| 6.EE.A.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| 6.EE.A.2 | Write, 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.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| 6.EE.A.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 6.EE.A.2c | Evaluate 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.3 | Apply 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.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| 6.EE.B.6 | Use variables to represent numbers and write expressions when solving a 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.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| 6.NS.B.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| 6.NS.C.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.C.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| 6.NS.C.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| 6.NS.C.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.C.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| 6.NS.C.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| 6.NS.C.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| 6.NS.C.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| 6.NS.C.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.NS.C.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.RP.A.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| 6.RP.A.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| 6.RP.A.3 | Use ratio and rate reasoning to solve 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.3a | Make 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.3b | Solve 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.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| 6.EE.A.2 | Write, 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.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| 6.EE.A.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.B.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.7 | Solve 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.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.B.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.8 | Write 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.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| 6.NS.C.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| 6.EE.C.9 | Use 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.C.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| 6.NS.C.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.C.8 | Solve 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.3 | Use 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.3a | Make 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 |
| 6.SP.A.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| 6.SP.A.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.A.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 6.SP.B.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| 6.SP.B.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.B.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.A.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| 6.G.A.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.G.A.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| 6.G.A.4 | Represent 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.8 | Solve 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.EE.A.1 | Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. | Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms |
| 7.EE.A.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| 7.EE.B.3 | Solve 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| 7.NS.A.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| 7.NS.A.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.A.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| 7.NS.A.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| 7.NS.A.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.A.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| 7.NS.A.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| 7.EE.B.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.A.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| 7.NS.A.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.A.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| 7.NS.A.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| 7.NS.A.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.B.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.A.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| 7.NS.A.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.A.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.A.2b | Understand 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.A.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| 7.NS.A.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.B.3 | Solve 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.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| 7.RP.A.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| 7.RP.A.2c | Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| 7.RP.A.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| 7.EE.A.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.A.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.A.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.RP.A.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.A.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.A.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| 7.SP.A.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| 7.SP.A.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| 7.SP.B.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 7.SP.B.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| 7.SP.C.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.C.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.C.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.C.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.C.8 | Find 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.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.C.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.C.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| 7.G.A.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| 7.G.A.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| 7.G.A.3 | Describe 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.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| 7.G.B.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.6 | Solve 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.EE.B.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.6 | Use 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.8 | Analyze 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.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| 8.EE.C.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| 8.EE.C.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| 8.F.A.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| 8.SP.A.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| 8.SP.A.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.A.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.A.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| 8.G.A.1 | Verify 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.1a | Lines 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.1b | Angles 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.1c | Parallel 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.2 | Understand 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.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| 8.G.A.4 | Understand 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.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| 8.G.B.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.B.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.B.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| 8.G.C.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.EE.A.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| 6.EE.A.2 | Write, 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.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| 6.EE.A.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 6.EE.A.2c | Evaluate 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.3 | Apply 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.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| 6.EE.B.6 | Use variables to represent numbers and write expressions when solving a 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.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| 6.NS.B.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| 6.NS.C.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.C.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| 6.NS.C.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| 6.NS.C.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.C.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| 6.NS.C.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| 6.NS.C.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| 6.NS.C.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| 6.NS.C.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.NS.C.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.RP.A.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| 6.RP.A.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| 6.RP.A.3 | Use ratio and rate reasoning to solve 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.3a | Make 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.3b | Solve 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.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| 6.EE.A.2 | Write, 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.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| 6.EE.A.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.B.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.7 | Solve 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.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.B.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.8 | Write 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.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| 6.NS.C.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| 6.EE.C.9 | Use 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.C.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| 6.NS.C.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.C.8 | Solve 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.3 | Use 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.3a | Make 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 |
| 6.SP.A.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| 6.SP.A.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.A.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 6.SP.B.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| 6.SP.B.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.B.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.A.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| 6.G.A.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.G.A.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| 6.G.A.4 | Represent 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.8 | Solve 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.EE.A.1 | Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. | Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms |
| 7.EE.A.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| 7.EE.B.3 | Solve 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| 7.NS.A.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| 7.NS.A.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.A.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| 7.NS.A.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| 7.NS.A.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.A.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| 7.NS.A.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| 7.EE.B.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.A.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| 7.NS.A.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.A.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| 7.NS.A.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| 7.NS.A.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.B.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.A.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| 7.NS.A.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.A.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.A.2b | Understand 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.A.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| 7.NS.A.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.B.3 | Solve 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.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| 7.RP.A.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| 7.RP.A.2c | Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| 7.RP.A.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| 7.EE.A.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.A.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.A.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.RP.A.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.A.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.A.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| 7.SP.A.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| 7.SP.A.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| 7.SP.B.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 7.SP.B.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| 7.SP.C.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.C.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.C.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.C.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.C.8 | Find 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.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.C.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.C.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| 7.G.A.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| 7.G.A.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| 7.G.A.3 | Describe 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.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| 7.G.B.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.6 | Solve 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.EE.B.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.6 | Use 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.8 | Analyze 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.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| 8.EE.C.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| 8.EE.C.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| 8.F.A.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| 8.SP.A.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| 8.SP.A.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.A.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.A.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| 8.G.A.1 | Verify 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.1a | Lines 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.1b | Angles 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.1c | Parallel 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.2 | Understand 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.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| 8.G.A.4 | Understand 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.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| 8.G.B.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.B.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.B.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| 8.G.C.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.EE.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| 6.EE.2 | Write, 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.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| 6.EE.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 6.EE.2c | Evaluate 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.3 | Apply 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.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| 6.EE.6 | Use variables to represent numbers and write expressions when solving a 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.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| 6.NS.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| 6.NS.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| 6.NS.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| 6.NS.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.RP.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| 6.RP.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| 6.RP.3 | Use ratio and rate reasoning to solve 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.3a | Make 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.3b | Solve 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.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| 6.EE.2 | Write, 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.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| 6.EE.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.7 | Solve 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.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.8 | Write 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.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| 6.EE.9 | Use 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.8 | Solve 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.3 | Use 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.3a | Make 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 |
| 6.SP.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| 6.SP.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 6.SP.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| 6.SP.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| 6.G.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.G.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| 6.G.4 | Represent 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.8 | Solve 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.EE.1 | Apply 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.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| 7.EE.3 | Solve 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| 7.NS.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| 7.NS.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| 7.NS.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2b | Understand 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| 7.RP.2c | Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| 7.RP.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| 7.EE.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.RP.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| 7.SP.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| 7.SP.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| 7.SP.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 7.SP.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| 7.SP.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.8 | Find 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.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| 7.G.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| 7.G.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| 7.G.3 | Describe 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.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| 7.G.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.6 | Solve 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.EE.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.6 | Use 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.8 | Analyze 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.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| 8.F.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| 8.SP.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| 8.SP.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| 8.G.1 | Verify 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.1a | Lines 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.1b | Angles 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.1c | Parallel 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.2 | Understand 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.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| 8.G.4 | Understand 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.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| 8.G.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| 8.G.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| MACC.6.EE.1.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| MACC.6.EE.1.2 | Write, 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.1.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| MACC.6.EE.1.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| MACC.6.EE.1.2c | Evaluate 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.1.3 | Apply 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.1.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| MACC.6.EE.2.6 | Use variables to represent numbers and write expressions when solving a 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.2.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| MACC.6.NS.2.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| MACC.6.NS.3.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.3.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| MACC.6.NS.3.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| MACC.6.NS.3.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| MACC.6.NS.3.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| MACC.6.NS.3.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| MACC.6.NS.3.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| MACC.6.NS.3.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| MACC.6.NS.3.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| MACC.6.NS.3.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| MACC.6.NS.3.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| MACC.6.RP.1.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| MACC.6.RP.1.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| MACC.6.RP.1.3 | Use ratio and rate reasoning to solve 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.1.3a | Make 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.1.3b | Solve 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.1.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| MACC.6.EE.1.2 | Write, read, and evaluate expressions in which letters stand for numbers. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| MACC.6.EE.1.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| MACC.6.EE.1.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MACC.6.EE.2.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.2.7 | Solve 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.2.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MACC.6.EE.2.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.2.8 | Write 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.3.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| MACC.6.NS.3.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| MACC.6.EE.3.9 | Use 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.3.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| MACC.6.NS.3.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| MACC.6.NS.3.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| MACC.6.NS.3.8 | Solve 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.1.3 | Use 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.1.3a | Make 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 |
| MACC.6.SP.1.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| MACC.6.SP.1.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| MACC.6.SP.1.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| MACC.6.SP.2.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| MACC.6.SP.2.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.2.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| MACC.6.SP.2.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.2.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.2.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.1.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| MACC.6.G.1.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| MACC.6.G.1.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| MACC.6.G.1.4 | Represent 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.3.8 | Solve 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 |
| MACC.7.EE.1.1 | Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. | Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms |
| MACC.7.EE.1.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| MACC.7.EE.2.3 | Solve 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.2.4 | Use 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.2.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| MACC.7.NS.1.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| MACC.7.NS.1.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| MACC.7.NS.1.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| MACC.7.NS.1.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| MACC.7.NS.1.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| MACC.7.NS.1.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| MACC.7.NS.1.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| MACC.7.EE.2.3 | Solve 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.1.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| MACC.7.NS.1.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| MACC.7.NS.1.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| MACC.7.NS.1.1d | Apply 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.1.2 | Apply 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.1.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| MACC.7.NS.1.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.1.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| MACC.7.NS.1.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| MACC.7.EE.2.3 | Solve 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.1.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| MACC.7.NS.1.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| MACC.7.NS.1.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| MACC.7.NS.1.1d | Apply 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.1.2 | Apply 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.1.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| MACC.7.NS.1.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.1.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| MACC.7.NS.1.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| MACC.7.NS.1.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| MACC.7.EE.2.3 | Solve 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.1.2 | Recognize 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.1.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| MACC.7.RP.1.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| MACC.7.RP.1.2c | Represent proportional relationships by equations. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| MACC.7.RP.1.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| MACC.7.EE.1.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.2.4 | Use 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.2.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.1.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MACC.7.NS.1.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MACC.7.NS.1.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MACC.7.RP.1.2 | Recognize 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.1.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| MACC.7.RP.1.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| MACC.7.RP.1.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| MACC.7.SP.1.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| MACC.7.SP.1.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| MACC.7.SP.2.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| MACC.7.SP.2.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.3.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| MACC.7.SP.3.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| MACC.7.SP.3.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| MACC.7.SP.3.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| MACC.7.SP.3.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| MACC.7.SP.3.8 | Find 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.3.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| MACC.7.SP.3.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| MACC.7.SP.3.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| MACC.7.G.1.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| MACC.7.G.1.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| MACC.7.G.1.3 | Describe 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.2.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| MACC.7.G.2.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.2.6 | Solve 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 |
| MACC.8.EE.2.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.2.6 | Use 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.3.8 | Analyze 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.3.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| MACC.8.EE.3.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| MACC.8.EE.3.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| MACC.8.F.1.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.1.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.1.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.2.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.2.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.1.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| MACC.8.SP.1.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| MACC.8.SP.1.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| MACC.8.SP.1.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| MACC.8.SP.1.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| MACC.8.G.1.1 | Verify 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.1.1a | Lines 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.1.1b | Angles 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.1.1c | Parallel 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.1.2 | Understand 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.1.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| MACC.8.G.1.4 | Understand 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.1.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| MACC.8.G.2.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| MACC.8.G.2.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| MACC.8.G.2.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| MACC.8.G.3.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| MCC6.EE.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| MCC6.EE.2 | Write, 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.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| MCC6.EE.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| MCC6.EE.2c | Evaluate 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.3 | Apply 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.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| MCC6.EE.6 | Use variables to represent numbers and write expressions when solving a 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.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| MCC6.NS.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| MCC6.NS.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| MCC6.NS.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| MCC6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| MCC6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| MCC6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| MCC6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| MCC6.NS.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| MCC6.NS.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| MCC6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| MCC6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| MCC6.RP.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| MCC6.RP.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| MCC6.RP.3 | Use ratio and rate reasoning to solve 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.3a | Make 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.3b | Solve 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.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| MCC6.EE.2 | Write, 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.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| MCC6.EE.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MCC6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.7 | Solve 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.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MCC6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.8 | Write 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.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| MCC6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| MCC6.EE.9 | Use 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| MCC6.NS.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| MCC6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| MCC6.NS.8 | Solve 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.3 | Use 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.3a | Make 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 |
| MCC6.SP.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| MCC6.SP.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| MCC6.SP.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| MCC6.SP.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| MCC6.SP.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| MCC6.SP.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| MCC6.G.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| MCC6.G.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| MCC6.G.4 | Represent 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.8 | Solve 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 |
| MCC7.EE.1 | Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. | Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms |
| MCC7.EE.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| MCC7.EE.3 | Solve 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| MCC7.NS.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| MCC7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| MCC7.NS.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| MCC7.NS.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| MCC7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| MCC7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| MCC7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| MCC7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| MCC7.NS.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| MCC7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| MCC7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| MCC7.NS.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| MCC7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| MCC7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| MCC7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| MCC7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| MCC7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| MCC7.NS.2b | Understand 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| MCC7.NS.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| MCC7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| MCC7.EE.3 | Solve 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.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| MCC7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| MCC7.RP.2c | Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| MCC7.RP.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| MCC7.EE.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MCC7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MCC7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| MCC7.RP.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| MCC7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| MCC7.RP.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| MCC7.SP.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| MCC7.SP.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| MCC7.SP.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| MCC7.SP.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| MCC7.SP.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| MCC7.SP.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| MCC7.SP.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| MCC7.SP.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| MCC7.SP.8 | Find 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.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| MCC7.SP.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| MCC7.SP.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| MCC7.G.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| MCC7.G.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| MCC7.G.3 | Describe 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.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| MCC7.G.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.6 | Solve 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 |
| MCC8.EE.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.6 | Use 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.8 | Analyze 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.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| MCC8.EE.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| MCC8.EE.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| MCC8.F.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| MCC8.SP.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| MCC8.SP.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| MCC8.SP.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| MCC8.SP.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| MCC8.G.1 | Verify 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.1a | Lines 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.1b | Angles 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.1c | Parallel 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.2 | Understand 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.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| MCC8.G.4 | Understand 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.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| MCC8.G.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| MCC8.G.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| MCC8.G.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| MCC8.G.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.EE.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| 6.EE.2 | Write, 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.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| 6.EE.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 6.EE.2c | Evaluate 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.3 | Apply 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.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| 6.EE.6 | Use variables to represent numbers and write expressions when solving a 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.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| 6.NS.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| 6.NS.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| 6.NS.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| 6.NS.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.RP.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| 6.RP.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| 6.RP.3 | Use ratio and rate reasoning to solve 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.3a | Make 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.3b | Solve 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.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| 6.EE.2 | Write, 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.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| 6.EE.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.7 | Solve 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.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.8 | Write 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.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| 6.EE.9 | Use 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.8 | Solve 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.3 | Use 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.3a | Make 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 |
| 6.SP.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| 6.SP.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 6.SP.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| 6.SP.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| 6.G.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.G.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| 6.G.4 | Represent 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.8 | Solve 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.EE.1 | Apply 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.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| 7.EE.3 | Solve 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| 7.NS.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| 7.NS.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| 7.NS.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2b | Understand 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| 7.RP.2c | Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| 7.RP.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| 7.EE.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.RP.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| 7.SP.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| 7.SP.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| 7.SP.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 7.SP.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| 7.SP.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.8 | Find 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.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| 7.G.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| 7.G.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| 7.G.3 | Describe 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.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| 7.G.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.6 | Solve 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.EE.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.6 | Use 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.8 | Analyze 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.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| 8.F.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| 8.SP.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| 8.SP.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| 8.G.1 | Verify 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.1a | Lines 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.1b | Angles 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.1c | Parallel 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.2 | Understand 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.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| 8.G.4 | Understand 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.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| 8.G.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| 8.G.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.EE.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| 6.EE.2 | Write, 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.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| 6.EE.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 6.EE.2c | Evaluate 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.3 | Apply 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.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| 6.EE.6 | Use variables to represent numbers and write expressions when solving a 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.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| 6.NS.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| 6.NS.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| 6.NS.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| 6.NS.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.RP.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| 6.RP.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| 6.RP.3 | Use ratio and rate reasoning to solve 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.3a | Make 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.3b | Solve 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.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| 6.EE.2 | Write, 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.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| 6.EE.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.7 | Solve 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.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.8 | Write 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.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| 6.EE.9 | Use 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.8 | Solve 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.3 | Use 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.3a | Make 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 |
| 6.SP.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| 6.SP.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 6.SP.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| 6.SP.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| 6.G.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.G.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| 6.G.4 | Represent 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.8 | Solve 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.EE.1 | Apply 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.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| 7.EE.3 | Solve 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| 7.NS.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| 7.NS.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| 7.NS.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2b | Understand 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| 7.RP.2c | Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| 7.RP.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| 7.EE.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.RP.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| 7.SP.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| 7.SP.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| 7.SP.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 7.SP.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| 7.SP.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.8 | Find 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.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| 7.G.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| 7.G.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| 7.G.3 | Describe 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.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| 7.G.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.6 | Solve 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.EE.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.6 | Use 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.8 | Analyze 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.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| 8.F.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| 8.SP.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| 8.SP.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| 8.G.1 | Verify 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.1a | Lines 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.1b | Angles 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.1c | Parallel 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.2 | Understand 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.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| 8.G.4 | Understand 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.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| 8.G.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| 8.G.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.EE.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| 6.EE.2 | Write, 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.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| 6.EE.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 6.EE.2c | Evaluate 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.3 | Apply 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.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| 6.EE.6 | Use variables to represent numbers and write expressions when solving a 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.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| 6.NS.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| 6.NS.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| 6.NS.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| 6.NS.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.RP.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| 6.RP.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| 6.RP.3 | Use ratio and rate reasoning to solve 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.3a | Make 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.3b | Solve 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.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| 6.EE.2 | Write, 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.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| 6.EE.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.7 | Solve 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.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.8 | Write 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.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| 6.EE.9 | Use 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.8 | Solve 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.3 | Use 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.3a | Make 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 |
| 6.SP.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| 6.SP.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 6.SP.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| 6.SP.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| 6.G.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.G.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| 6.G.4 | Represent 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.8 | Solve 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.EE.1 | Apply 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.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| 7.EE.3 | Solve 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| 7.NS.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| 7.NS.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| 7.NS.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2b | Understand 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| 7.RP.2c | Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| 7.RP.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| 7.EE.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.RP.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| 7.SP.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| 7.SP.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| 7.SP.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 7.SP.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| 7.SP.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.8 | Find 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.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| 7.G.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| 7.G.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| 7.G.3 | Describe 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.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| 7.G.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.6 | Solve 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.EE.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.6 | Use 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.8 | Analyze 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.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| 8.F.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| 8.SP.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| 8.SP.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| 8.G.1 | Verify 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.1a | Lines 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.1b | Angles 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.1c | Parallel 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.2 | Understand 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.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| 8.G.4 | Understand 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.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| 8.G.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| 8.G.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.EE.1 | Write and evaluate numerical expressions involving whole-number exponents. | Evaluating Expressions Using the Order of Operations|Writing Algebraic Expressions |
| 6.EE.2 | Write, 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.2a | Write expressions that record operations with numbers and with letters standing for numbers. | Writing Algebraic Expressions |
| 6.EE.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 6.EE.2c | Evaluate 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.3 | Apply 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.4 | Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). | Simplifying Expressions by Combining Like Terms |
| 6.EE.6 | Use variables to represent numbers and write expressions when solving a 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.2 | Fluently divide multi-digit numbers using the standard algorithm. | Using the Divisibility Rules |
| 6.NS.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Identifying the Greatest Common Factor (GCF)|Identifying the Least Common Multiple (LCM)|Listing Factors |
| 6.NS.5 | Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.6a | Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite. | Identifying Opposite Integers and Absolute Value |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Graphing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Integers|Graphing Integers|Identifying Opposite Integers and Absolute Value |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Integers |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Integers |
| 6.NS.7c | Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. | Identifying Opposite Integers and Absolute Value |
| 6.NS.7d | Distinguish comparisons of absolute value from statements about order. | Comparing Integers |
| 6.NS.7 | Understand ordering and absolute value of rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.NS.7b | Write, interpret, and explain statements of order for rational numbers in real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.RP.1 | Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. | Expressing Ratios in Simplified Form |
| 6.RP.2 | Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. | Expressing Unit Rates |
| 6.RP.3 | Use ratio and rate reasoning to solve 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.3a | Make 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.3b | Solve 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.3c | Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. | Applying the Percent Proportion|Solving Proportions to Find Missing Terms |
| 6.EE.2 | Write, 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.2b | Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. | Applying the Distributive Property to Write Equivalent Expressions with Variables|Writing One-Step Equations and Solving Word Problems |
| 6.EE.3 | Apply the properties of operations to generate equivalent expressions. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.7 | Solve 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.4 | Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 6.EE.5 | Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. | Applying Knowledge of 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.8 | Write 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.7 | Understand ordering and absolute value of rational numbers | Graphing Inequalities on a Number Line |
| 6.NS.7a | Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. | Graphing Inequalities on a Number Line |
| 6.EE.9 | Use 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.6 | Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.6b | Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point |
| 6.NS.6c | Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 6.NS.8 | Solve 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.3 | Use 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.3a | Make 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 |
| 6.SP.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. | Selecting a Representative Sampling Method for a Population |
| 6.SP.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.3 | Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 6.SP.4 | Display numerical data in plots on a number line, including dot plots, histograms, and box plots. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| 6.SP.5 | Summarize numerical data sets in relation to their context, such as by: | Analyzing and Comparing Data 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.5a | Reporting the number of observations. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Describing a Set of Data (Mean, Median, Mode, Range)|Representing a Set of Data Using a Data Display |
| 6.SP.5b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. | Analyzing and Comparing Data 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.5c | Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. | Analyzing and Comparing Data 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.5d | Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. | Analyzing and Comparing Data 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.G.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. | Finding Area of Polygons Using Composing and Decomposing Techniques |
| 6.G.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. | Solving Problems Involving Volume |
| 6.G.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. | Solving Problems Involving Polygons on the Coordinate Plane |
| 6.G.4 | Represent 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.8 | Solve 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.EE.1 | Apply 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.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Writing Algebraic Expressions |
| 7.EE.3 | Solve 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| 7.NS.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Identifying Properties of Addition and Multiplication |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.1d | Apply properties of operations as strategies to add and subtract rational numbers. | Identifying Properties of Addition and Multiplication |
| 7.NS.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Using the Divisibility Rules |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Identifying Properties of Addition and Multiplication |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | Evaluating Expressions Using the Order of Operations |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Integers with Like Signs|Adding Integers with Unlike Signs|Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1a | Describe situations in which opposite quantities combine to make 0. | Adding Integers with Unlike Signs|Identifying Opposite Integers and Absolute Value |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing 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.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Combining Integer Subtraction and Addition|Identifying Opposite Integers and Absolute Value|Subtracting Integers |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying and Dividing Integers |
| 7.NS.2b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators|Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1b | Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. | Adding Decimal Numbers|Adding Like Denominators|Adding Unlike Denominators |
| 7.NS.1c | Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. | Subtracting Decimal Numbers|Subtracting Like Denominators|Subtracting Unlike Denominators |
| 7.NS.1d | Apply 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.2 | Apply 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.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2b | Understand 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.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| 7.NS.2d | Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. | Converting Fractions to Decimals|Writing an Integer as a Fraction or Decimal |
| 7.NS.3 | Solve real-world and mathematical problems involving the four operations with rational numbers.1 | 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 |
| 7.EE.3 | Solve 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.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Determining Proportional Relationships Between Quantities |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Solving Practical Problems Using Ratios |
| 7.RP.2c | Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. | Applying the Percent Proportion|Identify Actual Measurements and Scale Factors|Solving Proportions to Find Missing Terms |
| 7.RP.3 | Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. | Applying the Percent Proportion|Computing Simple Interest|Identifying the Percent of Change |
| 7.EE.2 | Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. | Applying Knowledge of 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.4 | Use 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.4a | Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. | Applying Knowledge of 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.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2a | Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.NS.2c | Apply properties of operations as strategies to multiply and divide rational numbers. | Applying the Distributive Property to Solve Equations|Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 7.RP.2 | Recognize 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.2a | Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. | Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2b | Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. | Determining the Constant of Variation|Identifying a Graph of a Direct Variation|Identifying a Table of a Direct Variation |
| 7.RP.2d | Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. | Finding the Rate of Change |
| 7.SP.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. | Selecting a Representative Sampling Method for a Population |
| 7.SP.2 | Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. | Drawing Inferences About a Population |
| 7.SP.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. | Analyzing and Comparing Data Sets|Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 7.SP.4 | Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. | Analyzing and Comparing Data 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.5 | Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | Determining the Likelihood of Events |
| 7.SP.6 | Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. | Determining the Likelihood of Events|Identifying and Comparing Probabilities|Identifying the Theoretical Probability of an Event|Making Predictions |
| 7.SP.7b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. | Identifying and Comparing Probabilities|Making Predictions |
| 7.SP.8 | Find 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.8a | Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8b | Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes), identify the outcomes in the sample space which compose the event. | Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space |
| 7.SP.8c | Design and use a simulation to generate frequencies for compound events. | Finding the Probabilities of Compound Events |
| 7.G.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. | Solving Problems Involving Scale Drawings of Geometric Figures |
| 7.G.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. | Analyzing Triangles|Identifying and Classifying Polygons |
| 7.G.3 | Describe 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.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. | Solving Problems Involving Area and Circumference of a Circle |
| 7.G.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a 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.6 | Solve 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.EE.5 | Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. | Determining the Constant of 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.6 | Use 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.8 | Analyze 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.8a | Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. | Determining if an Ordered Pair is a Solution to the System|Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8b | Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. | Determining the Solution(s) of the System of Linear Equations|Solving Systems of Two Linear Equations Algebraically|Solving Systems of Two Linear Equations Graphically |
| 8.EE.8c | Solve real-world and mathematical problems leading to two linear equations in two variables. | Solving Systems of Two Linear Equations Algebraically |
| 8.F.1 | Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. | Determining if a Relation is a 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.2 | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). | Calculating the Slope of a Graphed 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.3 | Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. | Classifying Functions as Linear or 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.4 | Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. | Calculating the Slope of a Graphed 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.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. | Calculating the Slope of a Graphed 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.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Finding the Rate of 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 |
| 8.SP.1 | Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. | Interpreting Scatter Plots to Investigate Patterns of Association |
| 8.SP.2 | Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.3 | Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. | Solving Problems Involving Bivariate Measurement Data |
| 8.SP.4 | Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a 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 |
| 8.G.1 | Verify 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.1a | Lines 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.1b | Angles 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.1c | Parallel 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.2 | Understand 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.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. | Using the Coordinate Plane to Demonstrate Transformations |
| 8.G.4 | Understand 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.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. | Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| 8.G.6 | Explain a proof of the Pythagorean Theorem and its converse. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 8.G.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. | Finding Distance Between Two Points on the Coordinate Plane |
| 8.G.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. | Solving Problems Involving Volume |
| 111.26.b.3 | The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. (Number and operations) | Evaluating Expressions Using the Order of Operations |
| 111.26.b.3.E | Multiply and divide positive rational numbers fluently. | Evaluating Expressions Using the Order of Operations |
| 111.26.b.4 | The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality) | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Writing Algebraic Expressions |
| 111.26.b.4.A | Compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships; | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Writing Algebraic Expressions |
| 111.26.b.7 | The 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.A | Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization; | Evaluating Expressions Using the Order of Operations |
| 111.26.b.7.B | Distinguish between expressions and equations verbally, numerically, and algebraically; | Evaluating Expressions|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 111.26.b.7.D | Generate equivalent expressions using the properties of operations : such as the inverse, identity, commutative, associative, and distributive properties. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms |
| 111.26.b.2 | The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations) | Comparing Integers|Graphing Integers|Identifying Integers |
| 111.26.b.2.B | Identify a number, its opposite, and its absolute value; | Identifying Opposite Integers and Absolute Value |
| 111.26.b.2.C | Locate, compare, and order integers and rational numbers | Comparing Integers|Graphing Integers|Identifying Integers |
| 111.26.b.3 | The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. (Number and operations) | Adding Integers with Like 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.C | Represent 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.D | Add, 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.E | Multiply and divide positive rational numbers fluently. | Combining Integer Multiplication and Division|Multiplying and Dividing Integers |
| 111.26.b.9 | The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships) | Comparing Integers|Graphing Integers |
| 111.26.b.9.B | Represent solutions for one-variable, one-step equations and inequalities on number lines; | Comparing Integers|Graphing Integers |
| 111.26.b.2 | The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations) | Classifying Numbers|Comparing Fractions and Decimals|Defining Rational and Irrational|Identifying Parts of a Fraction |
| 111.26.b.2.A | Classify whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers; | Classifying Numbers|Defining Rational and Irrational |
| 111.26.b.2.C | Locate, compare, and order integers and rational numbers | Comparing Fractions and Decimals |
| 111.26.b.2.E | Extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0. | Identifying Parts of a Fraction |
| 111.26.b.3 | The student applies mathematical process standards to represent addition, subtraction, multiplication, and division while solving problems and justifying solutions. (Number and operations) | Adding Decimal 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.A | Recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values; | Dividing Fractions |
| 111.26.b.3.E | Multiply and divide positive rational numbers fluently. | Dividing Decimal Numbers|Dividing Fractions|Multiplying Decimal Numbers|Multiplying Fractions |
| 111.26.b.4 | The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality) | Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions|Writing Fractions in Simplest Form|Writing an Integer as a Fraction or Decimal |
| 111.26.b.4.C | Give examples of ratios as multiplicative comparisons of two quantities describing the same attribute; | Writing Fractions in Simplest Form |
| 111.26.b.4.F | Represent benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers; | Converting Fractions to Decimals |
| 111.26.b.4.G | Generate equivalent forms of fractions, decimals, and percents using 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.5 | The 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.C | Use 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 |
| 111.26.b.2 | The student applies mathematical process standards to represent and use rational numbers in a variety of forms. (Number and operations) | Comparing Fractions, Decimals, Percents, and Ratios|Expressing Ratios in Simplified Form |
| 111.26.b.2.C | Locate, compare, and order integers and rational numbers | Comparing Fractions, Decimals, Percents, and Ratios |
| 111.26.b.2.D | Order a set of rational numbers arising from mathematical and real-world contexts; | Comparing Fractions, Decimals, Percents, and Ratios |
| 111.26.b.2.E | Extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0. | Expressing Ratios in Simplified Form |
| 111.26.b.4 | The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. (Proportionality) | Comparing Fractions, Decimals, Percents, and 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.B | Apply 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.C | Give examples of ratios as multiplicative comparisons of two quantities describing the same attribute; | Expressing Ratios in Simplified Form |
| 111.26.b.4.D | Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients; | Expressing Ratios in Simplified Form|Expressing Unit Rates|Solving Practical Problems Using Ratios |
| 111.26.b.4.E | Represent 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.F | Represent benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers; | Comparing Fractions, Decimals, Percents, and 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.G | Generate 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.H | Convert units within a measurement system, including the use of proportions and unit rates. | Identify Actual Measurements and Scale Factors |
| 111.26.b.5 | The student applies mathematical process standards to solve problems involving proportional relationships. (Proportionality) | Applying the Percent 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.A | Represent 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.B | Solve 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.C | Use 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 |
| 111.26.b.10 | The 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.A | Model 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.B | Determine 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.4 | The 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.A | Compare two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships; | Applying Knowledge of Two-Step Equations to Solve Word Problems|Writing One-Step Equations and Solving Word Problems |
| 111.26.b.6 | The student applies mathematical process standards to use multiple representations to describe algebraic relationships. (Expressions, equations, and relationships) | Writing One-Step Equations and Solving Word Problems |
| 111.26.b.6.C | Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. | Writing One-Step Equations and Solving Word Problems |
| 111.26.b.7 | The student applies mathematical process standards to develop concepts of expressions and equations. (Expressions, equations, and relationships) | Applying Knowledge of 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.B | Distinguish 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.C | Determine 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.D | Generate equivalent expressions using the properties of operations : such as the inverse, identity, commutative, associative, and distributive properties. | Applying the Distributive Property to Write Equivalent Expressions with Variables |
| 111.26.b.9 | The student applies mathematical process standards to use equations and inequalities to represent situations. (Expressions, equations, and relationships) | Solving 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.A | Write 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.C | Write corresponding real-world problems given one-variable, one-step equations or inequalities. | Writing One-Step Equations and Solving Word Problems |
| 111.26.b.10 | The 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.A | Model 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.B | Determine 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.9 | The 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.A | Write 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.B | Represent 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.C | Write corresponding real-world problems given one-variable, one-step equations or inequalities. | Solving Word Problems Using Inequalities|Writing Mathematical Sentences Using Inequalities |
| 111.26.b.11 | The student applies mathematical process standards to use coordinate geometry to identify locations on a plane. The student is expected to graph points in all four quadrants using ordered pairs of rational numbers. (Measurement and data) | Identifying Quadrants and Graphing an Ordered Pair|Identifying an Ordered Pair Given a Graphed Point|Identifying the x- and y-Intercepts |
| 111.26.b.6 | The student applies mathematical process standards to use multiple representations to describe algebraic relationships. (Expressions, equations, and relationships) | Graphing a Linear Function in Two Variables Using Tables|Identifying Domain and Range|Identifying Independent and Dependent Variables|Representing Relations and Functions in Different Forms |
| 111.26.b.6.A | Identify independent and dependent quantities from tables and graphs; | Identifying Domain and Range|Identifying Independent and Dependent Variables |
| 111.26.b.6.C | Represent a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. | Graphing a Linear Function in Two Variables Using Tables|Representing Relations and Functions in Different Forms |
| 111.26.b.12 | The student applies mathematical process standards to use numerical or graphical representations to analyze problems. (Measurement and data) | Analyzing and Comparing Data 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.A | Represent 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.B | Use 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.C | Summarize numeric data with numerical summaries, including the mean and median (measures of center) and the range and interquartile range (IQR) (measures of spread), and use these summaries to describe the center, spread, and shape of the data distribution; | Analyzing and Comparing Data 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.D | Summarize categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and use these summaries to describe the data distribution. | Analyzing and Comparing Data 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.13 | The 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.A | Interpret 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.B | Distinguish 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 |
| 111.26.b.10 | The 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.A | Model 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.8 | The 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.A | Extend previous knowledge of triangles and their properties to include the sum of angles of a triangle, the relationship between the lengths of sides and measures of angles in a triangle, and determining when three lengths form a triangle; | Analyzing Triangles|Classifying Triangles Using Angles and Sides|Identifying and Classifying Polygons|Solving Equations for an Unknown Angle in a Figure |
| 111.26.b.8.B | Model area formulas for parallelograms, trapezoids, and triangles by decomposing and rearranging parts of these shapes; | Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Polygons on the Coordinate Plane |
| 111.26.b.8.C | Write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers; | Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Polygons on the Coordinate Plane |
| 111.26.b.8.D | Determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. | Finding Area of Polygons Using Composing and Decomposing Techniques|Finding the Missing Dimension of a Rectangle or Triangle|Solving Problems Involving Polygons on the Coordinate Plane |
| 111.27.b.3 | The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. (Number and operations) | Evaluating Expressions Using the Order of Operations |
| 111.27.b.3.A | Add, subtract, multiply, and divide rational numbers fluently; | Evaluating Expressions Using the Order of Operations |
| 111.27.b.3.B | Apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers. | Evaluating Expressions Using the Order of Operations |
| 111.27.b.4 | The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality) | Evaluating Expressions |
| 111.27.b.4.A | Represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt; | Evaluating Expressions |
| 111.27.b.2 | The student applies mathematical process standards to represent and use rational numbers in a variety of forms. The student is expected to extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers. (Number and operations) | Classifying Numbers|Defining Rational and Irrational |
| 111.27.b.3 | The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. (Number and operations) | Adding Decimal 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.A | Add, 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.B | Apply 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.4 | The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality) | Converting Fractions to Decimals |
| 111.27.b.4.D | Solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems; | Converting Fractions to Decimals |
| 111.27.b.13.A | Calculate the sales tax for a given purchase and calculate income tax for earned wages ; | Applying the Percent Proportion |
| 111.27.b.13.E | Calculate and compare simple interest and compound interest earnings; | Computing Simple Interest |
| 111.27.b.4 | The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality) | Applying the Percent 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.B | Calculate unit rates from rates in mathematical and real-world problems; | Expressing Unit Rates|Solving Practical Problems Using Ratios |
| 111.27.b.4.C | Determine 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.D | Solve 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.5 | The 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.A | Generalize 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.C | Solve mathematical and real-world problems involving similar shape and scale drawings. | Identify Actual Measurements and Scale Factors |
| 111.27.b.10.A | Write 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.B | Represent 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.C | Write 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.A | Model 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.B | Determine 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 |
| 111.27.b.4 | The student applies mathematical process standards to represent and solve problems involving proportional relationships. (Proportionality) | Determining Whether Functions are Linear or Nonlinear Using 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.A | Represent 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.C | Determine 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.7 | The student applies mathematical process standards to represent linear relationships using multiple representations. The student is expected to represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b. (Expressions, equations, and relationships) | Calculating the Slope of a Graphed 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) |
| 111.27.b.12.A | Compare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads; | Analyzing and Comparing Data 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.B | Use 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.C | Compare two populations based on data in random samples from these populations, including informal comparative inferences about differences between the two populations. | Analyzing and Comparing Data Sets |
| 111.27.b.6 | The student applies mathematical process standards to use probability and statistics to describe or solve problems involving proportional relationships. (Proportionality) | Finding the Probabilities of Compound 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.A | Represent 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.B | Select and use different simulations to represent simple and compound events with and without technology | 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.C | Make 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.D | Make 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.E | Find 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.F | Use 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.G | Solve 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.H | Solve 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.I | Determine 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 |
| 111.27.b.11.C | Write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships. | Analyzing Triangles|Identifying Angle Relationships Formed by Parallel Lines and a Transversal|Solving Equations for an Unknown Angle in a Figure |
| 111.27.b.5 | The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. (Proportionality) | Describing the Sequence of Transformations of Two Similar 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.A | Generalize 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.B | Describe pi () as the ratio of the circumference of a circle to its diameter; | Solving Problems Involving Area and Circumference of a Circle |
| 111.27.b.5.C | Solve mathematical and 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.8 | Use models to determine the approximate formulas for the circumference and area of a circle and connect the models to the actual formulas. | Solving Problems Involving Area and Circumference of a Circle |
| 111.27.b.8.A | The student applies mathematical process standards to develop geometric relationships with volume. (Expressions, equations, and relationships) | Solving Problems Involving Volume |
| 111.27.b.8.B | Model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas; | Solving Problems Involving Volume |
| 111.27.b.8.C | Explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connect that relationship to the formulas; | Solving Problems Involving Volume |
| 111.27.b.9.A | Solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids; | Solving Problems Involving Volume |
| 111.27.b.9.B | Determine the circumference and area of circles; | Solving Problems Involving Area and Circumference of a Circle |
| 111.27.b.9.C | Determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles; | Finding Area of Polygons Using Composing and Decomposing Techniques|Solving Problems Involving Area and Circumference of a Circle|Solving Problems Involving Polygons on the Coordinate Plane |
| 111.27.b.9.D | Solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape's net. | Investigating Surface Area and Cross-Sections of Solids|Solving Problems Involving Surface Area |
| 111.28.b.12 | The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. (Personal financial literacy) | Computing Simple Interest |
| 111.28.b.12.A | Solve real-world problems comparing how interest rate and loan length affect the cost of credit; | Computing Simple Interest |
| 111.28.b.12.D | Calculate and compare simple interest and compound interest earnings; | Computing Simple Interest |
| 111.28.b.2 | The student applies mathematical process standards to represent and use real numbers in a variety of forms. | Comparing Fractions, Decimals, Percents, and Ratios |
| 111.28.b.2.D | Order a set of real numbers arising from mathematical and real-world contexts. | Comparing Fractions, Decimals, Percents, and Ratios |
| 111.28.b.4 | The 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.A | Use 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.B | Graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship | Calculating 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.C | Use 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.5 | The 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.A | Represent 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.B | Represent 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.E | Solve 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.F | Distinguish 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.G | Identify 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.H | Identify examples of proportional and non-proportional functions that arise from mathematical and real-world problems | Classifying 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.I | Write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations. | Determining Slope and 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.9 | The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations. (Expressions, equations, and relationships) | Determining if an Ordered Pair is a Solution to the 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 |
| 111.28.b.11 | The student applies mathematical process standards to use statistical procedures to describe data. (Measurement and data) | Computing the Interquartile Range and Mean Absolute Deviation of a Data Set|Interpreting Scatter Plots to Investigate Patterns of Association|Selecting a Representative Sampling Method for a Population |
| 111.28.b.11.A | Construct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data; | Interpreting Scatter Plots to Investigate Patterns of Association |
| 111.28.b.11.B | Determine the mean absolute deviation and use this quantity as a measure of the average distance data are from the mean using a data set of no more than 10 data points. | Computing the Interquartile Range and Mean Absolute Deviation of a Data Set |
| 111.28.b.11.C | Simulate generating random samples of the same size from a population with known characteristics to develop the notion of a random sample being representative of the population from which it was selected. | Selecting a Representative Sampling Method for a Population |
| 111.28.b.5 | The student applies mathematical process standards to use proportional and 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.C | Contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation; | Interpreting Scatter Plots to Investigate Patterns of Association|Solving Problems Involving Bivariate Measurement Data |
| 111.28.b.5.D | Use a trend line that approximates the linear relationship between bivariate sets of data to make predictions; | Solving Problems Involving Bivariate Measurement Data |
| 111.28.b.10 | The 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.A | Generalize 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.B | Differentiate 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.C | Explain 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.D | Model 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.3 | The 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.A | Generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation; | Describing the Sequence of Transformations of Two Similar Figures|Solving Problems Involving Scale Drawings of Geometric Figures|Using the Coordinate Plane to Demonstrate Transformations |
| 111.28.b.3.B | Compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane | Describing the Sequence of Transformations of Two Similar Figures|Using the Coordinate Plane to Demonstrate Transformations |
| 111.28.b.3.C | Use an algebraic representation to explain the effect of a given positive rational scale factor applied to 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.6 | The student applies mathematical process standards to develop mathematical relationships and make connections to geometric formulas. (Expressions, equations, and relationships) | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane|Solving Problems Involving Volume |
| 111.28.b.6.A | Describe the volume formula V = Bh of a cylinder in terms of its base area and its height; | Solving Problems Involving Volume |
| 111.28.b.6.C | Use models and diagrams to explain the Pythagorean theorem. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane |
| 111.28.b.7 | The student applies mathematical process standards to use geometry to solve problems. (Expressions, equations, and relationships) | Determining Unknown Side Lengths in Right Triangles of Polygons and 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.A | Solve problems involving the volume of cylinders, cones, and spheres; | Solving Problems Involving Volume |
| 111.28.b.7.B | Use previous knowledge of surface area to make connections to the formulas for lateral and total surface area and determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders; | Investigating Surface Area and Cross-Sections of Solids|Solving Problems Involving Surface Area |
| 111.28.b.7.C | Use the Pythagorean Theorem and its converse to solve problems | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids |
| 111.28.b.7.D | Determine the distance between two points on a coordinate plane using the Pythagorean Theorem. | Finding Distance Between Two Points on the Coordinate Plane |
| 111.28.b.8 | The student applies mathematical process standards to use 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.D | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. and | Analyzing Triangles|Identifying Angle Relationships Formed by Parallel Lines and a Transversal |
| 6.19c | Investigate and recognize the inverse property for multiplication. | Dividing Fractions |
| 6.2b | Identify a given fraction, decimal or percent from a representation; | Adding Like 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 |
| 6.2c | Demonstrate equivalent relationships among fractions, decimals, and percents; | Converting Fractions to Decimals|Converting Mixed Numbers and Improper Fractions |
| 6.2d | Compare and order fractions, decimals, and percents. | Comparing Fractions and Decimals |
| 6.4 | Demonstrate multiple representations of multiplication and division of fractions. | Dividing Fractions|Multiplying Fractions |
| 6.6a | Multiply 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 |
| 6.6b | Estimate 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 |
| 6.7 | Solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimals | Adding 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 |
| 6.1 | Describe and compare data, using ratios, and use appropriate notations, such as a/b , a to b, and a:b. | Expressing Ratios in Simplified Form|Expressing Unit Rates|Solving Practical Problems Using Ratios |
| 6.2a | Investigate and describe fractions, decimals and percents as ratios; | Expressing Ratios in Simplified Form |
| 6.2b | Identify a given fraction, decimal or percent from a representation; | Expressing a Fraction as a Percent |
| 6.2c | Demonstrate equivalent relationships among fractions, decimals, and percents; | Expressing a Decimal Number as a Percent|Expressing a Fraction as a Percent|Expressing a Percent as a Decimal Number|Expressing a Percent as a Fraction |
| 6.2d | Compare and order fractions, decimals, and percents. | Comparing Fractions, Decimals, Percents, and Ratios |
| 6.14a | Construct circle graphs; | Representing a Set of Data Using a Data Display |
| 6.14b | Draw conclusions and make predictions, using circle graphs; | Analyzing and Describing the Distribution of a Data Set (Data Display) |
| 6.14c | Compare and contrast graphs that present information from the same data set. | Analyzing and Describing the Distribution of a Data Set (Data Display)|Representing a Set of Data Using a Data Display |
| 6.15a | Describe mean as balance point; | Describing a Set of Data (Mean, Median, Mode, Range) |
| 6.15b | Decide which measure of center is appropriate for a given purpose. | Analyzing and Comparing Data Sets |
| 6.16a | Compare and contrast dependent and independent events; | Identifying the Outcomes in a Sample Space |
| 6.16b | Determine probabilities for dependent and independent events. | Determining the Likelihood of Events|Finding the Probabilities of Compound Events|Identifying the Outcomes in a Sample Space|Identifying the Theoretical Probability of an Event |
| 6.2b | Identify a given fraction, decimal or percent from a representation; | Representing a Set of Data Using a Data Display |
| 6.10a | Define pi (π) as the ratio of the circumference of a circle to its diameter; | Solving Problems Involving Area and Circumference of a Circle |
| 6.10b | Solve practical problems involving circumference and area of a circle, given the diameter or radius; | Solving Problems Involving Area and Circumference of a Circle |
| 6.10c | Solve practical problems involving area and perimeter; | Finding Area of Polygons Using Composing and Decomposing 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 |
| 6.10d | Describe 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 |
| 6.12 | Determine congruence of segments, angles, and polygons. | Analyzing Triangles|Identifying and Classifying Polygons|Using Congruency Statements to Identify Corresponding Parts of a Polygon |
| 6.13 | Describe and identify properties of quadrilaterals. | Identifying Polygons|Identifying Quadrilaterals|Identifying and Classifying Polygons |
| 7.5a | Describe volume and surface area of cylinders; | Solving Problems Involving Volume |
| 7.5b | Solve practical problems involving the volume and surface area of rectangular prisms and cylinders; | Investigating Surface Area and Cross-Sections of Solids|Solving Problems Involving Surface Area|Solving Problems Involving Volume |
| 7.5c | Describe how changing one measured attribute of a rectangular prism affects its volume and surface area. | Solving Problems Involving Surface Area|Solving Problems Involving Volume |
| 7.6 | Determine whether plane figures quadrilaterals and triangles are similar and write proportions to express the relationships between corresponding sides of similar figures. | Describing the Sequence of Transformations of Two Similar Figures|Solving Problems Involving Scale Drawings of Geometric Figures |
| 7.7 | Compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid. | Identifying Polygons|Identifying Quadrilaterals|Identifying and Classifying Polygons |
| 7.8 | Given a polygon in the coordinate plane,represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane. | Describing the Sequence of Transformations of Two Similar 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 |
| 8.15a | Solve multistep linear equations in one variable on one and two sides of the equation; | Simplifying Expressions by Combining Like Terms |
| 8.15b | Solve two-step linear inequalities and graph the results on a number line; | Simplifying Expressions by Combining Like Terms |
| 8.15c | Identify properties of operations used to solve an equation. | Identifying Properties of Addition and Multiplication |
| 8.1a | Simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers; | Evaluating Expressions Using the Order of Operations|Identifying Properties of Addition and Multiplication|Simplifying Expressions by Combining Like Terms|Writing Algebraic Expressions |
| 8.4 | Apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. | Evaluating Expressions|Evaluating Expressions Using the Order of Operations |
| 8.10a | Verify the Pythagorean Theorem | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane |
| 8.10b | Apply the Pythagorean Theorem. | Determining Unknown Side Lengths in Right Triangles of Polygons and Solids|Finding Distance Between Two Points on the Coordinate Plane |
| 8.11 | Solve practical area and perimeter problems involving composite plane figures. | Finding Area of Polygons Using Composing and Decomposing 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 |
| 8.6a | Verify by measuring and describe the relationships among vertical angles, adjacent angles, supplementary angles, and complementary angles | Identifying Angle Relationships Formed by Parallel Lines and a Transversal|Solving Equations for an Unknown Angle in a Figure |
| 8.6b | Measure angles of less than 360°. | Classifying Angles |
| 8.7a | Investigate and solve practical problems involving volume and surface area of prisms, cylinders, cones, and pyramids; | Investigating Surface Area and Cross-Sections of Solids|Solving Problems Involving Surface Area|Solving Problems Involving Volume |
| 8.7b | Describe how changing one measured attribute of the figure affects the volume and surface area. | Solving Problems Involving Surface Area|Solving Problems Involving Volume |
| 8.8a | Apply transformations to plane 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 |
| 8.8b | Identify 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 |
| 8.9 | Construct a three-dimensional model, given the top or bottom, side, and front views. | Investigating Surface Area and Cross-Sections of Solids |
