ATLTGames

Pi and the Lost Function Bundle
$47.97
Includes:
  • Pi and the Lost Function Game (for Windows or Mac)
  • Full Access to the Online Content Tool (OCT)
  • 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:

Additional Information for Educators


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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 - 2010

6

Whole Numbers

6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
6.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 Expressions
6.EE.A.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
6.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 Expressions
6.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 Operations
6.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 Terms
6.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 Terms
6.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 Expressions
6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
6.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 Factors

Integers

6.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 Integers
6.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 Value
6.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 Value
6.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 Integers
6.NS.C.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
6.NS.C.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
6.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 Value
6.NS.C.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

6.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 Fractions
6.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 Numbers
6.NS.C.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
6.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 Monomials
6.NS.C.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

6.NS.C.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form
6.RP.A.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 Rates
6.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 Terms
6.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 Ratios
6.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 Ratios
6.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 Terms

Equations

6.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 Problems
6.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 Problems
6.EE.A.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
6.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 Problems
6.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 Problems
6.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 Variables

Inequalities

6.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 Inequalities
6.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 Inequalities
6.NS.C.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
6.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 Line

Graphing

6.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 Variables
6.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-Intercepts
6.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 Point
6.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-Intercepts
6.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-Values
6.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 Patterns
6.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 Patterns

Probability & Statistics

6.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 Population
6.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 Display
6.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 Set
6.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 Display
6.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 Population
6.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 Display
6.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 Population
6.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 Set
6.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 Set

Geometry

6.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 Techniques
6.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 Volume
6.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 Plane
6.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 Solids
6.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 Plane

7

Whole Numbers

7.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 Terms
7.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 Expressions
7.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 Operations
7.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 Operations
7.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 Operations
7.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 Multiplication
7.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 Multiplication
7.NS.A.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
7.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 Rules
7.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 Multiplication
7.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 Multiplication
7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

7.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 Integers
7.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 Integers
7.NS.A.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
7.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 Value
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
7.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 Integers

Fractions & Decimals

7.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 Decimal
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Decimal
7.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 Fractions
7.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 Decimal
7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
7.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 Decimal
7.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 Denominators

Exponents & Roots

7.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 Notation

Rates, Ratios, & Proportions

7.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 Fraction
7.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 Terms
7.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 Quantities
7.RP.A.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios
7.RP.A.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 Terms
7.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 Change

Equations

7.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 Problems
7.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 Problems
7.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 Problems
7.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 Variables
7.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 Variables
7.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 Variables

Inequalities

7.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 Inequalities
7.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 Inequalities
7.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 Graphs

Graphing

7.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 Variation
7.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 Variation
7.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 Variation
7.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 Change

Probability & Statistics

7.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 Population
7.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 Population
7.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 Set
7.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 Events
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Space
7.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 Space
7.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 Space
7.SP.C.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

7.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 Figures
7.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 Polygons
7.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 Solids
7.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 Circle
7.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 Figure
7.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 Volume

8

Whole Numbers

8.EE.C.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
8.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 Terms

Fractions & Decimals

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.Classifying Numbers
8.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 Decimal

Exponents & Roots

8.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 Square
8.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 Notation
8.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 Roots

Rates, Ratios, & Proportions

8.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 Terms

Equations

8.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 Problems
8.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 Problems

Graphing

8.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 Variation
8.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 Variation
8.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 Graphically
8.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 Graphically
8.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 Graphically
8.EE.C.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
8.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 Forms
8.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 Forms
8.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 Variation
8.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 Variation
8.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 Tables
8.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 Variation

Probability & Statistics

8.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 Association
8.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 Data
8.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 Data
8.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 Table

Geometry

8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Polygon
8.G.A.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
8.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 Figures
8.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 Transversal
8.G.B.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
8.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 Solids
8.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 Plane
8.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 Volume

Arkansas Common Core State Standards - 2012

6

Whole Numbers

6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
6.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 Expressions
6.EE.A.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
6.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 Expressions
6.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 Operations
6.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 Terms
6.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 Terms
6.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 Expressions
6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
6.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 Factors

Integers

6.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 Integers
6.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 Value
6.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 Value
6.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 Integers
6.NS.C.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
6.NS.C.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
6.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 Value
6.NS.C.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

6.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 Fractions
6.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 Numbers
6.NS.C.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
6.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 Monomials
6.NS.C.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

6.NS.C.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
6.NS.C.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form
6.RP.A.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 Rates
6.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 Terms
6.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 Ratios
6.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 Ratios
6.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 Terms

Equations

6.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 Problems
6.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 Problems
6.EE.A.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
6.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 Problems
6.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 Problems
6.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 Variables

Inequalities

6.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 Inequalities
6.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 Inequalities
6.NS.C.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
6.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 Line

Graphing

6.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 Variables
6.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-Intercepts
6.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 Point
6.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-Intercepts
6.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-Values
6.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 Patterns
6.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 Patterns

Probability & Statistics

6.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 Population
6.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 Display
6.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 Set
6.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 Display
6.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 Population
6.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 Display
6.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 Population
6.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 Set
6.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 Set

Geometry

6.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 Techniques
6.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 Volume
6.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 Plane
6.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 Solids
6.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 Plane

7

Whole Numbers

7.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 Terms
7.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 Expressions
7.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 Operations
7.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 Operations
7.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 Operations
7.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 Multiplication
7.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 Multiplication
7.NS.A.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
7.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 Rules
7.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 Multiplication
7.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 Multiplication
7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

7.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 Integers
7.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 Integers
7.NS.A.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
7.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 Value
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
7.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 Integers

Fractions & Decimals

7.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 Decimal
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Decimal
7.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 Fractions
7.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 Decimal
7.NS.A.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
7.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 Decimal
7.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 Denominators

Exponents & Roots

7.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 Notation

Rates, Ratios, & Proportions

7.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 Fraction
7.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 Terms
7.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 Quantities
7.RP.A.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios
7.RP.A.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 Terms
7.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 Change

Equations

7.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 Problems
7.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 Problems
7.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 Problems
7.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 Variables
7.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 Variables
7.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 Variables

Inequalities

7.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 Inequalities
7.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 Inequalities
7.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 Graphs

Graphing

7.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 Variation
7.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 Variation
7.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 Variation
7.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 Change

Probability & Statistics

7.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 Population
7.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 Population
7.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 Set
7.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 Events
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Space
7.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 Space
7.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 Space
7.SP.C.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

7.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 Figures
7.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 Polygons
7.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 Solids
7.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 Circle
7.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 Figure
7.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 Volume

8

Whole Numbers

8.EE.C.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
8.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 Terms

Fractions & Decimals

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.Classifying Numbers
8.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 Decimal

Exponents & Roots

8.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 Square
8.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 Notation
8.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 Roots

Rates, Ratios, & Proportions

8.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 Terms

Equations

8.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 Problems
8.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 Problems

Graphing

8.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 Variation
8.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 Variation
8.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 Graphically
8.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 Graphically
8.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 Graphically
8.EE.C.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
8.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 Forms
8.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 Forms
8.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 Variation
8.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 Variation
8.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 Tables
8.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 Variation

Probability & Statistics

8.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 Association
8.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 Data
8.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 Data
8.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 Table

Geometry

8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Polygon
8.G.A.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
8.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 Figures
8.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 Transversal
8.G.B.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
8.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 Solids
8.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 Plane
8.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 Volume

California Common Core State Standards - 2013

6

Whole Numbers

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
6.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 Expressions
6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
6.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 Expressions
6.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 Operations
6.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 Terms
6.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 Terms
6.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 Expressions
6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
6.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 Factors

Integers

6.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 Integers
6.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 Value
6.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 Value
6.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 Integers
6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
6.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 Value
6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

6.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 Fractions
6.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 Numbers
6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
6.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 Monomials
6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form
6.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 Rates
6.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 Terms
6.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 Ratios
6.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 Ratios
6.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 Terms

Equations

6.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 Problems
6.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 Problems
6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
6.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 Problems
6.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 Problems
6.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 Variables

Inequalities

6.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 Inequalities
6.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 Inequalities
6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number Line

Graphing

6.EE.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 Variables
6.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-Intercepts
6.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 Point
6.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-Intercepts
6.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-Values
6.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 Patterns
6.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 Patterns

Probability & Statistics

6.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 Population
6.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 Display
6.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 Set
6.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 Display
6.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 Population
6.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 Display
6.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 Population
6.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 Set
6.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 Set

Geometry

6.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 Techniques
6.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 Volume
6.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 Plane
6.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 Solids
6.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 Plane

7

Whole Numbers

7.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 Terms
7.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 Expressions
7.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 Operations
7.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 Operations
7.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 Operations
7.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 Multiplication
7.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 Multiplication
7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
7.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 Rules
7.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 Multiplication
7.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 Multiplication
7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

7.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 Integers
7.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 Integers
7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
7.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 Value
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
7.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 Integers

Fractions & Decimals

7.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 Decimal
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Decimal
7.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 Fractions
7.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 Decimal
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
7.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 Decimal
7.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 Denominators

Exponents & Roots

7.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 Notation

Rates, Ratios, & Proportions

7.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 Fraction
7.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 Terms
7.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 Quantities
7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios
7.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 Terms
7.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 Change

Equations

7.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 Problems
7.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 Problems
7.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 Problems
7.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 Variables
7.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 Variables
7.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 Variables

Inequalities

7.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 Inequalities
7.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 Inequalities
7.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 Graphs

Graphing

7.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 Variation
7.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 Variation
7.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 Variation
7.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 Change

Probability & Statistics

7.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 Population
7.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 Population
7.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 Set
7.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 Events
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Space
7.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 Space
7.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 Space
7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

7.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 Figures
7.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 Polygons
7.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 Solids
7.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 Circle
7.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 Figure
7.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 Volume

8

Whole Numbers

8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
8.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 Terms

Fractions & Decimals

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.Classifying Numbers
8.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 Decimal

Exponents & Roots

8.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 Square
8.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 Notation
8.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 Roots

Rates, Ratios, & Proportions

8.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 Terms

Equations

8.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 Problems
8.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 Problems

Graphing

8.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 Variation
8.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 Variation
8.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 Graphically
8.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 Graphically
8.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 Graphically
8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
8.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 Forms
8.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 Forms
8.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 Variation
8.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 Variation
8.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 Tables
8.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 Variation

Probability & Statistics

8.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 Association
8.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 Data
8.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 Data
8.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 Table

Geometry

8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Polygon
8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
8.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 Figures
8.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 Transversal
8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
8.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 Solids
8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane
8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving Volume

Florida Common Core State Standards - 2010

6

Whole Numbers

MACC.6.EE.1.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
MACC.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 Expressions
MACC.6.EE.1.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
MACC.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 Expressions
MACC.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 Operations
MACC.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 Terms
MACC.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 Terms
MACC.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 Expressions
MACC.6.NS.2.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
MACC.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 Factors

Integers

MACC.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 Integers
MACC.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 Value
MACC.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 Value
MACC.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 Integers
MACC.6.NS.3.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
MACC.6.NS.3.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
MACC.6.NS.3.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
MACC.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 Value
MACC.6.NS.3.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

MACC.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 Fractions
MACC.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 Numbers
MACC.6.NS.3.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
MACC.6.NS.3.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

MACC.6.EE.1.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
MACC.6.EE.1.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
MACC.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 Monomials
MACC.6.NS.3.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
MACC.6.NS.3.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

MACC.6.NS.3.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
MACC.6.NS.3.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
MACC.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 Form
MACC.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 Rates
MACC.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 Terms
MACC.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 Ratios
MACC.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 Ratios
MACC.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 Terms

Equations

MACC.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 Problems
MACC.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 Problems
MACC.6.EE.1.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
MACC.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 Problems
MACC.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 Problems
MACC.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 Variables

Inequalities

MACC.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 Inequalities
MACC.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 Inequalities
MACC.6.NS.3.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
MACC.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 Line

Graphing

MACC.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 Variables
MACC.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-Intercepts
MACC.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 Point
MACC.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-Intercepts
MACC.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-Values
MACC.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 Patterns
MACC.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 Patterns

Probability & Statistics

MACC.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 Population
MACC.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 Display
MACC.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 Set
MACC.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 Display
MACC.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 Population
MACC.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 Display
MACC.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 Population
MACC.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 Set
MACC.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 Set

Geometry

MACC.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 Techniques
MACC.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 Volume
MACC.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 Plane
MACC.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 Solids
MACC.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 Plane

7

Whole Numbers

MACC.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 Terms
MACC.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 Expressions
MACC.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 Operations
MACC.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 Operations
MACC.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 Operations
MACC.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 Multiplication
MACC.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 Multiplication
MACC.7.NS.1.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
MACC.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 Rules
MACC.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 Multiplication
MACC.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 Multiplication
MACC.7.NS.1.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

MACC.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 Integers
MACC.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 Integers
MACC.7.NS.1.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
MACC.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 Value
MACC.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 Integers
MACC.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 Integers
MACC.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 Integers
MACC.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 Integers
MACC.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 Integers
MACC.7.NS.1.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
MACC.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 Integers

Fractions & Decimals

MACC.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 Decimal
MACC.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 Denominators
MACC.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 Denominators
MACC.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 Denominators
MACC.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 Denominators
MACC.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 Decimal
MACC.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 Fractions
MACC.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 Decimal
MACC.7.NS.1.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
MACC.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 Decimal
MACC.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 Denominators

Exponents & Roots

MACC.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 Notation

Rates, Ratios, & Proportions

MACC.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 Fraction
MACC.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 Terms
MACC.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 Quantities
MACC.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 Ratios
MACC.7.RP.1.2cRepresent proportional relationships by equations.Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms
MACC.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 Change

Equations

MACC.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 Problems
MACC.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 Problems
MACC.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 Problems
MACC.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 Variables
MACC.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 Variables
MACC.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 Variables

Inequalities

MACC.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 Inequalities
MACC.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 Inequalities
MACC.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 Graphs

Graphing

MACC.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 Variation
MACC.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 Variation
MACC.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 Variation
MACC.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 Change

Probability & Statistics

MACC.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 Population
MACC.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 Population
MACC.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 Set
MACC.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 Events
MACC.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 Predictions
MACC.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 Predictions
MACC.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 Predictions
MACC.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 Predictions
MACC.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 Space
MACC.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 Space
MACC.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 Space
MACC.7.SP.3.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

MACC.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 Figures
MACC.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 Polygons
MACC.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 Solids
MACC.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 Circle
MACC.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 Figure
MACC.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 Volume

8

Whole Numbers

MACC.8.EE.3.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
MACC.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 Terms

Fractions & Decimals

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.Classifying Numbers
MACC.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 Decimal

Exponents & Roots

MACC.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 Square
MACC.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 Notation
MACC.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 Roots

Rates, Ratios, & Proportions

MACC.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 Terms

Equations

MACC.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 Problems
MACC.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 Problems

Graphing

MACC.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 Variation
MACC.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 Variation
MACC.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 Graphically
MACC.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 Graphically
MACC.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 Graphically
MACC.8.EE.3.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
MACC.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 Forms
MACC.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 Forms
MACC.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 Variation
MACC.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 Variation
MACC.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 Tables
MACC.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 Variation

Probability & Statistics

MACC.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 Association
MACC.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 Data
MACC.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 Data
MACC.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 Table

Geometry

MACC.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 Transformations
MACC.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 Transformations
MACC.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 Transformations
MACC.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 Transformations
MACC.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 Polygon
MACC.8.G.1.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
MACC.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 Figures
MACC.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 Transversal
MACC.8.G.2.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
MACC.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 Solids
MACC.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 Plane
MACC.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 Volume

Georgia Common Core Performance Standards - 2011

6

Whole Numbers

MCC6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
MCC6.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 Expressions
MCC6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
MCC6.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 Expressions
MCC6.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 Operations
MCC6.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 Terms
MCC6.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 Terms
MCC6.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 Expressions
MCC6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
MCC6.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 Factors

Integers

MCC6.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 Integers
MCC6.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 Value
MCC6.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 Value
MCC6.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 Integers
MCC6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
MCC6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
MCC6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
MCC6.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 Value
MCC6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

MCC6.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 Fractions
MCC6.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 Numbers
MCC6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
MCC6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

MCC6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
MCC6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
MCC6.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 Monomials
MCC6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
MCC6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

MCC6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
MCC6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
MCC6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form
MCC6.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 Rates
MCC6.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 Terms
MCC6.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 Ratios
MCC6.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 Ratios
MCC6.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 Terms

Equations

MCC6.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 Problems
MCC6.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 Problems
MCC6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
MCC6.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 Problems
MCC6.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 Problems
MCC6.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 Variables

Inequalities

MCC6.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 Inequalities
MCC6.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 Inequalities
MCC6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
MCC6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number Line

Graphing

MCC6.EE.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 Variables
MCC6.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-Intercepts
MCC6.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 Point
MCC6.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-Intercepts
MCC6.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-Values
MCC6.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 Patterns
MCC6.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 Patterns

Probability & Statistics

MCC6.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 Population
MCC6.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 Display
MCC6.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 Set
MCC6.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 Display
MCC6.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 Population
MCC6.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 Display
MCC6.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 Population
MCC6.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 Set
MCC6.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 Set

Geometry

MCC6.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 Techniques
MCC6.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 Volume
MCC6.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 Plane
MCC6.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 Solids
MCC6.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 Plane

7

Whole Numbers

MCC7.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 Terms
MCC7.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 Expressions
MCC7.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 Operations
MCC7.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 Operations
MCC7.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 Operations
MCC7.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 Multiplication
MCC7.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 Multiplication
MCC7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
MCC7.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 Rules
MCC7.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 Multiplication
MCC7.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 Multiplication
MCC7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

MCC7.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 Integers
MCC7.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 Integers
MCC7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
MCC7.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 Value
MCC7.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 Integers
MCC7.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 Integers
MCC7.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 Integers
MCC7.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 Integers
MCC7.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 Integers
MCC7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
MCC7.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 Integers

Fractions & Decimals

MCC7.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 Decimal
MCC7.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 Denominators
MCC7.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 Denominators
MCC7.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 Denominators
MCC7.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 Denominators
MCC7.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 Decimal
MCC7.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 Fractions
MCC7.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 Decimal
MCC7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
MCC7.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 Decimal
MCC7.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 Denominators

Exponents & Roots

MCC7.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 Notation

Rates, Ratios, & Proportions

MCC7.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 Fraction
MCC7.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 Terms
MCC7.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 Quantities
MCC7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios
MCC7.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 Terms
MCC7.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 Change

Equations

MCC7.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 Problems
MCC7.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 Problems
MCC7.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 Problems
MCC7.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 Variables
MCC7.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 Variables
MCC7.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 Variables

Inequalities

MCC7.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 Inequalities
MCC7.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 Inequalities
MCC7.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 Graphs

Graphing

MCC7.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 Variation
MCC7.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 Variation
MCC7.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 Variation
MCC7.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 Change

Probability & Statistics

MCC7.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 Population
MCC7.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 Population
MCC7.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 Set
MCC7.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 Events
MCC7.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 Predictions
MCC7.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 Predictions
MCC7.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 Predictions
MCC7.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 Predictions
MCC7.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 Space
MCC7.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 Space
MCC7.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 Space
MCC7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

MCC7.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 Figures
MCC7.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 Polygons
MCC7.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 Solids
MCC7.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 Circle
MCC7.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 Figure
MCC7.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 Volume

8

Whole Numbers

MCC8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
MCC8.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 Terms

Fractions & Decimals

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.Classifying Numbers
MCC8.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 Decimal

Exponents & Roots

MCC8.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 Square
MCC8.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 Notation
MCC8.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 Roots

Rates, Ratios, & Proportions

MCC8.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 Terms

Equations

MCC8.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 Problems
MCC8.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 Problems

Graphing

MCC8.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 Variation
MCC8.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 Variation
MCC8.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 Graphically
MCC8.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 Graphically
MCC8.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 Graphically
MCC8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
MCC8.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 Forms
MCC8.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 Forms
MCC8.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 Variation
MCC8.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 Variation
MCC8.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 Tables
MCC8.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 Variation

Probability & Statistics

MCC8.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 Association
MCC8.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 Data
MCC8.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 Data
MCC8.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 Table

Geometry

MCC8.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 Transformations
MCC8.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 Transformations
MCC8.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 Transformations
MCC8.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 Transformations
MCC8.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 Polygon
MCC8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
MCC8.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 Figures
MCC8.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 Transversal
MCC8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
MCC8.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 Solids
MCC8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane
MCC8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving Volume

Michigan Common Core State Standards - 2010

6

Whole Numbers

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
6.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 Expressions
6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
6.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 Expressions
6.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 Operations
6.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 Terms
6.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 Terms
6.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 Expressions
6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
6.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 Factors

Integers

6.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 Integers
6.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 Value
6.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 Value
6.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 Integers
6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
6.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 Value
6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

6.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 Fractions
6.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 Numbers
6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
6.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 Monomials
6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form
6.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 Rates
6.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 Terms
6.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 Ratios
6.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 Ratios
6.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 Terms

Equations

6.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 Problems
6.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 Problems
6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
6.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 Problems
6.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 Problems
6.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 Variables

Inequalities

6.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 Inequalities
6.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 Inequalities
6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number Line

Graphing

6.EE.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 Variables
6.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-Intercepts
6.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 Point
6.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-Intercepts
6.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-Values
6.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 Patterns
6.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 Patterns

Probability & Statistics

6.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 Population
6.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 Display
6.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 Set
6.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 Display
6.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 Population
6.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 Display
6.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 Population
6.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 Set
6.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 Set

Geometry

6.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 Techniques
6.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 Volume
6.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 Plane
6.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 Solids
6.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 Plane

7

Whole Numbers

7.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 Terms
7.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 Expressions
7.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 Operations
7.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 Operations
7.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 Operations
7.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 Multiplication
7.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 Multiplication
7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
7.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 Rules
7.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 Multiplication
7.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 Multiplication
7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

7.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 Integers
7.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 Integers
7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
7.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 Value
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
7.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 Integers

Fractions & Decimals

7.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 Decimal
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Decimal
7.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 Fractions
7.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 Decimal
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
7.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 Decimal
7.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 Denominators

Exponents & Roots

7.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 Notation

Rates, Ratios, & Proportions

7.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 Fraction
7.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 Terms
7.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 Quantities
7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios
7.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 Terms
7.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 Change

Equations

7.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 Problems
7.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 Problems
7.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 Problems
7.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 Variables
7.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 Variables
7.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 Variables

Inequalities

7.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 Inequalities
7.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 Inequalities
7.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 Graphs

Graphing

7.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 Variation
7.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 Variation
7.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 Variation
7.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 Change

Probability & Statistics

7.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 Population
7.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 Population
7.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 Set
7.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 Events
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Space
7.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 Space
7.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 Space
7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

7.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 Figures
7.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 Polygons
7.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 Solids
7.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 Circle
7.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 Figure
7.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 Volume

8

Whole Numbers

8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
8.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 Terms

Fractions & Decimals

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.Classifying Numbers
8.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 Decimal

Exponents & Roots

8.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 Square
8.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 Notation
8.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 Roots

Rates, Ratios, & Proportions

8.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 Terms

Equations

8.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 Problems
8.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 Problems

Graphing

8.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 Variation
8.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 Variation
8.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 Graphically
8.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 Graphically
8.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 Graphically
8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
8.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 Forms
8.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 Forms
8.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 Variation
8.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 Variation
8.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 Tables
8.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 Variation

Probability & Statistics

8.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 Association
8.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 Data
8.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 Data
8.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 Table

Geometry

8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Polygon
8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
8.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 Figures
8.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 Transversal
8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
8.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 Solids
8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane
8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving Volume

New York Common Core State Standards - 2012

6

Whole Numbers

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
6.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 Expressions
6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
6.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 Expressions
6.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 Operations
6.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 Terms
6.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 Terms
6.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 Expressions
6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
6.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 Factors

Integers

6.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 Integers
6.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 Value
6.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 Value
6.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 Integers
6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
6.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 Value
6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

6.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 Fractions
6.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 Numbers
6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
6.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 Monomials
6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form
6.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 Rates
6.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 Terms
6.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 Ratios
6.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 Ratios
6.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 Terms

Equations

6.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 Problems
6.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 Problems
6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
6.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 Problems
6.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 Problems
6.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 Variables

Inequalities

6.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 Inequalities
6.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 Inequalities
6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number Line

Graphing

6.EE.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 Variables
6.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-Intercepts
6.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 Point
6.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-Intercepts
6.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-Values
6.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 Patterns
6.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 Patterns

Probability & Statistics

6.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 Population
6.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 Display
6.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 Set
6.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 Display
6.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 Population
6.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 Display
6.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 Population
6.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 Set
6.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 Set

Geometry

6.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 Techniques
6.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 Volume
6.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 Plane
6.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 Solids
6.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 Plane

7

Whole Numbers

7.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 Terms
7.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 Expressions
7.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 Operations
7.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 Operations
7.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 Operations
7.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 Multiplication
7.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 Multiplication
7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
7.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 Rules
7.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 Multiplication
7.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 Multiplication
7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

7.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 Integers
7.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 Integers
7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
7.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 Value
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
7.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 Integers

Fractions & Decimals

7.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 Decimal
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Decimal
7.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 Fractions
7.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 Decimal
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
7.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 Decimal
7.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 Denominators

Exponents & Roots

7.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 Notation

Rates, Ratios, & Proportions

7.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 Fraction
7.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 Terms
7.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 Quantities
7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios
7.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 Terms
7.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 Change

Equations

7.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 Problems
7.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 Problems
7.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 Problems
7.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 Variables
7.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 Variables
7.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 Variables

Inequalities

7.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 Inequalities
7.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 Inequalities
7.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 Graphs

Graphing

7.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 Variation
7.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 Variation
7.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 Variation
7.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 Change

Probability & Statistics

7.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 Population
7.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 Population
7.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 Set
7.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 Events
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Space
7.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 Space
7.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 Space
7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

7.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 Figures
7.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 Polygons
7.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 Solids
7.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 Circle
7.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 Figure
7.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 Volume

8

Whole Numbers

8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
8.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 Terms

Fractions & Decimals

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.Classifying Numbers
8.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 Decimal

Exponents & Roots

8.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 Square
8.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 Notation
8.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 Roots

Rates, Ratios, & Proportions

8.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 Terms

Equations

8.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 Problems
8.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 Problems

Graphing

8.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 Variation
8.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 Variation
8.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 Graphically
8.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 Graphically
8.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 Graphically
8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
8.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 Forms
8.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 Forms
8.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 Variation
8.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 Variation
8.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 Tables
8.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 Variation

Probability & Statistics

8.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 Association
8.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 Data
8.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 Data
8.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 Table

Geometry

8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Polygon
8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
8.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 Figures
8.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 Transversal
8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
8.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 Solids
8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane
8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving Volume

North Carolina Common Core State Standards - 2012

6

Whole Numbers

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
6.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 Expressions
6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
6.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 Expressions
6.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 Operations
6.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 Terms
6.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 Terms
6.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 Expressions
6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
6.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 Factors

Integers

6.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 Integers
6.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 Value
6.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 Value
6.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 Integers
6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
6.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 Value
6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

6.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 Fractions
6.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 Numbers
6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
6.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 Monomials
6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form
6.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 Rates
6.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 Terms
6.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 Ratios
6.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 Ratios
6.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 Terms

Equations

6.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 Problems
6.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 Problems
6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
6.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 Problems
6.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 Problems
6.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 Variables

Inequalities

6.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 Inequalities
6.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 Inequalities
6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number Line

Graphing

6.EE.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 Variables
6.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-Intercepts
6.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 Point
6.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-Intercepts
6.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-Values
6.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 Patterns
6.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 Patterns

Probability & Statistics

6.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 Population
6.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 Display
6.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 Set
6.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 Display
6.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 Population
6.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 Display
6.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 Population
6.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 Set
6.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 Set

Geometry

6.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 Techniques
6.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 Volume
6.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 Plane
6.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 Solids
6.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 Plane

7

Whole Numbers

7.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 Terms
7.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 Expressions
7.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 Operations
7.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 Operations
7.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 Operations
7.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 Multiplication
7.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 Multiplication
7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
7.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 Rules
7.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 Multiplication
7.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 Multiplication
7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

7.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 Integers
7.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 Integers
7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
7.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 Value
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
7.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 Integers

Fractions & Decimals

7.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 Decimal
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Decimal
7.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 Fractions
7.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 Decimal
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
7.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 Decimal
7.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 Denominators

Exponents & Roots

7.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 Notation

Rates, Ratios, & Proportions

7.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 Fraction
7.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 Terms
7.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 Quantities
7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios
7.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 Terms
7.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 Change

Equations

7.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 Problems
7.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 Problems
7.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 Problems
7.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 Variables
7.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 Variables
7.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 Variables

Inequalities

7.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 Inequalities
7.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 Inequalities
7.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 Graphs

Graphing

7.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 Variation
7.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 Variation
7.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 Variation
7.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 Change

Probability & Statistics

7.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 Population
7.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 Population
7.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 Set
7.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 Events
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Space
7.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 Space
7.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 Space
7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

7.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 Figures
7.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 Polygons
7.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 Solids
7.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 Circle
7.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 Figure
7.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 Volume

8

Whole Numbers

8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
8.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 Terms

Fractions & Decimals

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.Classifying Numbers
8.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 Decimal

Exponents & Roots

8.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 Square
8.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 Notation
8.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 Roots

Rates, Ratios, & Proportions

8.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 Terms

Equations

8.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 Problems
8.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 Problems

Graphing

8.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 Variation
8.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 Variation
8.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 Graphically
8.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 Graphically
8.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 Graphically
8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
8.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 Forms
8.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 Forms
8.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 Variation
8.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 Variation
8.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 Tables
8.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 Variation

Probability & Statistics

8.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 Association
8.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 Data
8.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 Data
8.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 Table

Geometry

8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Polygon
8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
8.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 Figures
8.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 Transversal
8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
8.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 Solids
8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane
8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving Volume

South Carolina Common Core State Standards - 2010

6

Whole Numbers

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions
6.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 Expressions
6.EE.2aWrite expressions that record operations with numbers and with letters standing for numbers.Writing Algebraic Expressions
6.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 Expressions
6.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 Operations
6.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 Terms
6.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 Terms
6.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 Expressions
6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Using the Divisibility Rules
6.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 Factors

Integers

6.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 Integers
6.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 Value
6.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 Value
6.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 Integers
6.NS.7Understand ordering and absolute value of rational numbersComparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Integers
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Integers
6.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 Value
6.NS.7dDistinguish comparisons of absolute value from statements about order.Comparing Integers

Fractions & Decimals

6.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 Fractions
6.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 Numbers
6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions and Decimals
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Evaluating All Powers|Identifying Bases and Exponents
6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Factoring Monomials
6.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 Monomials
6.NS.7Understand ordering and absolute value of rational numbersComparing Numbers in Scientific Notation
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

6.NS.7Understand ordering and absolute value of rational numbersComparing Fractions, Decimals, Percents, and Ratios
6.NS.7bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios
6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Expressing Ratios in Simplified Form
6.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 Rates
6.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 Terms
6.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 Ratios
6.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 Ratios
6.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 Terms

Equations

6.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 Problems
6.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 Problems
6.EE.3Apply the properties of operations to generate equivalent expressions.Applying the Distributive Property to Write Equivalent Expressions with Variables
6.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 Problems
6.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 Problems
6.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 Variables

Inequalities

6.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 Inequalities
6.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 Inequalities
6.NS.7Understand ordering and absolute value of rational numbersGraphing Inequalities on a Number Line
6.NS.7aInterpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Graphing Inequalities on a Number Line

Graphing

6.EE.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 Variables
6.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-Intercepts
6.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 Point
6.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-Intercepts
6.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-Values
6.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 Patterns
6.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 Patterns

Probability & Statistics

6.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 Population
6.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 Display
6.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 Set
6.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 Display
6.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 Population
6.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 Display
6.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 Population
6.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 Set
6.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 Set

Geometry

6.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 Techniques
6.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 Volume
6.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 Plane
6.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 Solids
6.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 Plane

7

Whole Numbers

7.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 Terms
7.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 Expressions
7.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 Operations
7.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 Operations
7.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 Operations
7.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 Multiplication
7.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 Multiplication
7.NS.1dApply properties of operations as strategies to add and subtract rational numbers.Identifying Properties of Addition and Multiplication
7.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 Rules
7.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 Multiplication
7.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 Multiplication
7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.1Evaluating Expressions Using the Order of Operations

Integers

7.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 Integers
7.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 Integers
7.NS.1aDescribe situations in which opposite quantities combine to make 0.Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value
7.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 Value
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.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 Integers
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
7.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 Integers

Fractions & Decimals

7.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 Decimal
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Denominators
7.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 Decimal
7.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 Fractions
7.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 Decimal
7.NS.2cApply properties of operations as strategies to multiply and divide rational numbers.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
7.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 Decimal
7.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 Denominators

Exponents & Roots

7.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 Notation

Rates, Ratios, & Proportions

7.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 Fraction
7.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 Terms
7.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 Quantities
7.RP.2bIdentify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.Solving Practical Problems Using Ratios
7.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 Terms
7.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 Change

Equations

7.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 Problems
7.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 Problems
7.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 Problems
7.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 Variables
7.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 Variables
7.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 Variables

Inequalities

7.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 Inequalities
7.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 Inequalities
7.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 Graphs

Graphing

7.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 Variation
7.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 Variation
7.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 Variation
7.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 Change

Probability & Statistics

7.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 Population
7.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 Population
7.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 Set
7.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 Events
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Predictions
7.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 Space
7.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 Space
7.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 Space
7.SP.8cDesign and use a simulation to generate frequencies for compound events.Finding the Probabilities of Compound Events

Geometry

7.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 Figures
7.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 Polygons
7.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 Solids
7.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 Circle
7.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 Figure
7.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 Volume

8

Whole Numbers

8.EE.7Solve linear equations in one variable.Simplifying Expressions by Combining Like Terms
8.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 Terms

Fractions & Decimals

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.Classifying Numbers
8.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 Decimal

Exponents & Roots

8.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 Square
8.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 Notation
8.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 Roots

Rates, Ratios, & Proportions

8.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 Terms

Equations

8.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 Problems
8.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 Problems

Graphing

8.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 Variation
8.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 Variation
8.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 Graphically
8.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 Graphically
8.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 Graphically
8.EE.8cSolve real-world and mathematical problems leading to two linear equations in two variables.Solving Systems of Two Linear Equations Algebraically
8.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 Forms
8.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 Forms
8.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 Variation
8.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 Variation
8.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 Tables
8.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 Variation

Probability & Statistics

8.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 Association
8.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 Data
8.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 Data
8.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 Table

Geometry

8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Transformations
8.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 Polygon
8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Using the Coordinate Plane to Demonstrate Transformations
8.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 Figures
8.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 Transversal
8.G.6Explain a proof of the Pythagorean Theorem and its converse.Determining Unknown Side Lengths in Right Triangles of Polygons and Solids
8.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 Solids
8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Finding Distance Between Two Points on the Coordinate Plane
8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Solving Problems Involving Volume

Texas Essential Knowledge and Skills for Mathematics - 2012

6

Whole Numbers

111.26.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 Operations
111.26.b.3.EMultiply and divide positive rational numbers fluently.Evaluating Expressions Using the Order of Operations
111.26.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 Expressions
111.26.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 Expressions
111.26.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 Expressions
111.26.b.7.AGenerate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization;Evaluating Expressions Using the Order of Operations
111.26.b.7.BDistinguish between expressions and equations verbally, numerically, and algebraically;Evaluating Expressions|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions
111.26.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 Terms

Integers

111.26.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 Integers
111.26.b.2.BIdentify a number, its opposite, and its absolute value;Identifying Opposite Integers and Absolute Value
111.26.b.2.CLocate, compare, and order integers and rational numbersComparing Integers|Graphing Integers|Identifying Integers
111.26.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 Integers
111.26.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 Integers
111.26.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 Integers
111.26.b.3.EMultiply and divide positive rational numbers fluently.Combining Integer Multiplication and Division|Multiplying and Dividing Integers
111.26.b.9The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships)Comparing Integers|Graphing Integers
111.26.b.9.BRepresent solutions for one-variable, one-step equations and inequalities on number lines;Comparing Integers|Graphing Integers

Fractions & Decimals

111.26.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 Fraction
111.26.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 Irrational
111.26.b.2.CLocate, compare, and order integers and rational numbersComparing Fractions and Decimals
111.26.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 Fraction
111.26.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 Denominators
111.26.b.3.ARecognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values;Dividing Fractions
111.26.b.3.EMultiply and divide positive rational numbers fluently.Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions
111.26.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 Decimal
111.26.b.4.CGive examples of ratios as multiplicative comparisons of two quantities describing the same attribute;Writing Fractions in Simplest Form
111.26.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 Decimals
111.26.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 Decimal
111.26.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 Decimal
111.26.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 Decimal

Exponents & Roots

111.26.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 Notation
111.26.b.2.CLocate, compare, and order integers and rational numbersComparing Numbers in Scientific Notation
111.26.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 Numbers
111.26.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 Numbers

Rates, Ratios, & Proportions

111.26.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 Form
111.26.b.2.CLocate, compare, and order integers and rational numbersComparing Fractions, Decimals, Percents, and Ratios
111.26.b.2.DOrder a set of rational numbers arising from mathematical and real-world contexts;Comparing Fractions, Decimals, Percents, and Ratios
111.26.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 Form
111.26.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 Ratios
111.26.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 Ratios
111.26.b.4.CGive examples of ratios as multiplicative comparisons of two quantities describing the same attribute;Expressing Ratios in Simplified Form
111.26.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 Ratios
111.26.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 Fraction
111.26.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 Fraction
111.26.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 Fraction
111.26.b.4.HConvert units within a measurement system, including the use of proportions and unit rates.Identify Actual Measurements and Scale Factors
111.26.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 Terms
111.26.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 Terms
111.26.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 Change
111.26.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 Fraction

Equations

111.26.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 Problems
111.26.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 Problems
111.26.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 Problems
111.26.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 Problems
111.26.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 Problems
111.26.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 Problems
111.26.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 Problems
111.26.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 Problems
111.26.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 Problems
111.26.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 Problems
111.26.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 Variables
111.26.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 Problems
111.26.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 Problems
111.26.b.9.CWrite corresponding real-world problems given one-variable, one-step equations or inequalities.Writing One-Step Equations and Solving Word Problems

Inequalities

111.26.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 Inequalities
111.26.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 Inequalities
111.26.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 Division
111.26.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 Inequalities
111.26.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 Inequalities
111.26.b.9.BRepresent solutions for one-variable, one-step equations and inequalities on number lines;Graphing Inequalities on a Number Line|Writing Inequalities Using Graphs
111.26.b.9.CWrite corresponding real-world problems given one-variable, one-step equations or inequalities.Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities

Graphing

111.26.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-Intercepts
111.26.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 Forms
111.26.b.6.AIdentify independent and dependent quantities from tables and graphs;Identifying Domain and Range|Identifying Independent and Dependent Variables
111.26.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 Forms

Probability & Statistics

111.26.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 Display
111.26.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 Display
111.26.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 Display
111.26.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 Display
111.26.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 Display
111.26.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 Set

Geometry

111.26.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 Volume
111.26.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 Volume
111.26.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 Plane
111.26.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 Figure
111.26.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 Plane
111.26.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 Plane
111.26.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 Plane

7

Whole Numbers

111.27.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 Operations
111.27.b.3.AAdd, subtract, multiply, and divide rational numbers fluently;Evaluating Expressions Using the Order of Operations
111.27.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 Operations
111.27.b.4The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)Evaluating Expressions
111.27.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 Expressions

Integers

111.27.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 Integers
111.27.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 Integers
111.27.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 Addition

Fractions & Decimals

111.27.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 Irrational
111.27.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 Denominators
111.27.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 Denominators
111.27.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 Numbers
111.27.b.4The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality)Converting Fractions to Decimals
111.27.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 Decimals

Rates, Ratios, & Proportions

111.27.b.13.ACalculate the sales tax for a given purchase and calculate income tax for earned wages ;Applying the Percent Proportion
111.27.b.13.ECalculate and compare simple interest and compound interest earnings;Computing Simple Interest
111.27.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 Terms
111.27.b.4.BCalculate unit rates from rates in mathematical and real-world problems;Expressing Unit Rates|Solving Practical Problems Using Ratios
111.27.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 Terms
111.27.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 Terms
111.27.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 Factors
111.27.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 Factors
111.27.b.5.CSolve mathematical and real-world problems involving similar shape and scale drawings.Identify Actual Measurements and Scale Factors

Equations

111.27.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 Equations
111.27.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 Problems
111.27.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 Subtraction
111.27.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 Subtraction

Inequalities

111.27.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 Problems
111.27.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 Graphs
111.27.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 Problems
111.27.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 Subtraction
111.27.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 Subtraction

Graphing

111.27.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 Patterns
111.27.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 Patterns
111.27.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 Variation
111.27.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 & Statistics

111.27.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 Set
111.27.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 Population
111.27.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 Sets
111.27.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 Predictions
111.27.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 Event
111.27.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 Predictions
111.27.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 Predictions
111.27.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 Predictions
111.27.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 Predictions
111.27.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 Population
111.27.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 Display
111.27.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 Predictions
111.27.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 Predictions

Geometry

111.27.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 Figure
111.27.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 Transformations
111.27.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 Transformations
111.27.b.5.BDescribe pi () as the ratio of the circumference of a circle to its diameter;Solving Problems Involving Area and Circumference of a Circle
111.27.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 Transformations
111.27.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 Circle
111.27.b.8.AThe student applies mathematical process standards to develop geometric relationships with volume. (Expressions, equations, and relationships)Solving Problems Involving Volume
111.27.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 Volume
111.27.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 Volume
111.27.b.9.ASolve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids;Solving Problems Involving Volume
111.27.b.9.BDetermine the circumference and area of circles;Solving Problems Involving Area and Circumference of a Circle
111.27.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 Plane
111.27.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 Area

8

Integers

111.28.b.2The student applies mathematical process standards to represent and use real numbers in a variety of forms.Comparing Integers
111.28.b.2.DOrder a set of real numbers arising from mathematical and real-world contexts.Comparing Integers

Fractions & Decimals

111.28.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 Irrational
111.28.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 Irrational
111.28.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 Irrational
111.28.b.2.DOrder a set of real numbers arising from mathematical and real-world contexts.Comparing Fractions and Decimals

Exponents & Roots

111.28.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 Notation
111.28.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 Square
111.28.b.2.CConvert between standard decimal notation and scientific notationConverting from Scientific Notation to Standard Form|Writing in Scientific Notation
111.28.b.2.DOrder a set of real numbers arising from mathematical and real-world contexts.Comparing Numbers in Scientific Notation

Rates, Ratios, & Proportions

111.28.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 Interest
111.28.b.12.ASolve real-world problems comparing how interest rate and loan length affect the cost of credit;Computing Simple Interest
111.28.b.12.DCalculate and compare simple interest and compound interest earnings;Computing Simple Interest
111.28.b.2The student applies mathematical process standards to represent and use real numbers in a variety of forms.Comparing Fractions, Decimals, Percents, and Ratios
111.28.b.2.DOrder a set of real numbers arising from mathematical and real-world contexts.Comparing Fractions, Decimals, Percents, and Ratios

Equations

111.28.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 Sides
111.28.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 Sides

Graphing

111.28.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 Points
111.28.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 Variation
111.28.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 Variation
111.28.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 Variation
111.28.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 Variation
111.28.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 Variation
111.28.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 Forms
111.28.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 Variation
111.28.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 Graphically

Probability & Statistics

111.28.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 Population
111.28.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 Association
111.28.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 Set
111.28.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 Population
111.28.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 Data
111.28.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 Data
111.28.b.5.DUse a trend line that approximates the linear relationship between bivariate sets of data to make predictions;Solving Problems Involving Bivariate Measurement Data

Geometry

111.28.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 Transformations
111.28.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 Transformations
111.28.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 Transformations
111.28.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 Transformations
111.28.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 Transformations
111.28.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 Transformations
111.28.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 Transformations
111.28.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 Transformations
111.28.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 Transformations
111.28.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 Volume
111.28.b.6.ADescribe the volume formula V = Bh of a cylinder in terms of its base area and its height;Solving Problems Involving Volume
111.28.b.6.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 Plane
111.28.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 Volume
111.28.b.7.ASolve problems involving the volume of cylinders, cones, and spheres;Solving Problems Involving Volume
111.28.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 Area
111.28.b.7.CUse the Pythagorean Theorem and its converse to solve problemsDetermining Unknown Side Lengths in Right Triangles of Polygons and Solids
111.28.b.7.DDetermine the distance between two points on a coordinate plane using the Pythagorean Theorem.Finding Distance Between Two Points on the Coordinate Plane
111.28.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 Transversal
111.28.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 Transversal

Virginia Mathematics Standards of Learning - 2009

6

Whole Numbers

6.18Solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.Simplifying Expressions by Combining Like Terms
6.19aInvestigate and recognize the identity properties for addition and multiplication; Identifying Properties of Addition and Multiplication
6.19bInvestigate and recognize the multiplicative property of zero; Identifying Properties of Addition and Multiplication
6.8Evaluate whole number numerical expressions, using the order of operations.Evaluating Expressions|Evaluating Expressions Using the Order of Operations

Integers

6.2dCompare and order fractions, decimals, and percents.Comparing Integers
6.3aIdentify and represent integers;Identifying Integers|Modeling Real Life Using Integers
6.3bOrder and compare integers;Comparing Integers|Graphing Integers
6.3cIdentify and describe absolute value of integers.Graphing Integers|Identifying Opposite Integers and Absolute Value

Fractions & Decimals

6.19cInvestigate and recognize the inverse property for multiplication.Dividing Fractions
6.2bIdentify 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 Form
6.2cDemonstrate equivalent relationships among fractions, decimals, and percents;Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions
6.2dCompare and order fractions, decimals, and percents.Comparing Fractions and Decimals
6.4Demonstrate multiple representations of multiplication and division of fractions.Dividing Fractions|Multiplying Fractions
6.6aMultiply 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 Decimal
6.6bEstimate 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 Decimal
6.7Solve 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 Decimal

Exponents & Roots

6.2cDemonstrate equivalent relationships among fractions, decimals, and percents;Converting from Scientific Notation to Standard Form
6.2dCompare and order fractions, decimals, and percents.Comparing Numbers in Scientific Notation
6.5Investigate and describe concepts of positive exponents and perfect squares.Evaluating All Powers|Identifying Bases and Exponents|Identifying the Square Root of a Perfect Square
6.8Evaluate whole number numerical expressions, using the order of operations.Evaluating All Powers

Rates, Ratios, & Proportions

6.1Describe 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 Ratios
6.2aInvestigate and describe fractions, decimals and percents as ratios;Expressing Ratios in Simplified Form
6.2bIdentify a given fraction, decimal or percent from a representation;Expressing a Fraction as a Percent
6.2cDemonstrate 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 Fraction
6.2dCompare and order fractions, decimals, and percents.Comparing Fractions, Decimals, Percents, and Ratios

Equations

6.18Solve 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 Problems

Inequalities

6.20Graph inequalities on a number line.Graphing Inequalities on a Number Line|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities

Graphing

6.11aIdentify the coordinates of a point in a coordinate plane; Identifying an Ordered Pair Given a Graphed Point
6.11bGraph ordered pairs in a coordinate plane. Identifying Quadrants and Graphing an Ordered Pair|Identifying the Distance Between Two Points with Identical x- or y-Values
6.12Determine congruence of segments, angles, and polygons.Identifying the Distance Between Two Points with Identical x- or y-Values
6.17Identify and extend geometric and arithmetic sequences.Predicting Values in Tables Using Numerical Patterns

Probability & Statistics

6.14aConstruct circle graphs; Representing a Set of Data Using a Data Display
6.14bDraw 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 Display
6.15aDescribe 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 Sets
6.16aCompare and contrast dependent and independent events;Identifying the Outcomes in a Sample Space
6.16bDetermine 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 Event
6.2bIdentify a given fraction, decimal or percent from a representation;Representing a Set of Data Using a Data Display

Geometry

6.10aDefine pi (π) as the ratio of the circumference of a circle to its diameter;Solving Problems Involving Area and Circumference of a Circle
6.10bSolve practical problems involving circumference and area of a circle, given the diameter or radius; Solving Problems Involving Area and Circumference of a Circle
6.10cSolve 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 Plane
6.10dDescribe 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 Volume
6.12Determine congruence of segments, angles, and polygons.Analyzing Triangles|Identifying and Classifying Polygons|Using Congruency Statements to Identify Corresponding Parts of a Polygon
6.13Describe and identify properties of quadrilaterals.Identifying Polygons|Identifying Quadrilaterals|Identifying and Classifying Polygons

7

Whole Numbers

7.13aWrite verbal expressions as algebraic expressions and sentences as equations and vice versa;Writing Algebraic Expressions
7.13bEvaluate algebraic expressions for given replacement values of the variables. Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Simplifying Expressions by Combining Like Terms
7.16aApply the following properties of operations with real numbers: the commutative and associative properties for addition and multiplication;Identifying Properties of Addition and Multiplication
7.16cApply the following properties of operations with real numbers: the additive and multiplicative identity properties;Identifying Properties of Addition and Multiplication
7.16eApply the following properties of operations with real numbers: the multiplicative property of zero.Identifying Properties of Addition and Multiplication

Integers

7.16dApply the following properties of operations with real numbers: the additive and multiplicative inverse properties;Adding Integers with Unlike Signs
7.1eIdentify and describe absolute value for rational numbers.Comparing Integers|Identifying Opposite Integers and Absolute Value
7.3aModel 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 Integers
7.3bAdd, 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 Integers

Fractions & Decimals

7.16dApply the following properties of operations with real numbers: the additive and multiplicative inverse properties;Dividing Fractions

Exponents & Roots

7.1aInvestigate and describe the concept of negative exponents for powers of ten;Representing Expressions with Only Positive Exponents
7.1bDetermine scientific notation for numbers greater than zero;Converting from Scientific Notation to Standard Form|Writing in Scientific Notation
7.1cCompare and order fractions, decimals, percents and numbers written in scientific notation;Comparing Numbers in Scientific Notation
7.1dDetermine square roots;Identifying the Square Root of a Perfect Square

Rates, Ratios, & Proportions

7.4Solve 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 Terms

Equations

7.13aWrite verbal expressions as algebraic expressions and sentences as equations and vice versa;Writing One-Step Equations and Solving Word Problems
7.14aSolve 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 Sides
7.14bSolve 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 Problems
7.16bApply 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 Variables

Inequalities

7.15aSolve 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 Inequalities
7.15bGraph solutions to inequalities on the number line. Graphing Inequalities on a Number Line|Writing Inequalities Using Graphs|Writing Mathematical Sentences Using Inequalities

Graphing

7.12Represent relationships with tables, graphs, rules, and words.Determining if a Relation is a Function|Identifying Domain and Range|Representing Relations and Functions in Different Forms
7.2Describe and represent arithmetic and geometric sequences using variable expressions. Predicting Values in Tables Using Numerical Patterns

Probability & Statistics

7.10Determine the probability of compound events, using the Fundamental (Basic) Counting Principle. Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space
7.11aGiven 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 Display
7.11bCompare 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 Display
7.9Investigate 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 Predictions

Geometry

7.5aDescribe volume and surface area of cylinders;Solving Problems Involving Volume
7.5bSolve 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 Volume
7.5cDescribe how changing one measured attribute of a rectangular prism affects its volume and surface area.Solving Problems Involving Surface Area|Solving Problems Involving Volume
7.6Determine 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 Figures
7.7Compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid.Identifying Polygons|Identifying Quadrilaterals|Identifying and Classifying Polygons
7.8Given 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 Transformations

8

Whole Numbers

8.15aSolve multistep linear equations in one variable on one and two sides of the equation;Simplifying Expressions by Combining Like Terms
8.15bSolve two-step linear inequalities and graph the results on a number line;Simplifying Expressions by Combining Like Terms
8.15cIdentify properties of operations used to solve an equation.Identifying Properties of Addition and Multiplication
8.1aSimplify 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 Expressions
8.4Apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Evaluating Expressions|Evaluating Expressions Using the Order of Operations

Integers

8.15cIdentify properties of operations used to solve an equation.Adding Integers with Unlike Signs
8.1bCompare and order decimals, fractions, percents, and numbers written in scientific notation.Comparing Integers
8.2Describe orally and in writing the relationships between the subsets of the real number system.Identifying Integers

Fractions & Decimals

8.15cIdentify properties of operations used to solve an equation.Dividing Fractions
8.1aSimplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;Classifying Numbers|Defining Rational and Irrational
8.1bCompare and order decimals, fractions, percents, and numbers written in scientific notation.Comparing Fractions and Decimals|Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions
8.2Describe orally and in writing the relationships between the subsets of the real number system.Classifying Numbers|Defining Rational and Irrational

Exponents & Roots

8.1aSimplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers;Evaluating All Powers
8.1bCompare 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 Notation
8.4Apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Identifying the Square Root of a Perfect Square
8.5aDetermine whether a given number is a perfect squareIdentifying the Square Root of a Perfect Square
8.5bFind the two consecutive whole numbers between which a square root lies.Estimating Square Roots

Rates, Ratios, & Proportions

8.1bCompare 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 Fraction
8.3aSolve 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 Terms
8.3bDetermine the percent increase or decrease for a given situation.Identifying the Percent of Change

Equations

8.15aSolve 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 Sides
8.15cIdentify properties of operations used to solve an equation.Applying the Distributive Property to Solve Equations
8.1aSimplify 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 Variables

Inequalities

8.15bSolve 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 Subtraction

Graphing

8.14Make 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 Forms
8.16Graph 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 Variation
8.17Identify the domain, range, independent variable or dependent variable in a given situation. Identifying Domain and Range|Identifying Independent and Dependent Variables

Probability & Statistics

8.12Determine the probability of independent and dependent events with and without replacement.Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space
8.13aMake comparisons, predictions, and inferences, using information displayed in graphs;Interpreting Scatter Plots to Investigate Patterns of Association|Solving Problems Involving Bivariate Measurement Data

Geometry

8.10aVerify the Pythagorean TheoremDetermining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane
8.10bApply the Pythagorean Theorem. Determining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane
8.11Solve 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 Plane
8.6aVerify 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 Figure
8.6bMeasure angles of less than 360°.Classifying Angles
8.7aInvestigate 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 Volume
8.7bDescribe how changing one measured attribute of the figure affects the volume and surface area.Solving Problems Involving Surface Area|Solving Problems Involving Volume
8.8aApply 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 Transformations
8.8bIdentify 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 Transformations
8.9Construct a three-dimensional model, given the top or bottom, side, and front views.Investigating Surface Area and Cross-Sections of Solids

Story

Story Trailer
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