.3Solve multi-step real-life 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction7.RP.2Recognize and represent proportional relationships between quantities.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms7.RP.2aDecide 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 Quantities7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios7.RP.2cRepresent 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 Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms7.RP.3Use 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 Proportion|Computing Simple Interest|Identifying the Percent of ChangeEquations7.EE.2Understand 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 Two-Step Equations to Solve Word Problems|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.EE.4aSolve 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 Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2aUnderstand 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 real-world contexts.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalities7.EE.2Understand 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 Two-Step Inequalities to Solve Word Problems|Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities7.EE.4bSolve 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 Two-Step Inequalities to Solve Word Problems|Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Solving Word Problems Using Inequalities|Writing Inequalities Using GraphsGraphing7.RP.2Recognize and represent proportional relationships between quantities.Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2aDecide 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 Variation|Identifying a Table of a Direct Variation7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation7.RP.2dExplain 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 ChangeProbability & Statistics7.SP.1Understand 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 Population7.SP.2Use 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 Population7.SP.3Informally 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 Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set7.SP.4Use 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)7.SP.5Understand 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 Events7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Identifying and Comparing Probabilities|Making Predictions7.SP.7Develop 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 Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7aDevelop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions7.SP.7bDevelop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.Identifying and Comparing Probabilities|Making Predictions7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space7.SP.8aUnderstand 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 Events|Identifying the Outcomes in a Sample Space7.SP.8bRepresent 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 Events|Identifying the Outcomes in a Sample Space7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound EventsGeometry7.G.1Solve 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 Figures7.G.2Draw (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 Triangles|Identifying and Classifying Polygons7.G.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Investigating Surface Area and Cross-Sections of Solids7.G.4Know 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 Circle7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Solving Equations for an Unknown Angle in a Figure7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Surface Area|Solving Problems Involving Volume8Whole Numbers8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms8.EE.7bSolve 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 TermsFractions & Decimals8.EE.2Use 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 Numbers8.NS.1Know 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 Numbers|Converting Fractions to Decimals|Defining Rational and Irrational|Writing an Integer as a Fraction or DecimalExponents & Roots8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Evaluating All Powers|Factoring Monomials|Finding the Prime Factorization of a Number|Identifying Bases and Exponents|Identifying Prime and Composite Numbers|Laws of Exponents|Representing Expressions with Only Positive Exponents|Simplifying Monomial Expressions (Division)|Simplifying Monomial Expressions (Multiplication)|Simplifying Monomial Expressions (Powers of Powers)8.EE.2Use 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 Cube|Identifying the Square Root of a Perfect Square8.EE.3Use 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 Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific Notation8.NS.2Use 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 RootsRates, Ratios, & Proportions8.EE.5Graph 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 Quantities|Expressing Unit Rates|Solving Proportions to Find Missing TermsEquations8.EE.7Solve linear equations in one variable.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word Problems8.EE.7bSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides|Writing One-Step Equations and Solving Word ProblemsGraphing8.EE.5Graph 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 Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing an Equation for Direct Variation8.EE.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.EE.8Analyze and solve pairs of simultaneous linear equations.Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8aUnderstand 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically8.EE.8bSolve 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 Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically8.F.1Understand 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 Function|Identifying Domain and Range|Identifying Independent and Dependent Variables|Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Representing Relations and Functions in Different Forms8.F.2Compare 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 Line|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different Forms8.F.3Interpret 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 Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Identifying a Graph of a Direct Variation8.F.4Construct 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 Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.F.5Describe 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 Line|Classifying Functions as Linear or Nonlinear|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Tables8.SP.3Use 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 Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Predicting Values Using a Direct Variation|Writing an Equation for Direct VariationProbability & Statistics8.SP.1Construct 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 Association8.SP.2Know 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 Data8.SP.3Use 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 Data8.SP.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way 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 Two-Way TableGeometry8.G.1Verify experimentally the properties of rotations, reflections, and translations:Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1aLines are taken to lines, and line segments to line segments of the same length.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1bAngles are taken to angles of the same measure.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.1cParallel lines are taken to parallel lines.Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.G.2Understand that a two-dimensional 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 Polygon8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations8.G.4Understand that a two-dimensional 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 two-dimensional figures, describe a sequence that exhibits the similarity between them.Describing the Sequence of Transformations of Two Similar Figures8.G.5Use 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 angle-angle criterion for similarity of triangles.Identifying Angle Relationships Formed by Parallel Lines and a Transversal8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving VolumeTexas Essential Knowledge and Skills for Mathematics - 20126Whole Numbers111.26.b.3The 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 Operations111.26.b.3.EMultiply and divide positive rational numbers fluently.Evaluating Expressions Using the Order of Operations111.26.b.4The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality)Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Writing Algebraic Expressions111.26.b.4.ACompare 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 Operations|Identifying Properties of Addition and Multiplication|Writing Algebraic Expressions111.26.b.7The student applies mathematical process standards to develop concepts of expressions and equations. (Expressions, equations, and relationships)Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions111.26.b.7.AGenerate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization;Evaluating Expressions Using the Order of Operations111.26.b.7.BDistinguish between expressions and equations verbally, numerically, and algebraically;Evaluating Expressions|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions111.26.b.7.DGenerate equivalent expressions using the properties of operations : such as the inverse, identity, commutative, associative, and distributive properties.Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like TermsIntegers111.26.b.2The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations)Comparing Integers|Graphing Integers|Identifying Integers111.26.b.2.BIdentify a number, its opposite, and its absolute value;Identifying Opposite Integers and Absolute Value111.26.b.2.CLocate, compare, and order integers and rational numbersComparing Integers|Graphing Integers|Identifying Integers111.26.b.3The 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 Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Modeling Real Life Using Integers|Multiplying and Dividing Integers|Subtracting Integers111.26.b.3.CRepresent integer operations with concrete models and connect the actions with the models to standardized algorithms;Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Modeling Real Life Using Integers|Multiplying and Dividing Integers111.26.b.3.DAdd, subtract, multiply, and divide integers fluently;Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting Integers111.26.b.3.EMultiply and divide positive rational numbers fluently.Combining Integer Multiplication and Division|Multiplying and Dividing Integers111.26.b.9The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships)Comparing Integers|Graphing Integers111.26.b.9.BRepresent solutions for one-variable, one-step equations and inequalities on number lines;Comparing Integers|Graphing IntegersFractions & Decimals111.26.b.2The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations)Classifying Numbers|Comparing Fractions and Decimals|Defining Rational and Irrational|Identifying Parts of a Fraction111.26.b.2.AClassify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers;Classifying Numbers|Defining Rational and Irrational111.26.b.2.CLocate, compare, and order integers and rational numbersComparing Fractions and Decimals111.26.b.2.EExtend 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 Fraction111.26.b.3The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. (Number and operations)Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators111.26.b.3.ARecognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values;Dividing Fractions111.26.b.3.EMultiply and divide positive rational numbers fluently.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions111.26.b.4The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality)Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal111.26.b.4.CGive examples of ratios as multiplicative comparisons of two quantities describing the same attribute;Writing Fractions in Simplest Form111.26.b.4.FRepresent 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 Decimals111.26.b.4.GGenerate equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money;Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal111.26.b.5The student applies mathematical process standards to solve problems involving proportional relationships. (Proportionality)Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal111.26.b.5.CUse equivalent fractions, decimals, and percents to show equal parts of the same whole.Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or DecimalExponents & Roots111.26.b.2The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations)Comparing Numbers in Scientific Notation111.26.b.2.CLocate, compare, and order integers and rational numbersComparing Numbers in Scientific Notation111.26.b.7The student applies mathematical process standards to develop concepts of expressions and equations. (Expressions, equations, and relationships)Evaluating All Powers|Finding the Prime Factorization of a Number|Identifying Prime and Composite Numbers111.26.b.7.AGenerate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization;Evaluating All Powers|Finding the Prime Factorization of a Number|Identifying Prime and Composite NumbersRates, Ratios, & Proportions111.26.b.2The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations)Comparing Fractions, Decimals, Percents, and Ratios|Expressing Ratios in Simplified Form111.26.b.2.CLocate, compare, and order integers and rational numbersComparing Fractions, Decimals, Percents, and Ratios111.26.b.2.DOrder a set of rational numbers arising from mathematical and real-world contexts;Comparing Fractions, Decimals, Percents, and Ratios111.26.b.2.EExtend 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 Form111.26.b.4The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality)Comparing Fractions, Decimals, Percents, and Ratios|Expressing Ratios in Simplified Form|Expressing Unit Rates|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios111.26.b.4.BApply qualitative and quantitative reasoning to solve prediction and comparison of real-world problems involving ratios and rates;Expressing Ratios in Simplified Form|Expressing Unit Rates|Solving Practical Problems Using Ratios111.26.b.4.CGive examples of ratios as multiplicative comparisons of two quantities describing the same attribute;Expressing Ratios in Simplified Form111.26.b.4.DGive examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients;Expressing Ratios in Simplified Form|Expressing Unit Rates|Solving Practical Problems Using Ratios111.26.b.4.ERepresent ratios and percents with concrete models, fractions, and decimals;Expressing Ratios in Simplified Form|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction111.26.b.4.FRepresent 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 Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction111.26.b.4.GGenerate equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money;Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction111.26.b.4.HConvert units within a measurement system, including the use of proportions and unit rates.Identify Actual Measurements and Scale Factors111.26.b.5The student applies mathematical process standards to solve problems involving proportional relationships. (Proportionality)Applying the Percent Proportion|Computing Simple Interest|Determining Proportional Relationships Between Quantities|Expressing Ratios in Simplified Form|Expressing Unit Rates|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction|Identify Actual Measurements and Scale Factors|Identifying the Percent of Change|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms111.26.b.5.ARepresent mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions;Determining Proportional Relationships Between Quantities|Expressing Ratios in Simplified Form|Expressing Unit Rates|Identify Actual Measurements and Scale Factors|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms111.26.b.5.BSolve real-world 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 Proportion|Computing Simple Interest|Identifying the Percent of Change111.26.b.5.CUse equivalent fractions, decimals, and percents to show equal parts of the same whole.Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a FractionEquations111.26.b.10The student applies mathematical process standards to use equations and inequalities to solve problems. (Expressions, equations, and relationships)Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems111.26.b.10.AModel and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts;Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems111.26.b.10.BDetermine if the given value(s) make(s) one-variable, one-step equations or inequalities true.Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems111.26.b.4The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality)Applying Knowledge of Two-Step Equations to Solve Word Problems|Writing One-Step Equations and Solving Word Problems111.26.b.4.ACompare 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 Two-Step Equations to Solve Word Problems|Writing One-Step Equations and Solving Word Problems111.26.b.6The student applies mathematical process standards to use multiple representations to describe algebraic relationships. (Expressions, equations, and relationships)Writing One-Step Equations and Solving Word Problems111.26.b.6.CRepresent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.Writing One-Step Equations and Solving Word Problems111.26.b.7The student applies mathematical process standards to develop concepts of expressions and equations. (Expressions, equations, and relationships)Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems111.26.b.7.BDistinguish between expressions and equations verbally, numerically, and algebraically;Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Write Equivalent Expressions with Variables|Solving One-Step Equations Involving Addition or Subtraction|Writing One-Step Equations and Solving Word Problems111.26.b.7.CDetermine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations;Applying the Distributive Property to Solve Equations|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Writing One-Step Equations and Solving Word Problems111.26.b.7.DGenerate 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 Variables111.26.b.9The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships)Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems111.26.b.9.AWrite one-variable, one-step equations and inequalities to represent constraints or conditions within problems;Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word Problems111.26.b.9.CWrite corresponding real-world problems given one-variable, one-step equations or inequalities.Writing One-Step Equations and Solving Word ProblemsInequalities111.26.b.10The student applies mathematical process standards to use equations and inequalities to solve problems. (Expressions, equations, and relationships)Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Word Problems Using Inequalities111.26.b.10.AModel and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts;Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Word Problems Using Inequalities111.26.b.10.BDetermine if the given value(s) make(s) one-variable, one-step equations or inequalities true.Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division111.26.b.9The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships)Graphing Inequalities on a Number Line|Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities111.26.b.9.AWrite one-variable, one-step equations and inequalities to represent constraints or conditions within problems;Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Word Problems Using Inequalities|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities111.26.b.9.BRepresent solutions for one-variable, one-step equations and inequalities on number lines;Graphing Inequalities on a Number Line|Writing Inequalities Using Graphs111.26.b.9.CWrite corresponding real-world problems given one-variable, one-step equations or inequalities.Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using InequalitiesGraphing111.26.b.11The 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 Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts111.26.b.6The 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 Tables|Identifying Domain and Range|Identifying Independent and Dependent Variables|Representing Relations and Functions in Different Forms111.26.b.6.AIdentify independent and dependent quantities from tables and graphs;Identifying Domain and Range|Identifying Independent and Dependent Variables111.26.b.6.CRepresent 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 Tables|Representing Relations and Functions in Different FormsProbability & Statistics111.26.b.12The student applies mathematical process standards to use numerical or graphical representations to analyze problems. (Measurement and data)Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display111.26.b.12.ARepresent numeric data graphically, including dot plots, stem-and-leaf plots, histograms, and box plots;Representing a Set of Data Using a Data Display111.26.b.12.BUse the graphical representation of numeric data to describe the center, spread, and shape of the data distribution;Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)111.26.b.12.CSummarize 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 Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Describing a Set of Data (Mean, Median, Mode, Range)111.26.b.12.DSummarize 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 Sets|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 Display111.26.b.13The student applies mathematical process standards to use numerical or graphical representations to solve problems. (Measurement and data)Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Representing a Set of Data Using a Data Display111.26.b.13.AInterpret numeric data summarized in dot plots, stem-and-leaf plots, histograms, and box plots;Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Representing a Set of Data Using a Data Display111.26.b.13.BDistinguish between situations that yield data with and without variability.Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data SetGeometry111.26.b.10The student applies mathematical process standards to use equations and inequalities to solve problems. (Expressions, equations, and relationships)Analyzing Triangles|Finding the Missing Dimension of a Rectangle or Triangle|Identifying Angle Relationships Formed by Parallel Lines and a Transversal|Solving Equations for an Unknown Angle in a Figure|Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Surface Area|Solving Problems Involving Volume111.26.b.10.AModel and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts;Analyzing Triangles|Finding the Missing Dimension of a Rectangle or Triangle|Identifying Angle Relationships Formed by Parallel Lines and a Transversal|Solving Equations for an Unknown Angle in a Figure|Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Surface Area|Solving Problems Involving Volume111.26.b.8The student applies mathematical process standards to use geometry to represent relationships and solve problems. (Expressions, equations, and relationships)Analyzing Triangles|Classifying Triangles Using Angles and Sides|Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Identifying and Classifying Polygons|Solving Equations for an Unknown Angle in a Figure|Solving Problems Involving Polygons on the Coordinate Plane111.26.b.8.AExtend 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 Triangles|Classifying Triangles Using Angles and Sides|Identifying and Classifying Polygons|Solving Equations for an Unknown Angle in a Figure111.26.b.8.BModel area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes;Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Polygons on the Coordinate Plane111.26.b.8.CWrite 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 Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Polygons on the Coordinate Plane111.26.b.8.DDetermine 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 Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Polygons on the Coordinate Plane7Whole Numbers111.27.b.3The 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 Operations111.27.b.3.AAdd, subtract, multiply, and divide rational numbers fluently;Evaluating Expressions Using the Order of Operations111.27.b.3.BApply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.Evaluating Expressions Using the Order of Operations111.27.b.4The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)Evaluating Expressions111.27.b.4.ARepresent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt;Evaluating ExpressionsIntegers111.27.b.3The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. (Number and operations) Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting Integers111.27.b.3.AAdd, subtract, multiply, and divide rational numbers fluently;Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting Integers111.27.b.3.BApply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and AdditionFractions & Decimals111.27.b.2The 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 Numbers|Defining Rational and Irrational111.27.b.3The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. (Number and operations) Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators111.27.b.3.AAdd, subtract, multiply, and divide rational numbers fluently;Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators111.27.b.3.BApply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.Adding Like Denominators|Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions|Subtracting Decimal Numbers111.27.b.4The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)Converting Fractions to Decimals111.27.b.4.DSolve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems;Converting Fractions to DecimalsRates, Ratios, & Proportions111.27.b.13.ACalculate the sales tax for a given purchase and calculate income tax for earned wages ;Applying the Percent Proportion111.27.b.13.ECalculate and compare simple interest and compound interest earnings;Computing Simple Interest111.27.b.4The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Computing Simple Interest|Determining Proportional Relationships Between Quantities|Expressing Ratios in Simplified Form|Expressing Unit Rates|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction|Identify Actual Measurements and Scale Factors|Identifying the Percent of Change|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms111.27.b.4.BCalculate unit rates from rates in mathematical and real-world problems;Expressing Unit Rates|Solving Practical Problems Using Ratios111.27.b.4.CDetermine the constant of proportionality (k = y/x) within mathematical and real-world problems;Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms111.27.b.4.DSolve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems;Applying the Percent Proportion|Comparing Fractions, Decimals, Percents, and Ratios|Computing Simple Interest|Expressing Ratios in Simplified Form|Expressing Unit Rates|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction|Identifying the Percent of Change|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms111.27.b.5The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. (Proportionality)Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors111.27.b.5.AGeneralize the critical attributes of similarity, including ratios within and between similar shapes;Determining Proportional Relationships Between Quantities|Identify Actual Measurements and Scale Factors111.27.b.5.CSolve mathematical and real-world problems involving similar shape and scale drawings.Identify Actual Measurements and Scale FactorsEquations111.27.b.10.AWrite one-variable, two-step equations and inequalities to represent constraints or conditions within problems;Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations111.27.b.10.CWrite a corresponding real-world problem given a one-variable, two-step equation or inequality.Applying Knowledge of Two-Step Equations to Solve Word Problems111.27.b.11.AModel and solve one-variable, two-step equations and inequalities;Applying Knowledge of Two-Step Equations to Solve Word Problems|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction111.27.b.11.BDetermine if the given value(s) make(s) one-variable, two-step equations and inequalities true;Applying Knowledge of Two-Step Equations to Solve Word Problems|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or SubtractionInequalities111.27.b.10.AWrite one-variable, two-step equations and inequalities to represent constraints or conditions within problems;Applying Knowledge of Two-Step Inequalities to Solve Word Problems111.27.b.10.BRepresent solutions for one-variable, two-step equations and inequalities on number lines;Graphing Inequalities on a Number Line|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction|Writing Inequalities Using Graphs111.27.b.10.CWrite a corresponding real-world problem given a one-variable, two-step equation or inequality.Applying Knowledge of Two-Step Inequalities to Solve Word Problems111.27.b.11.AModel and solve one-variable, two-step equations and inequalities;Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or Subtraction111.27.b.11.BDetermine if the given value(s) make(s) one-variable, two-step equations and inequalities true;Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or SubtractionGraphing111.27.b.4The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Finding the Rate of Change|Graphing a Linear Function in Two Variables Using Tables|Identifying a Table of a Direct Variation|Predicting Values Using a Direct Variation|Predicting Values in Tables Using Numerical Patterns111.27.b.4.ARepresent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt;Determining Whether Functions are Linear or Nonlinear Using Patterns|Finding the Rate of Change|Graphing a Linear Function in Two Variables Using Tables|Predicting Values in Tables Using Numerical Patterns111.27.b.4.CDetermine the constant of proportionality (k = y/x) within mathematical and real-world problems;Determining the Constant of Variation|Identifying a Table of a Direct Variation|Predicting Values Using a Direct Variation111.27.b.7The 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 Line|Classifying Functions as Linear or Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying Independent and Dependent Variables|Identifying the Slope of a Line Using Two Points|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)Probability & Statistics111.27.b.12.ACompare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads;Analyzing and Comparing Data Sets|Analyzing and Describing the Distribution of a Data Set (Data Display)|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set111.27.b.12.BUse data from a random sample to make inferences about a population;Drawing Inferences About a Population|Selecting a Representative Sampling Method for a Population111.27.b.12.CCompare 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 Sets111.27.b.6The student applies mathematical process standards to use probability and statistics to describe or solve problems involving proportional relationships. (Proportionality) Finding the Probabilities of Compound Events|Identifying and Comparing Probabilities|Identifying the Outcomes in a Sample Space|Identifying the Theoretical Probability of an Event|Making Predictions111.27.b.6.ARepresent sample spaces for simple and compound events using lists and tree diagrams;Identifying the Outcomes in a Sample Space|Identifying the Theoretical Probability of an Event111.27.b.6.BSelect and use different simulations to represent simple and compound events with and without technologyFinding the Probabilities of Compound Events|Identifying and Comparing Probabilities|Identifying the Outcomes in a Sample Space|Identifying the Theoretical Probability of an Event|Making Predictions111.27.b.6.CMake predictions and determine solutions using experimental data for simple and compound events;Finding the Probabilities of Compound Events|Identifying and Comparing Probabilities|Making Predictions111.27.b.6.DMake predictions and determine solutions using theoretical probability for simple and compound events;Finding the Probabilities of Compound Events|Identifying and Comparing Probabilities|Identifying the Outcomes in a Sample Space|Identifying the Theoretical Probability of an Event|Making Predictions111.27.b.6.EFind the probabilities of a simple event and its complement and describe the relationship between the two;Identifying the Theoretical Probability of an Event|Making Predictions111.27.b.6.FUse data from a random sample to make inferences about a population;Drawing Inferences About a Population|Selecting a Representative Sampling Method for a Population111.27.b.6.GSolve problems using data represented in bar graphs, dot plots, and circle graphs, including part-to-whole and part-to-part comparisons and equivalents;Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display111.27.b.6.HSolve problems using qualitative and quantitative predictions and comparisons from simple experiments;Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions111.27.b.6.IDetermine experimental and theoretical probabilities related to simple and compound events using data and sample spaces.Finding the Probabilities of Compound Events|Identifying and Comparing Probabilities|Identifying the Outcomes in a Sample Space|Identifying the Theoretical Probability of an Event|Making PredictionsGeometry111.27.b.11.CWrite and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.Analyzing Triangles|Identifying Angle Relationships Formed by Parallel Lines and a Transversal|Solving Equations for an Unknown Angle in a Figure111.27.b.5The 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 Figures|Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations111.27.b.5.AGeneralize the critical attributes of similarity, including ratios within and between similar shapes;Describing the Sequence of Transformations of Two Similar Figures|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations111.27.b.5.BDescribe pi () as the ratio of the circumference of a circle to its diameter;Solving Problems Involving Area and Circumference of a Circle111.27.b.5.CSolve mathematical and real-world problems involving similar shape and scale drawings.Describing the Sequence of Transformations of Two Similar Figures|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations111.27.b.8Use 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 Circle111.27.b.8.AThe student applies mathematical process standards to develop geometric relationships with volume. (Expressions, equations, and relationships)Solving Problems Involving Volume111.27.b.8.BModel 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 Volume111.27.b.8.CExplain 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 Volume111.27.b.9.ASolve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids;Solving Problems Involving Volume111.27.b.9.BDetermine the circumference and area of circles;Solving Problems Involving Area and Circumference of a Circle111.27.b.9.CDetermine 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 Techniques|Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Polygons on the Coordinate Plane111.27.b.9.DSolve 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 Cross-Sections of Solids|Solving Problems Involving Surface Area8Integers111.28.b.2The student applies mathematical process standards to represent and use real numbers in a variety of forms.Comparing Integers111.28.b.2.DOrder a set of real numbers arising from mathematical and real-world contexts.Comparing IntegersFractions & Decimals111.28.b.2The student applies mathematical process standards to represent and use real numbers in a variety of forms.Classifying Numbers|Comparing Fractions and Decimals|Defining Rational and Irrational111.28.b.2.AExtend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers;Classifying Numbers|Defining Rational and Irrational111.28.b.2.BApproximate 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 Irrational111.28.b.2.DOrder a set of real numbers arising from mathematical and real-world contexts.Comparing Fractions and DecimalsExponents & Roots111.28.b.2The student applies mathematical process standards to represent and use real numbers in a variety of forms.Comparing Numbers in Scientific Notation|Converting from Scientific Notation to Standard Form|Estimating Square Roots|Identifying the Square Root of a Perfect Square|Writing in Scientific Notation111.28.b.2.BApproximate 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 Roots|Identifying the Square Root of a Perfect Square111.28.b.2.CConvert between standard decimal  notation and scientific notationConverting from Scientific Notation to Standard Form|Writing in Scientific Notation111.28.b.2.DOrder a set of real numbers arising from mathematical and real-world contexts.Comparing Numbers in Scientific NotationRates, Ratios, & Proportions111.28.b.12The 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 Interest111.28.b.12.ASolve real-world problems comparing how interest rate and loan length affect the cost of credit;Computing Simple Interest111.28.b.12.DCalculate and compare simple interest and compound interest earnings;Computing Simple Interest111.28.b.2The student applies mathematical process standards to represent and use real numbers in a variety of forms.Comparing Fractions, Decimals, Percents, and Ratios111.28.b.2.DOrder a set of real numbers arising from mathematical and real-world contexts.Comparing Fractions, Decimals, Percents, and RatiosEquations111.28.b.8The student applies mathematical process standards to use one-variable equations or inequalities in problem situations.  (Expressions, equations, and relationships) Applying the Distributive Property to Solve Equations|Solving an Equation with Variables on Two Sides111.28.b.8.CModel and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants;Applying the Distributive Property to Solve Equations|Solving an Equation with Variables on Two SidesGraphing111.28.b.4The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope.  (Proportionality)  Calculating the Slope of a Graphed Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining the Constant of Variation|Finding the Rate of Change|Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying a Graph of a Direct Variation|Identifying the Slope of a Line Using Two Points|Identifying the x- and y-Intercepts|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)111.28.b.4.AUse similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y-values to the change in x-values, (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 Line|Identifying the Slope of a Line Using Two Points111.28.b.4.BGraph proportional relationships, interpreting the unit rate as the slope of the line that models the relationshipCalculating the Slope of a Graphed Line|Determining the Constant of Variation|Finding the Rate of Change|Identifying a Graph of a Direct Variation111.28.b.4.CUse data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.Calculating the Slope of a Graphed Line|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Finding the Rate of Change|Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying the Slope of a Line Using Two Points|Identifying the x- and y-Intercepts|Predicting Values in Tables Using Numerical Patterns|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)111.28.b.5The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions. (Proportionality)Calculating the Slope of a Graphed Line|Classifying Functions as Linear or Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining if a Relation is a Function|Determining the Constant of Variation|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying Domain and Range|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation|Predicting Values Using a Direct Variation|Representing Relations and Functions in Different Forms|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation111.28.b.5.ARepresent linear proportional situations with tables, graphs, and equations in the form of y = kx;Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation|Predicting Values Using a Direct Variation|Writing an Equation for Direct Variation111.28.b.5.BRepresent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0;Calculating the Slope of a Graphed Line|Classifying Functions as Linear or Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)111.28.b.5.ESolve problems involving direct variation;Determining the Constant of Variation|Predicting Values Using a Direct Variation|Writing an Equation for Direct Variation111.28.b.5.FDistinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0;Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation111.28.b.5.GIdentify functions using sets of ordered pairs, tables, mappings, and graphs;Determining if a Relation is a Function|Identifying Domain and Range|Representing Relations and Functions in Different Forms111.28.b.5.HIdentify examples of proportional and non-proportional functions that arise from mathematical and real-world problemsClassifying Functions as Linear or Nonlinear|Determining if a Relation is a Function|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation111.28.b.5.IWrite 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 y-Intercept of Linear Equations in Slope-Intercept Form|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)111.28.b.9The 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 System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations GraphicallyProbability & Statistics111.28.b.11The 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 Set|Interpreting Scatter Plots to Investigate Patterns of Association|Selecting a Representative Sampling Method for a Population111.28.b.11.AConstruct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data;Interpreting Scatter Plots to Investigate Patterns of Association111.28.b.11.BDetermine 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 Set111.28.b.11.CSimulate 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 Population111.28.b.5The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions. (Proportionality)Interpreting Scatter Plots to Investigate Patterns of Association|Solving Problems Involving Bivariate Measurement Data111.28.b.5.CContrast 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 Association|Solving Problems Involving Bivariate Measurement Data111.28.b.5.DUse a trend line that approximates the linear relationship between bivariate sets of data to make predictions;Solving Problems Involving Bivariate Measurement DataGeometry111.28.b.10The student applies mathematical process standards to develop transformational geometry concepts. (Two-dimensional shapes) Describing the Sequence of Transformations of Two Similar Figures|Identifying Properties of Rotations, Reflections, and Translations|Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Polygons on the Coordinate Plane|Solving Problems Involving Scale Drawings of Geometric Figures|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations111.28.b.10.AGeneralize the properties of orientation and congruence of rotations, reflections, translations, and dilations of two-dimensional shapes on a coordinate plane;Describing the Sequence of Transformations of Two Similar Figures|Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations111.28.b.10.BDifferentiate between transformations that preserve congruence and those that do not;Describing the Sequence of Transformations of Two Similar Figures|Identifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations111.28.b.10.CExplain the effect of translations, reflections over the x- or y-axis, and rotations limited to 90, 180, 270, and 360 as applied to two-dimensional shapes on a coordinate plane using an algebraic representation;Identifying Properties of Rotations, Reflections, and Translations|Using the Coordinate Plane to Demonstrate Transformations111.28.b.10.DModel the effect on linear and area measurements of dilated two-dimensional shapes.Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Polygons on the Coordinate Plane|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations111.28.b.3The student applies mathematical process standards to use proportional relationships to describe dilations. (Proportionality) Describing the Sequence of Transformations of Two Similar Figures|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations111.28.b.3.AGeneralize 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 Figures|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations111.28.b.3.BCompare and contrast the attributes of a shape and its dilation(s) on a coordinate planeDescribing the Sequence of Transformations of Two Similar Figures|Using the Coordinate Plane to Demonstrate Transformations111.28.b.3.CUse an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation.Describing the Sequence of Transformations of Two Similar Figures|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations111.28.b.6The 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 Solids|Finding Distance Between Two Points on the Coordinate Plane|Solving Problems Involving Volume111.28.b.6.ADescribe the volume formula V = Bh of a cylinder in terms of its base area and its height;Solving Problems Involving Volume111.28.b.6.CUse models and diagrams to explain the Pythagorean theorem.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane111.28.b.7The 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 Solids|Finding Distance Between Two Points on the Coordinate Plane|Investigating Surface Area and Cross-Sections of Solids|Solving Problems Involving Surface Area|Solving Problems Involving Volume111.28.b.7.ASolve problems involving the volume of cylinders, cones, and spheres;Solving Problems Involving Volume111.28.b.7.BUse 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 Cross-Sections of Solids|Solving Problems Involving Surface Area111.28.b.7.CUse the Pythagorean Theorem and its converse to solve problemsDetermining Unknown Side Lengths in Right Triangles of Polygons and Solids111.28.b.7.DDetermine the distance between two points on a coordinate plane using the Pythagorean Theorem.Finding Distance Between Two Points on the Coordinate Plane111.28.b.8The student applies mathematical process standards to use one-variable equations or inequalities in problem situations.  (Expressions, equations, and relationships) Analyzing Triangles|Identifying Angle Relationships Formed by Parallel Lines and a Transversal111.28.b.8.DUse 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 angle-angle criterion for similarity of triangles. andAnalyzing Triangles|Identifying Angle Relationships Formed by Parallel Lines and a TransversalVirginia Mathematics Standards of Learning - 20096Whole Numbers6.18Solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.Simplifying Expressions by Combining Like Terms6.19aInvestigate and recognize the identity properties for addition and multiplication; Identifying Properties of Addition and Multiplication6.19bInvestigate and recognize the multiplicative property of zero; Identifying Properties of Addition and Multiplication6.8Evaluate whole number numerical expressions, using the order of operations.Evaluating Expressions|Evaluating Expressions Using the Order of OperationsIntegers6.2dCompare and order fractions, decimals, and percents.Comparing Integers6.3aIdentify and represent integers;Identifying Integers|Modeling Real Life Using Integers6.3bOrder and compare integers;Comparing Integers|Graphing Integers6.3cIdentify and describe absolute value of integers.Graphing Integers|Identifying Opposite Integers and Absolute ValueFractions & Decimals6.19cInvestigate and recognize the inverse property for multiplication.Dividing Fractions6.2bIdentify a given fraction, decimal or percent from a representation;Adding Like Denominators|Adding Unlike Denominators|Converting Mixed Numbers and Improper Fractions|Defining Rational and Irrational|Identifying Place Value and Rounding Decimal Numbers|Writing Fractions in Simplest Form6.2cDemonstrate equivalent relationships among fractions, decimals, and percents;Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions6.2dCompare and order fractions, decimals, and percents.Comparing Fractions and Decimals6.4Demonstrate multiple representations of multiplication and division of fractions.Dividing Fractions|Multiplying Fractions6.6aMultiply and divide fractions and mixed numbers;Converting Mixed Numbers and Improper Fractions|Dividing Fractions|Multiplying Fractions|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal6.6bEstimate solutions and then solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.Adding Like Denominators|Adding Unlike Denominators|Converting Mixed Numbers and Improper Fractions|Dividing Fractions|Multiplying Fractions|Subtracting Like Denominators|Subtracting Unlike Denominators|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal6.7Solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimalsAdding Decimal Numbers|Dividing Decimal Numbers|Identifying Place Value and Rounding Decimal Numbers|Multiplying Decimal Numbers|Subtracting Decimal Numbers|Writing an Integer as a Fraction or DecimalExponents & Roots6.2cDemonstrate equivalent relationships among fractions, decimals, and percents;Converting from Scientific Notation to Standard Form6.2dCompare and order fractions, decimals, and percents.Comparing Numbers in Scientific Notation6.5Investigate and describe concepts of positive exponents and perfect squares.Evaluating All Powers|Identifying Bases and Exponents|Identifying the Square Root of a Perfect Square6.8Evaluate whole number numerical expressions, using the order of operations.Evaluating All PowersRates, Ratios, & Proportions6.1Describe and compare data, using ratios, and use appropriate notations, such as  a/b , a to b, and a:b.Expressing Ratios in Simplified Form|Expressing Unit Rates|Solving Practical Problems Using Ratios6.2aInvestigate and describe fractions, decimals and percents as ratios;Expressing Ratios in Simplified Form6.2bIdentify a given fraction, decimal or percent from a representation;Expressing a Fraction as a Percent6.2cDemonstrate equivalent relationships among fractions, decimals, and percents;Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction6.2dCompare and order fractions, decimals, and percents.Comparing Fractions, Decimals, Percents, and RatiosEquations6.18Solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Writing One-Step Equations and Solving Word ProblemsInequalities6.20Graph inequalities on a number line.Graphing Inequalities on a Number Line|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using InequalitiesGraphing6.11aIdentify the coordinates of a point in a coordinate plane; Identifying an Ordered Pair Given a Graphed Point6.11bGraph ordered pairs in a coordinate plane. Identifying Quadrants and Graphing an Ordered Pair|Identifying the Distance Between Two Points with Identical x- or y-Values6.12Determine congruence of segments, angles, and polygons.Identifying the Distance Between Two Points with Identical x- or y-Values6.17Identify and extend geometric and arithmetic sequences.Predicting Values in Tables Using Numerical PatternsProbability & Statistics6.14aConstruct circle graphs; Representing a Set of Data Using a Data Display6.14bDraw conclusions and make predictions, using circle graphs;Analyzing and Describing the Distribution of a Data Set (Data Display)6.14cCompare 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 Display6.15aDescribe mean as balance point;Describing a Set of Data (Mean, Median, Mode, Range)6.15bDecide which measure of center is appropriate for a given purpose. Analyzing and Comparing Data Sets6.16aCompare and contrast dependent and independent events;Identifying the Outcomes in a Sample Space6.16bDetermine probabilities for dependent and independent events. Determining the Likelihood of Events|Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space|Identifying the Theoretical Probability of an Event6.2bIdentify a given fraction, decimal or percent from a representation;Representing a Set of Data Using a Data DisplayGeometry6.10aDefine pi (π) as the ratio of the circumference of a circle to its diameter;Solving Problems Involving Area and Circumference of a Circle6.10bSolve practical problems involving circumference and area of a circle, given the diameter or radius; Solving Problems Involving Area and Circumference of a Circle6.10cSolve practical problems involving area and perimeter; Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Polygons on the Coordinate Plane6.10dDescribe and determine the volume and surface area of a rectangular prism.Investigating Surface Area and Cross-Sections of Solids|Solving Problems Involving Surface Area|Solving Problems Involving Volume6.12Determine congruence of segments, angles, and polygons.Analyzing Triangles|Identifying and Classifying Polygons|Using Congruency Statements to Identify Corresponding Parts of a Polygon6.13Describe and identify properties of quadrilaterals.Identifying Polygons|Identifying Quadrilaterals|Identifying and Classifying Polygons7Whole Numbers7.13aWrite verbal expressions as algebraic expressions and sentences as equations and vice versa;Writing Algebraic Expressions7.13bEvaluate algebraic expressions for given replacement values of the variables. Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms7.16aApply the following properties of operations with real numbers: the commutative and associative properties for addition and multiplication;Identifying Properties of Addition and Multiplication7.16cApply the following properties of operations with real numbers: the additive and multiplicative identity properties;Identifying Properties of Addition and Multiplication7.16eApply the following properties of operations with real numbers: the multiplicative property of zero.Identifying Properties of Addition and MultiplicationIntegers7.16dApply the following properties of operations with real numbers: the additive and multiplicative inverse properties;Adding Integers with Unlike Signs7.1eIdentify and describe absolute value for rational numbers.Comparing Integers|Identifying Opposite Integers and Absolute Value7.3aModel addition, subtraction, multiplication and division of integers;Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Identifying Integers|Modeling Real Life Using Integers|Multiplying and Dividing Integers|Subtracting Integers7.3bAdd, subtract, multiply, and divide integers.Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Multiplication and Division|Combining Integer Subtraction and Addition|Multiplying and Dividing Integers|Subtracting IntegersFractions & Decimals7.16dApply the following properties of operations with real numbers: the additive and multiplicative inverse properties;Dividing FractionsExponents & Roots7.1aInvestigate and describe the concept of negative exponents for powers of ten;Representing Expressions with Only Positive Exponents7.1bDetermine scientific notation for numbers greater than zero;Converting from Scientific Notation to Standard Form|Writing in Scientific Notation7.1cCompare and order fractions, decimals, percents and numbers written in scientific notation;Comparing Numbers in Scientific Notation7.1dDetermine square roots;Identifying the Square Root of a Perfect SquareRates, Ratios, & Proportions7.4Solve single-step and multistep practical problems, using proportional reasoning.Applying the Percent Proportion|Determining Proportional Relationships Between Quantities|Expressing Unit Rates|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing TermsEquations7.13aWrite verbal expressions as algebraic expressions and sentences as equations and vice versa;Writing One-Step Equations and Solving Word Problems7.14aSolve one- and two-step linear equations in one variable; Applying the Distributive Property to Solve Equations|Solving One-Step Equations Involving Addition or Subtraction|Solving One-Step Equations Involving Multiplication or Division|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides7.14bSolve practical problems requiring the solution of one- and two-step linear equations.Applying Knowledge of Two-Step Equations to Solve Word Problems|Writing One-Step Equations and Solving Word Problems7.16bApply the following properties of operations with real numbers: the distributive property;Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with VariablesInequalities7.15aSolve one-step inequalities in one variable; Solving One-Step Inequalities Involving Addition or Subtraction|Solving One-Step Inequalities Involving Multiplication or Division|Solving Word Problems Using Inequalities7.15bGraph solutions to inequalities on the number line. Graphing Inequalities on a Number Line|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using InequalitiesGraphing7.12Represent relationships with tables, graphs, rules, and words.Determining if a Relation is a Function|Identifying Domain and Range|Representing Relations and Functions in Different Forms7.2Describe and represent arithmetic and geometric sequences using variable expressions. Predicting Values in Tables Using Numerical PatternsProbability & Statistics7.10Determine the probability of compound events, using the Fundamental (Basic) Counting Principle. Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space7.11aGiven 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 Display7.11bCompare 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 Display7.9Investigate and describe the difference between the experimental probability and theoretical probability of an event.Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making PredictionsGeometry7.5aDescribe volume and surface area of cylinders;Solving Problems Involving Volume7.5bSolve practical problems involving the volume and surface area of rectangular prisms and cylinders; Investigating Surface Area and Cross-Sections of Solids|Solving Problems Involving Surface Area|Solving Problems Involving Volume7.5cDescribe how changing one measured attribute of a rectangular prism affects its volume and surface area.Solving Problems Involving Surface Area|Solving Problems Involving Volume7.6Determine 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 Figures|Solving Problems Involving Scale Drawings of Geometric Figures7.7Compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid.Identifying Polygons|Identifying Quadrilaterals|Identifying and Classifying Polygons7.8Given 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 Figures|Identifying Properties of Rotations, Reflections, and Translations|Solving Problems Involving Scale Drawings of Geometric Figures|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8Whole Numbers8.15aSolve multistep linear equations in one variable on one and two sides of the equation;Simplifying Expressions by Combining Like Terms8.15bSolve two-step linear inequalities and graph the results on a number line;Simplifying Expressions by Combining Like Terms8.15cIdentify properties of operations used to solve an equation.Identifying Properties of Addition and Multiplication8.1aSimplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions8.4Apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Evaluating Expressions|Evaluating Expressions Using the Order of OperationsIntegers8.15cIdentify properties of operations used to solve an equation.Adding Integers with Unlike Signs8.1bCompare and order decimals, fractions, percents, and numbers written in scientific notation.Comparing Integers8.2Describe orally and in writing the relationships between the subsets of the real number system.Identifying IntegersFractions & Decimals8.15cIdentify properties of operations used to solve an equation.Dividing Fractions8.1aSimplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;Classifying Numbers|Defining Rational and Irrational8.1bCompare and order decimals, fractions, percents, and numbers written in scientific notation.Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions8.2Describe orally and in writing the relationships between the subsets of the real number system.Classifying Numbers|Defining Rational and IrrationalExponents & Roots8.1aSimplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;Evaluating All Powers8.1bCompare and order decimals, fractions, percents, and numbers written in scientific notation.Comparing Numbers in Scientific Notation|Converting from Scientific Notation to Standard Form|Writing in Scientific Notation8.4Apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Identifying the Square Root of a Perfect Square8.5aDetermine whether a given number is a perfect squareIdentifying the Square Root of a Perfect Square8.5bFind the two consecutive whole numbers between which a square root lies.Estimating Square RootsRates, Ratios, & Proportions8.1bCompare and order decimals, fractions, percents, and numbers written in scientific notation.Comparing Fractions, Decimals, Percents, and Ratios|Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction8.3aSolve practical problems involving rational numbers, percents, ratios, and proportions;Applying the Percent Proportion|Computing Simple Interest|Expressing a Percent as a Fraction|Solving Practical Problems Using Ratios|Solving Proportions to Find Missing Terms8.3bDetermine the percent increase or decrease for a given situation.Identifying the Percent of ChangeEquations8.15aSolve multistep linear equations in one variable on one and two sides of the equation;Applying Knowledge of Two-Step Equations to Solve Word Problems|Applying the Distributive Property to Solve Equations|Solving Two-Step Equations Involving Division with Addition or Subtraction|Solving Two-Step Equations Involving Multiplication with Addition or Subtraction|Solving an Equation with Variables on Two Sides8.15cIdentify properties of operations used to solve an equation.Applying the Distributive Property to Solve Equations8.1aSimplify 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 VariablesInequalities8.15bSolve two-step linear inequalities and graph the results on a number line;Applying Knowledge of Two-Step Inequalities to Solve Word Problems|Solving Two-Step Inequalities Involving Division with Addition or Subtraction|Solving Two-Step Inequalities Involving Multiplication with Addition or SubtractionGraphing8.14Make connections between any two representations (tables, graphs, words, and rules) of a given relationship.Determining if a Relation is a Function|Identifying Domain and Range|Representing Relations and Functions in Different Forms8.16Graph a linear equation in two variables.Calculating the Slope of a Graphed Line|Classifying Functions as Linear or Nonlinear|Determining Slope and y-Intercept of Linear Equations in Slope-Intercept Form|Determining Whether Functions are Linear or Nonlinear Using Patterns|Determining if a Relation is a Function|Determining the Constant of Variation|Finding the Rate of Change|Graphing Linear Equations in Slope-Intercept Form Using the Slope and y-Intercept |Graphing a Linear Function in Two Variables Using Intercepts|Graphing a Linear Function in Two Variables Using Tables|Identifying Quadrants and Graphing an Ordered Pair|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation|Identifying an Ordered Pair Given a Graphed Point|Identifying the Distance Between Two Points with Identical x- or y-Values|Identifying the Slope of a Line Using Two Points|Identifying the x- and y-Intercepts|Predicting Values Using a Direct Variation|Predicting Values in Tables Using Numerical Patterns|Representing Relations and Functions in Different Forms|Writing Equations in Slope-Intercept Form (Slope and a Point)|Writing Equations in Slope-Intercept Form (Slope and y-Intercept)|Writing Equations in Slope-Intercept Form (Two Points)|Writing an Equation for Direct Variation8.17Identify the domain, range, independent variable or dependent variable in a given situation. Identifying Domain and Range|Identifying Independent and Dependent VariablesProbability & Statistics8.12Determine the probability of independent and dependent events with and without replacement.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space8.13aMake comparisons, predictions, and inferences, using information displayed in graphs;Interpreting Scatter Plots to Investigate Patterns of Association|Solving Problems Involving Bivariate Measurement DataGeometry8.10aVerify the Pythagorean TheoremDetermining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane8.10bApply the Pythagorean Theorem. Determining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane8.11Solve practical area and perimeter problems involving composite plane figures.Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Polygons on the Coordinate Plane8.6aVerify 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 Transversal|Solving Equations for an Unknown Angle in a Figure8.6bMeasure angles of less than 360°.Classifying Angles8.7aInvestigate and solve practical problems involving volume and surface area of prisms, cylinders, cones, and pyramids;Investigating Surface Area and Cross-Sections of Solids|Solving Problems Involving Surface Area|Solving Problems Involving Volume8.7bDescribe how changing one measured attribute of the figure affects the volume and surface area.Solving Problems Involving Surface Area|Solving Problems Involving Volume8.8aApply transformations to plane figuresIdentifying Properties of Rotations, Reflections, and Translations|Using Congruency Statements to Identify Corresponding Parts of a Polygon|Using the Coordinate Plane to Demonstrate Transformations8.8bIdentify applications of transformations.Describing the Sequence of Transformations of Two Similar Figures|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations8.9Construct a three-dimensional model, given the top or bottom, side, and front views.Investigating Surface Area and Cross-Sections of Solids`

#### Story

Greenville’s population has mysteriously lost its ability to do math because the town’s mystical math orb has mysteriously disappeared.

The town’s only hope lies in the hands of Pi, the student’s character, who is drawn into quests to help the people re-learn their math skills. Pi quickly learns that the annual Corn Festival is a major contributor to Greenville’s prosperity.

If forgetting how to do math isn't catastrophe enough, this year's festival preparations begin to suffer a series of calamities and strange occurrences. By solving real-world mathematical dilemmas, Pi must save the festival, find the culprits, and be the hero.

#### Features

• Over 5,000 unique math problems
• A virtual math tablet, accessible at any time during the game, that contains tutoring aids including digitized text, instructional videos, and practice problems
• In-game assessments that challenge students and monitor their performance
• Immediate feedback concerning errors in challenges and practice problems
• A web-based reporting tool that provides detailed performance statistics
• Story and game dialog developed by renowned screen and video game writer, author, and professor, Lee Sheldon; game writer and designer, Phoebe Harris Elefante; and Emmy and Telly Award winning producer and writer, Graham Sheldon.
• Non-math related mini-games that keep the game fun and engaging, provide a ‘math brain break,’ and maintain students’ motivation to continue progressing through the curriculum

#### Documents

Document Description
Teachers Manual that discusses the game and its features in the context of classroom use. General topics covered include:
• Installation
• Usage
• Account Management
• Reporting
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&LT math game, Pi and the Lost Function, as an effective and efficient preparation tool for the different placement exams.

#### System Requirements

 PC System Requirements: Mac System Requirements: Windows® XP® (Updated with the latest Service Packs) with DirectX® 9.0c Celeron E3500 equivalent AMD Athlon® processor Intel G41 Express or 128 MB PCIe NVIDIA® GeForce® 6600 GT or ATI Radeon® 9800 PRO video card 6 GB available HD space 2 GB RAM Broadband Internet connection 1024X768 minimum display resolution Mac® OS X 10.5.8, 10.6.2 - 10.9 (not compatible with OS x 10.10 or newer) Intel® Processor NVIDIA® GeForce® 8600M GT or ATI Radeon® X1600 or better 6 GB available HD space 2 GB RAM Broadband Internet connection 1024X768 minimum display resolution