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#### MAT-08.AR

MAT-08.AR Domain

# (AR) Number and Operations

## Sub-Categories

• (OA) Operations and Algebraic Thinking
Learners will analyze patterns and relationships to generate and interpret numerical expressions.
• (RP) Ratios and Proportional Relationships
Learners will use ratios, rates, and proportions to model relationships and solve problems.
• (EE) Expressions and Equations
Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.
• (F) Functions
Learners will develop a foundational knowledge of functions and use them to model relationships between quantities.

## Calculation Method for Domains

Domains are larger groups of related standards. The Domain Grade is a calculation of all the related standards. Click on the standard name below each Domain to access the learning targets and rubrics/ proficiency scales for individual standards within the domain.

#### MAT-08.AR.EE.01

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (EE) Expressions and Equations Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.

## Progressions

Equations/Expressions

• MAT-01.AR.OA.06 Use the +, -, and = symbols accurately in an equation.
• MAT-02.AR.OA.02 Apply the properties of operations to solve addition and subtraction equations and justify thinking.
• MAT-03.AR.OA.02 Apply the properties of operations to solve multiplication and division equations and justify thinking.
• MAT-03.AR.OA.03 Solve word two-step authentic word problems using addition and subtraction within 1000, including equations with a letter as an unknown.
• MAT-03.AR.OA.04 Use strategies and visual models to solve authentic word problems with multiplication within 100, including unknowns, using grouping models and equations.
• MAT-03.AR.OA.05 Use strategies and visual models to solve authentic word problems with division within 100, including unknowns, using grouping models and equations.
• MAT-04.AR.OA.03 Solve multi-step authentic word problems using the four operations, including problems with interpreted remainders.
• MAT-04.AR.OA.05 Interpret multiplication equations as a comparison. Represent multiplicative comparisons as multiplication equations.
• MAT-05.AR.OA.02 Analyze problems using the order of operations to solve and evaluate expressions while justifying thinking.
• MAT-05.AR.OA.03 Write simple expressions that record calculations with numbers. Interpret numerical expressions without evaluating them.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• MAT-06.AR.EE.02 Read and evaluate algebraic expressions, including expressions with whole number exponents and grouping symbols. Write algebraic expressions to represent simple and authentic situations.
• MAT-06.AR.EE.04 Describe the concept of a solution of an equation or an inequality. Determine whether a given number is a solution to an equation or an inequality.
• MAT-06.AR.EE.05 Write and solve equations of the form of x + p = q and px = q for cases in which p and q are non-negative whole numbers or decimals, including authentic problems.
• MAT-07.AR.EE.02 Write and solve equations of the form px + q = r and p(x + q) = r , including in authentic problems.
• MAT-08.AR.EE.01 Explain the relationship between repeated multiplication and the properties of integer exponents. Apply a single exponent property to generate equivalent numeric and algebraic expressions that include numerical coefficients.
• MAT-08.AR.EE.02 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a non-negative rational number.
• MAT-08.AR.EE.06 Read, write, and evaluate numerical and algebraic expressions, including expressions involving absolute value. Solve and graph equations of the form |x| = r where r is a nonnegative rational number.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• MAT-09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
• MAT-12.AR.03 Interpret expressions that represent a quantity in context.
• MAT-12.AR.02 Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.

Equivalent Expressions

• MAT-06.AR.EE.03 Identify when two expressions are equivalent. Apply the properties of operations to generate equivalent expressions.
• MAT-07.AR.EE.01 Apply the properties of operations as strategies to add, subtract, factor, and expand linear expressions involving variables, integers, and/or non-negative fractions and decimals with an emphasis on writing equivalent expressions.
• MAT-08.AR.EE.01 Explain the relationship between repeated multiplication and the properties of integer exponents. Apply a single exponent property to generate equivalent numeric and algebraic expressions that include numerical coefficients.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• MAT-09.AR.02 Rearrange formulas to isolate a quantity or variable(s) of interest using the same reasoning as in solving equations.
• MAT-09.AR.07 Rearrange multi-variable formulas to highlight a quantity of interest.
• MAT-09.AR.F.06 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-12.NO.02 Perform operations on complex radical expressions to write a simplified equivalent expression.
• MAT-12.AR.01 Rearrange multi-variable formulas to highlight a quantity of interest.
• MAT-12.AR.02 Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.
• MAT-12.AR.04 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
• MAT-12.AR.05 Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
• MAT-12.AR.F.03 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-12.GM.01 Write the equation of a conic section given its special features. Convert between the standard form and general form equations of conic sections.
• MAT-12.GM.02 Identify key features of a conic section given its equation. Apply properties of conic sections in context.
• MAT-12.NO.12 Extend polynomial identities to the complex numbers.

#### MAT-08.AR.EE.02

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (EE) Expressions and Equations Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.

## Progressions

Equations/Expressions

• MAT-01.AR.OA.06 Use the +, -, and = symbols accurately in an equation.
• MAT-02.AR.OA.02 Apply the properties of operations to solve addition and subtraction equations and justify thinking.
• MAT-03.AR.OA.02 Apply the properties of operations to solve multiplication and division equations and justify thinking.
• MAT-03.AR.OA.03 Solve word two-step authentic word problems using addition and subtraction within 1000, including equations with a letter as an unknown.
• MAT-03.AR.OA.04 Use strategies and visual models to solve authentic word problems with multiplication within 100, including unknowns, using grouping models and equations.
• MAT-03.AR.OA.05 Use strategies and visual models to solve authentic word problems with division within 100, including unknowns, using grouping models and equations.
• MAT-04.AR.OA.03 Solve multi-step authentic word problems using the four operations, including problems with interpreted remainders.
• MAT-04.AR.OA.05 Interpret multiplication equations as a comparison. Represent multiplicative comparisons as multiplication equations.
• MAT-05.AR.OA.02 Analyze problems using the order of operations to solve and evaluate expressions while justifying thinking.
• MAT-05.AR.OA.03 Write simple expressions that record calculations with numbers. Interpret numerical expressions without evaluating them.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• MAT-06.AR.EE.02 Read and evaluate algebraic expressions, including expressions with whole number exponents and grouping symbols. Write algebraic expressions to represent simple and authentic situations.
• MAT-06.AR.EE.04 Describe the concept of a solution of an equation or an inequality. Determine whether a given number is a solution to an equation or an inequality.
• MAT-06.AR.EE.05 Write and solve equations of the form of x + p = q and px = q for cases in which p and q are non-negative whole numbers or decimals, including authentic problems.
• MAT-07.AR.EE.02 Write and solve equations of the form px + q = r and p(x + q) = r , including in authentic problems.
• MAT-08.AR.EE.01 Develop and know the properties of integer exponents. Apply a single exponent property to generate equivalent numerical and algebraic expressions that include numerical coefficients.
• MAT-08.AR.EE.02 Explain the relationship between repeated multiplication and the properties of integer exponents. Apply a single exponent property to generate equivalent numeric and algebraic expressions that include numerical coefficients.
• MAT-08.AR.EE.06 Read, write, and evaluate numerical and algebraic expressions, including expressions involving absolute value. Solve and graph equations of the form |x| = r where r is a nonnegative rational number.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• MAT-09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
• MAT-12.AR.03 Interpret expressions that represent a quantity in context.
• MAT-12.AR.02 Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.

#### MAT-08.AR.EE.03

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (EE) Expressions and Equations Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.

## Progressions

Linear Equations

• MAT-07.AR.EE.01 Apply the properties of operations as strategies to add, subtract, factor, and expand linear expressions involving variables, integers, and/or non-negative fractions and decimals with an emphasis on writing equivalent expressions.
• MAT-08.AR.EE.03 Explain the characteristics of a linear relationship, including identifying the slope and yintercept in tables, graphs, equations, and descriptions.
• MAT-08.AR.EE.04 Represent linear relationships using tables, graphs, equations, and descriptions when given a relationship in one of these forms.
• MAT-08.AR.EE.05 Solve linear equations with rational number coefficients and variables on both sides, including equations that require using the distributive property and/or combining and collecting like terms. Interpret the number of solutions. Give examples of linear equations in one variable with one solution, many solutions, or no solutions.
• MAT-09.AR.03 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.
• MAT-09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
• MAT-09.AR.05 Justify each step in solving a linear equation that may or may not have a solution.
• MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.
• MAT-09.AR.07 Solve a system of linear equations graphically and algebraically. Create and solve a system of linear equations in context and interpret the results.
• MAT-12.AR.05 Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
• MAT-12.AR.07 Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.
• MAT-12.AR.08 Create equations in two or more variables to represent relationships between quantities.
• Graph equations on coordinate axes with appropriate labels and scales.
• MAT-12.AR.09 Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.
• MAT-12.AR.12 Solve simple rational and radical equations in one variable and identify extraneous solutions.
• MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
• MAT-12.AR.15 Apply the Factor and Remainder Theorems to determine efficiently whether a liner expression is a factor of a polynomial expression.
• MAT-12.AR.16 Using graphs, technology, tables, or successive approximations, show that the solution(s) to the equation f(x) = g(x) is the x-value(s) that result in the y-values of f(x) and g(x) being the same.
• MAT-12.AR.17 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
• MAT-12.AR.18 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
• MAT-12.NO.13 Apply the Fundamental Theorem of Algebra to find all roots of a polynomial equation and determine the nature (i.e., integer, rational, irrational, real, complex) of the roots.
• MAT-12.AR.19 Solve a system of equations in three or more variables with matrices (using technology).
• MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.

Ratio and Proportional Relationships

• MAT-06.AR.RP.01 Describe the concept of a ratio relationship between two quantities using ratio language and visual models.
• MAT-06.AR.RP.03 Make and use tables of equivalent ratios, tape diagrams, double number line diagrams, and equations to reason about ratios, rates, and unit rates.
• MAT-07.AR.RP.02 Analyze the relationships between the dependent and independent variables of a proportional relationship using graphs and tables. 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, k) where k is the unit rate.
• MAT-07.AR.RP.03 Identify the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by an equation of the form y = kx, where k is the constant of proportionality, and describe the meaning of each variable (y, k, x) in the context of the situation.
• MAT-07.AR.RP.04 Use proportional relationships to solve multi-step problems involving ratios, percents, and scale drawings of geometric figures, including authentic problems.
• MAT-08.AR.EE.03 Explain the characteristics of a linear relationship, including identifying the slope and yintercept in tables, graphs, equations, and descriptions.
• MAT-08.AR.EE.04 Represent linear relationships using tables, graphs, equations, and descriptions when given a relationship in one of these forms.
• MAT-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• MAT-10.GM.15 Use transformations to decide if two given figures are similar. Apply the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• MAT-10.GM.16 Prove similarity theorems about triangles.
• MAT-10.GM.18 Recognize how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.
• MAT-10.GM.20 Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30°-60°-90° and 45°-45°-90°).
• MAT-10.GM.25 Explain and use the formulas for arc length and area of sectors of circles.
• MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
• MAT-10.GM.29 Determine the midpoint or endpoint of a line segment using coordinates. (+) Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
• MAT-10.GM.36 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; scaling a model).
• MAT-12.GM.03 Determine and apply appropriate formulas to solve right and non-right triangle problems in context.

#### MAT-08.AR.EE.04

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (EE) Expressions and Equations Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.

## Progressions

Linear Equations

• MAT-07.AR.EE.01 Apply the properties of operations as strategies to add, subtract, factor, and expand linear expressions involving variables, integers, and/or non-negative fractions and decimals with an emphasis on writing equivalent expressions.
• MAT-08.AR.EE.03 Explain the characteristics of a linear relationship, including identifying the slope and yintercept in tables, graphs, equations, and descriptions.
• MAT-08.AR.EE.04 Represent linear relationships using tables, graphs, equations, and descriptions when given a relationship in one of these forms.
• MAT-08.AR.EE.05 Solve linear equations with rational number coefficients and variables on both sides, including equations that require using the distributive property and/or combining and collecting like terms. Interpret the number of solutions. Give examples of linear equations in one variable with one solution, many solutions, or no solutions.
• MAT-09.AR.03 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.
• MAT-09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
• MAT-09.AR.05 Justify each step in solving a linear equation that may or may not have a solution.
• MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.
• MAT-09.AR.07 Solve a system of linear equations graphically and algebraically. Create and solve a system of linear equations in context and interpret the results.
• MAT-12.AR.05 Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
• MAT-12.AR.07 Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.
• MAT-12.AR.08 Create equations in two or more variables to represent relationships between quantities.
• Graph equations on coordinate axes with appropriate labels and scales.
• MAT-12.AR.09 Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.
• MAT-12.AR.12 Solve simple rational and radical equations in one variable and identify extraneous solutions.
• MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
• MAT-12.AR.15 Apply the Factor and Remainder Theorems to determine efficiently whether a liner expression is a factor of a polynomial expression.
• MAT-12.AR.16 Using graphs, technology, tables, or successive approximations, show that the solution(s) to the equation f(x) = g(x) is the x-value(s) that result in the y-values of f(x) and g(x) being the same.
• MAT-12.AR.17 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
• MAT-12.AR.18 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
• MAT-12.NO.13 Apply the Fundamental Theorem of Algebra to find all roots of a polynomial equation and determine the nature (i.e., integer, rational, irrational, real, complex) of the roots.
• MAT-12.AR.19 Solve a system of equations in three or more variables with matrices (using technology).
• MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.

Ratio and Proportional Relationships

• MAT-06.AR.RP.01 Describe the concept of a ratio relationship between two quantities using ratio language and visual models.
• MAT-06.AR.RP.03 Make and use tables of equivalent ratios, tape diagrams, double number line diagrams, and equations to reason about ratios, rates, and unit rates.
• MAT-07.AR.RP.02 Analyze the relationships between the dependent and independent variables of a proportional relationship using graphs and tables. 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, k) where k is the unit rate.
• MAT-07.AR.RP.03 Identify the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Represent proportional relationships by an equation of the form y = kx, where k is the constant of proportionality, and describe the meaning of each variable (y, k, x) in the context of the situation.
• MAT-07.AR.RP.04 Use proportional relationships to solve multi-step problems involving ratios, percents, and scale drawings of geometric figures, including authentic problems.
• MAT-08.AR.EE.03 Explain the characteristics of a linear relationship, including identifying the slope and yintercept in tables, graphs, equations, and descriptions.
• MAT-08.AR.EE.04 Represent linear relationships using tables, graphs, equations, and descriptions when given a relationship in one of these forms.
• MAT-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• MAT-10.GM.15 Use transformations to decide if two given figures are similar. Apply the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• MAT-10.GM.16 Prove similarity theorems about triangles.
• MAT-10.GM.18 Recognize how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.
• MAT-10.GM.20 Solve applied problems involving right triangles using trigonometric ratios, the Pythagorean Theorem, and special right triangles (30°-60°-90° and 45°-45°-90°).
• MAT-10.GM.25 Explain and use the formulas for arc length and area of sectors of circles.
• MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
• MAT-10.GM.29 Determine the midpoint or endpoint of a line segment using coordinates. (+) Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
• MAT-10.GM.36 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; scaling a model).
• MAT-12.GM.03 Determine and apply appropriate formulas to solve right and non-right triangle problems in context.

#### MAT-08.AR.EE.05

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (EE) Expressions and Equations Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.

## Progressions

Properties of Operations

• MAT-02.AR.OA.02 Apply the properties of operations to solve addition and subtraction equations and justify thinking.
• MAT-03.AR.OA.02 Apply the properties of operations to solve multiplication and division equations and justify thinking.
• MAT-04.AR.OA.02 Identify and apply the properties of operations for addition, subtraction, multiplication, and division and justify thinking.
• MAT-05.AR.OA.02 Analyze problems using the order of operations to solve and evaluate expressions while justifying thinking.
• MAT-06.AR.EE.03 Identify when two expressions are equivalent. Apply the properties of operations to generate equivalent expressions.
• MAT-07.AR.EE.01 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions involving variables, integers, and/or non-negative fractions and decimals with an emphasis on writing equivalent expressions.
• MAT-08.AR.EE.05 Solve linear equations with rational number coefficients and variables on both sides, including equations that require using the distributive property and/or combining and collecting like terms. Interpret the number of solutions. Give examples of linear equations in one variable with one solution, infinitely showing solutions or no solutions.
• MAT-09.AR.05 Justify each step in solving a linear equation that may or may not have a solution.

Linear Equations

• MAT-07.AR.EE.01 Apply the properties of operations as strategies to add, subtract, factor, and expand linear expressions involving variables, integers, and/or non-negative fractions and decimals with an emphasis on writing equivalent expressions.
• MAT-08.AR.EE.03 Explain the characteristics of a linear relationship, including identifying the slope and yintercept in tables, graphs, equations, and descriptions.
• MAT-08.AR.EE.04 Represent linear relationships using tables, graphs, equations, and descriptions when given a relationship in one of these forms.
• MAT-08.AR.EE.05 Solve linear equations with rational number coefficients and variables on both sides, including equations that require using the distributive property and/or combining and collecting like terms. Interpret the number of solutions. Give examples of linear equations in one variable with one solution, many solutions, or no solutions.
• MAT-09.AR.03 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.
• MAT-09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
• MAT-09.AR.05 Justify each step in solving a linear equation that may or may not have a solution.
• MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.
• MAT-09.AR.07 Solve a system of linear equations graphically and algebraically. Create and solve a system of linear equations in context and interpret the results.
• MAT-12.AR.05 Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
• MAT-12.AR.07 Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.
• MAT-12.AR.08 Create equations in two or more variables to represent relationships between quantities.
• Graph equations on coordinate axes with appropriate labels and scales.
• MAT-12.AR.09 Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.
• MAT-12.AR.12 Solve simple rational and radical equations in one variable and identify extraneous solutions.
• MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
• MAT-12.AR.15 Apply the Factor and Remainder Theorems to determine efficiently whether a liner expression is a factor of a polynomial expression.
• MAT-12.AR.16 Using graphs, technology, tables, or successive approximations, show that the solution(s) to the equation f(x) = g(x) is the x-value(s) that result in the y-values of f(x) and g(x) being the same.
• MAT-12.AR.17 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
• MAT-12.AR.18 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
• MAT-12.NO.13 Apply the Fundamental Theorem of Algebra to find all roots of a polynomial equation and determine the nature (i.e., integer, rational, irrational, real, complex) of the roots.
• MAT-12.AR.19 Solve a system of equations in three or more variables with matrices (using technology).
• MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.

#### MAT-08.AR.EE.06

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (EE) Expressions and Equations Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.

## Progressions

Equations/Expressions

• MAT-01.AR.OA.06 Use the +, -, and = symbols accurately in an equation.
• MAT-02.AR.OA.02 Apply the properties of operations to solve addition and subtraction equations and justify thinking.
• MAT-03.AR.OA.02 Apply the properties of operations to solve multiplication and division equations and justify thinking.
• MAT-03.AR.OA.03 Solve word two-step authentic word problems using addition and subtraction within 1000, including equations with a letter as an unknown.
• MAT-03.AR.OA.04 Use strategies and visual models to solve authentic word problems with multiplication within 100, including unknowns, using grouping models and equations.
• MAT-03.AR.OA.05 Use strategies and visual models to solve authentic word problems with division within 100, including unknowns, using grouping models and equations.
• MAT-04.AR.OA.03 Solve multi-step authentic word problems using the four operations, including problems with interpreted remainders.
• MAT-04.AR.OA.05 Interpret multiplication equations as a comparison. Represent multiplicative comparisons as multiplication equations.
• MAT-05.AR.OA.02 Analyze problems using the order of operations to solve and evaluate expressions while justifying thinking.
• MAT-05.AR.OA.03 Write simple expressions that record calculations with numbers. Interpret numerical expressions without evaluating them.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• MAT-06.AR.EE.02 Read and evaluate algebraic expressions, including expressions with whole number exponents and grouping symbols. Write algebraic expressions to represent simple and authentic situations.
• MAT-06.AR.EE.04 Describe the concept of a solution of an equation or an inequality. Determine whether a given number is a solution to an equation or an inequality.
• MAT-06.AR.EE.05 Write and solve equations of the form of x + p = q and px = q for cases in which p and q are non-negative whole numbers or decimals, including authentic problems.
• MAT-07.AR.EE.02 Write and solve equations of the form px + q = r and p(x + q) = r , including in authentic problems.
• MAT-08.AR.EE.01 Develop and know the properties of integer exponents. Apply a single exponent property to generate equivalent numerical and algebraic expressions that include numerical coefficients.
• MAT-08.AR.EE.02 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a non-negative rational number.
• MAT-08.AR.EE.06 Explain the relationship between repeated multiplication and the properties of integer exponents. Apply a single exponent property to generate equivalent numeric and algebraic expressions that include numerical coefficients.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• MAT-09.AR.04 Create linear and exponential equations in two or more variables to represent relationships between quantities. Graph equations on coordinate axes with appropriate labels and scales.
• MAT-12.AR.03 Interpret expressions that represent a quantity in context.
• MAT-12.AR.02 Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.

#### MAT-08.AR.EE.07

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (EE) Expressions and Equations Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.

## Progressions

Inequalities

• MAT-06.AR.EE.04 Describe the concept of a solution to an equation or an inequality. Determine whether a given number is a solution to an equation or an inequality.
• MAT-06.AR.EE.06 Write a statement of inequality of the form x > c or the form x < c to represent a constraint or condition. Recognize that inequalities of the form x > c or the form x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
• MAT-07.AR.EE.03 Write and solve one- and two-step inequalities where coefficients and solutions are integers and/or non-negative fractions and decimals, including authentic problems. Graph the solution set of the inequality and interpret it in the context of the problem.
• MAT-08.AR.EE.07 Solve and graph inequalities in one variable with rational number coefficients and variables on both sides, including equations that require using the distributive property and/or combining like terms.
• MAT-08.AR.EE.08 Graph linear inequalities in two variables on a coordinate plane. Interpret the possible solutions in the context of authentic problems.
• MAT-09.AR.03 Create equations and inequalities in two variables and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.
• MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.
• MAT-09.AR.07 Solve a system of linear equations graphically and algebraically. Create and solve a system of linear equations in context and interpret the results.
• MAT-09.AR.08 Graph the solution set to a two-variable system of linear inequalities. Create and graph the solution set to a two-variable system of linear inequalities in context.
• MAT-09.AR.09 Solve absolute value equations and inequalities in one or two variables.
• MAT-12.AR.07 Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.
• MAT-12.AR.09 Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.

#### MAT-08.AR.EE.08

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (EE) Expressions and Equations Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adapting approaches in novel situations.

## Progressions

Inequalities

• MAT-06.AR.EE.04 Describe the concept of a solution to an equation or an inequality. Determine whether a given number is a solution to an equation or an inequality.
• MAT-06.AR.EE.06 Write a statement of inequality of the form x > c or the form x < c to represent a constraint or condition. Recognize that inequalities of the form x > c or the form x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
• MAT-07.AR.EE.03 Write and solve one- and two-step inequalities where coefficients and solutions are integers and/or non-negative fractions and decimals, including authentic problems. Graph the solution set of the inequality and interpret it in the context of the problem.
• MAT-08.AR.EE.07 Solve and graph inequalities in one variable with rational number coefficients and variables on both sides, including equations that require using the distributive property and/or combining like terms.
• MAT-08.AR.EE.08 Graph linear inequalities in two variables on a coordinate plane. Interpret the possible solutions in the context of authentic problems.
• MAT-09.AR.03 Create equations and inequalities in two variables and use them to solve problems. Include equations arising from linear, quadratic, and exponential functions.
• MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.
• MAT-09.AR.07 Solve a system of linear equations graphically and algebraically. Create and solve a system of linear equations in context and interpret the results.
• MAT-09.AR.08 Graph the solution set to a two-variable system of linear inequalities. Create and graph the solution set to a two-variable system of linear inequalities in context.
• MAT-09.AR.09 Solve absolute value equations and inequalities in one or two variables.
• MAT-12.AR.07 Create equations and inequalities and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.
• MAT-12.AR.09 Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.

#### MAT-08.AR.F.01

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (F) Functions Learners will develop a foundational knowledge of functions and use them to model relationships between quantities.

## Progressions

Functional Relationships

• MAT-08.AR.F.01 Defend whether a relation is a function from various representations using appropriate function language.
• MAT-08.AR.F.02 Compare and contrast properties of two linear functions, each represented in a different way (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.03 Compare and contrast linear and non-linear functions represented in different ways (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.04 Model a linear relationship between two quantities by creating a table, graph, and equation. Interpret the rate of change and initial value of a linear function in terms of the situation it models.
• MAT-08.AR.F.05 Describe qualitatively the functional relationship between two quantities by analyzing a graph, including where the function is constant, increasing, or decreasing; linear or nonlinear; and discrete or continuous. Create a graph that exhibits the qualitative features of a function described.
• MAT-09.AR.F.01 Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.
• MAT-09.AR.F.02 Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of context.
• MAT-09.AR.F.03 Sketch key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.
• MAT-09.AR.F.04 Relate the domain of a linear, quadratic, or exponential function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-09.AR.F.05 Calculate and interpret the average rate of change of a linear, quadratic, or exponential function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-09.AR.F.06 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-09.AR.F.07 Compare key features of two linear, exponential, or quadratic functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
• MAT-09.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions.
• MAT-09.AR.F.10 Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse. Graph the inverse of a linear function.
• MAT-09.AR.F.11 Interpret the parameters of a linear, quadratic, or exponential function in terms of context.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).
• MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
• MAT-12.AR.F.01 Write a function that describes a relationship between two quantities.
• MAT-12.AR.F.02 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-12.AR.F.03 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-12.AR.F.04 Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.AR.F.05 Find inverse functions.
• MAT-12.AR.F.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.07 Compare key features of two functions, each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
• MAT-12.AR.F.08 Use tables, graphs, verbal discussions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
• MAT-12.AR.F.09 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-12.AR.F.11 Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.
• MAT-12.AR.F.12 Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity.
• MAT-12.AR.F.13 Determine whether a linear, quadratic, polynomial, exponential, logarithmic, or trigonometric model fits the situation. Determine an appropriate mathematical model in context (with or without technology).
• MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
• MAT-12.AR.F.16 Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π /3, π/4 and π/6.
• MAT-12.AR.F.17 Use the Pythagorean Identity sin²(θ) + cos²(θ) = 1 to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
• MAT-12.AR.F.18 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
• MAT-12.AR.F.19 Use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.
• MAT-12.AR.F.20 Use the unit circle to explain the symmetry (odd and even) and the periodicity of trigonometric functions.
• MAT-12.AR.F.21 Create a trigonometric function to model periodic phenomena.
• MAT-12.AR.F.22 Restrict the domain of a trigonometric function to construct its inverse.
• MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.
• MAT-12.AR.F.24 Know and apply the addition and subtraction formulas for sine, cosine, and tangent to solve problems.

#### MAT-08.AR.F.02

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (F) Functions Learners will develop a foundational knowledge of functions and use them to model relationships between quantities.

## Progressions

Functional Relationships

• MAT-08.AR.F.01 Defend whether a relation is a function from various representations using appropriate function language.
• MAT-08.AR.F.02 Compare and contrast properties of two linear functions, each represented in a different way (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.03 Compare and contrast linear and non-linear functions represented in different ways (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.04 Model a linear relationship between two quantities by creating a table, graph, and equation. Interpret the rate of change and initial value of a linear function in terms of the situation it models.
• MAT-08.AR.F.05 Describe qualitatively the functional relationship between two quantities by analyzing a graph, including where the function is constant, increasing, or decreasing; linear or nonlinear; and discrete or continuous. Create a graph that exhibits the qualitative features of a function described.
• MAT-09.AR.F.01 Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.
• MAT-09.AR.F.02 Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of context.
• MAT-09.AR.F.03 Sketch key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.
• MAT-09.AR.F.04 Relate the domain of a linear, quadratic, or exponential function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-09.AR.F.05 Calculate and interpret the average rate of change of a linear, quadratic, or exponential function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-09.AR.F.06 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-09.AR.F.07 Compare key features of two linear, exponential, or quadratic functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
• MAT-09.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions.
• MAT-09.AR.F.10 Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse. Graph the inverse of a linear function.
• MAT-09.AR.F.11 Interpret the parameters of a linear, quadratic, or exponential function in terms of context.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).
• MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
• MAT-12.AR.F.01 Write a function that describes a relationship between two quantities.
• MAT-12.AR.F.02 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-12.AR.F.03 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-12.AR.F.04 Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.AR.F.05 Find inverse functions.
• MAT-12.AR.F.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.07 Compare key features of two functions, each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
• MAT-12.AR.F.08 Use tables, graphs, verbal discussions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
• MAT-12.AR.F.09 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-12.AR.F.11 Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.
• MAT-12.AR.F.12 Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity.
• MAT-12.AR.F.13 Determine whether a linear, quadratic, polynomial, exponential, logarithmic, or trigonometric model fits the situation. Determine an appropriate mathematical model in context (with or without technology).
• MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
• MAT-12.AR.F.16 Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π /3, π/4 and π/6.
• MAT-12.AR.F.17 Use the Pythagorean Identity sin²(θ) + cos²(θ) = 1 to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
• MAT-12.AR.F.18 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
• MAT-12.AR.F.19 Use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.
• MAT-12.AR.F.20 Use the unit circle to explain the symmetry (odd and even) and the periodicity of trigonometric functions.
• MAT-12.AR.F.21 Create a trigonometric function to model periodic phenomena.
• MAT-12.AR.F.22 Restrict the domain of a trigonometric function to construct its inverse.
• MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.
• MAT-12.AR.F.24 Know and apply the addition and subtraction formulas for sine, cosine, and tangent to solve problems.

#### MAT-08.AR.F.03

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (F) Functions Learners will develop a foundational knowledge of functions and use them to model relationships between quantities.

## Progressions

Functional Relationships

• MAT-08.AR.F.01 Defend whether a relation is a function from various representations using appropriate function language.
• MAT-08.AR.F.02 Compare and contrast properties of two linear functions, each represented in a different way (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.03 Compare and contrast linear and non-linear functions represented in different ways (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.04 Model a linear relationship between two quantities by creating a table, graph, and equation. Interpret the rate of change and initial value of a linear function in terms of the situation it models.
• MAT-08.AR.F.05 Describe qualitatively the functional relationship between two quantities by analyzing a graph, including where the function is constant, increasing, or decreasing; linear or nonlinear; and discrete or continuous. Create a graph that exhibits the qualitative features of a function described.
• MAT-09.AR.F.01 Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.
• MAT-09.AR.F.02 Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of context.
• MAT-09.AR.F.03 Sketch key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.
• MAT-09.AR.F.04 Relate the domain of a linear, quadratic, or exponential function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-09.AR.F.05 Calculate and interpret the average rate of change of a linear, quadratic, or exponential function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-09.AR.F.06 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-09.AR.F.07 Compare key features of two linear, exponential, or quadratic functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
• MAT-09.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions.
• MAT-09.AR.F.10 Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse. Graph the inverse of a linear function.
• MAT-09.AR.F.11 Interpret the parameters of a linear, quadratic, or exponential function in terms of context.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).
• MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
• MAT-12.AR.F.01 Write a function that describes a relationship between two quantities.
• MAT-12.AR.F.02 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-12.AR.F.03 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-12.AR.F.04 Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.AR.F.05 Find inverse functions.
• MAT-12.AR.F.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.07 Compare key features of two functions, each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
• MAT-12.AR.F.08 Use tables, graphs, verbal discussions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
• MAT-12.AR.F.09 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-12.AR.F.11 Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.
• MAT-12.AR.F.12 Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity.
• MAT-12.AR.F.13 Determine whether a linear, quadratic, polynomial, exponential, logarithmic, or trigonometric model fits the situation. Determine an appropriate mathematical model in context (with or without technology).
• MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
• MAT-12.AR.F.16 Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π /3, π/4 and π/6.
• MAT-12.AR.F.17 Use the Pythagorean Identity sin²(θ) + cos²(θ) = 1 to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
• MAT-12.AR.F.18 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
• MAT-12.AR.F.19 Use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.
• MAT-12.AR.F.20 Use the unit circle to explain the symmetry (odd and even) and the periodicity of trigonometric functions.
• MAT-12.AR.F.21 Create a trigonometric function to model periodic phenomena.
• MAT-12.AR.F.22 Restrict the domain of a trigonometric function to construct its inverse.
• MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.
• MAT-12.AR.F.24 Know and apply the addition and subtraction formulas for sine, cosine, and tangent to solve problems.

#### MAT-08.AR.F.04

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (F) Functions Learners will develop a foundational knowledge of functions and use them to model relationships between quantities.

## Progressions

Functional Relationships

• MAT-08.AR.F.01 Defend whether a relation is a function from various representations using appropriate function language.
• MAT-08.AR.F.02 Compare and contrast properties of two linear functions, each represented in a different way (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.03 Compare and contrast linear and non-linear functions represented in different ways (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.04 Model a linear relationship between two quantities by creating a table, graph, and equation. Interpret the rate of change and initial value of a linear function in terms of the situation it models.
• MAT-08.AR.F.05 Describe qualitatively the functional relationship between two quantities by analyzing a graph, including where the function is constant, increasing, or decreasing; linear or nonlinear; and discrete or continuous. Create a graph that exhibits the qualitative features of a function described.
• MAT-09.AR.F.01 Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.
• MAT-09.AR.F.02 Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of context.
• MAT-09.AR.F.03 Sketch key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.
• MAT-09.AR.F.04 Relate the domain of a linear, quadratic, or exponential function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-09.AR.F.05 Calculate and interpret the average rate of change of a linear, quadratic, or exponential function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-09.AR.F.06 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-09.AR.F.07 Compare key features of two linear, exponential, or quadratic functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
• MAT-09.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions.
• MAT-09.AR.F.10 Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse. Graph the inverse of a linear function.
• MAT-09.AR.F.11 Interpret the parameters of a linear, quadratic, or exponential function in terms of context.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).
• MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
• MAT-12.AR.F.01 Write a function that describes a relationship between two quantities.
• MAT-12.AR.F.02 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-12.AR.F.03 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-12.AR.F.04 Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.AR.F.05 Find inverse functions.
• MAT-12.AR.F.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.07 Compare key features of two functions, each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
• MAT-12.AR.F.08 Use tables, graphs, verbal discussions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
• MAT-12.AR.F.09 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-12.AR.F.11 Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.
• MAT-12.AR.F.12 Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity.
• MAT-12.AR.F.13 Determine whether a linear, quadratic, polynomial, exponential, logarithmic, or trigonometric model fits the situation. Determine an appropriate mathematical model in context (with or without technology).
• MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
• MAT-12.AR.F.16 Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π /3, π/4 and π/6.
• MAT-12.AR.F.17 Use the Pythagorean Identity sin²(θ) + cos²(θ) = 1 to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
• MAT-12.AR.F.18 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
• MAT-12.AR.F.19 Use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.
• MAT-12.AR.F.20 Use the unit circle to explain the symmetry (odd and even) and the periodicity of trigonometric functions.
• MAT-12.AR.F.21 Create a trigonometric function to model periodic phenomena.
• MAT-12.AR.F.22 Restrict the domain of a trigonometric function to construct its inverse.
• MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.
• MAT-12.AR.F.24 Know and apply the addition and subtraction formulas for sine, cosine, and tangent to solve problems.

#### MAT-08.AR.F.05

 8th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  (F) Functions Learners will develop a foundational knowledge of functions and use them to model relationships between quantities.

## Progressions

Functional Relationships

• MAT-08.AR.F.01 Defend whether a relation is a function from various representations using appropriate function language.
• MAT-08.AR.F.02 Compare and contrast properties of two linear functions, each represented in a different way (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.03 Compare and contrast linear and non-linear functions represented in different ways (algebraically, graphically, numerically in tables, and/or by descriptions).
• MAT-08.AR.F.04 Model a linear relationship between two quantities by creating a table, graph, and equation. Interpret the rate of change and initial value of a linear function in terms of the situation it models.
• MAT-08.AR.F.05 Describe qualitatively the functional relationship between two quantities by analyzing a graph, including where the function is constant, increasing, or decreasing; linear or nonlinear; and discrete or continuous. Create a graph that exhibits the qualitative features of a function described.
• MAT-09.AR.F.01 Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.
• MAT-09.AR.F.02 Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of context.
• MAT-09.AR.F.03 Sketch key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.
• MAT-09.AR.F.04 Relate the domain of a linear, quadratic, or exponential function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-09.AR.F.05 Calculate and interpret the average rate of change of a linear, quadratic, or exponential function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-09.AR.F.06 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-09.AR.F.07 Compare key features of two linear, exponential, or quadratic functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
• MAT-09.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions.
• MAT-09.AR.F.10 Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse. Graph the inverse of a linear function.
• MAT-09.AR.F.11 Interpret the parameters of a linear, quadratic, or exponential function in terms of context.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (i.e., rigid versus non-rigid motion).
• MAT-10.GM.26 Recognize that the radian measure of an angle is the ratio of the length of the arc to the length of the radius of a circle.
• MAT-12.AR.F.01 Write a function that describes a relationship between two quantities.
• MAT-12.AR.F.02 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
• MAT-12.AR.F.03 Write a function defined by an expression in different but equivalent forms to reveal and explain the different properties of the function.
• MAT-12.AR.F.04 Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.AR.F.05 Find inverse functions.
• MAT-12.AR.F.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.07 Compare key features of two functions, each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
• MAT-12.AR.F.08 Use tables, graphs, verbal discussions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
• MAT-12.AR.F.09 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-12.AR.F.11 Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.
• MAT-12.AR.F.12 Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity.
• MAT-12.AR.F.13 Determine whether a linear, quadratic, polynomial, exponential, logarithmic, or trigonometric model fits the situation. Determine an appropriate mathematical model in context (with or without technology).
• MAT-12.AR.F.14 Write arithmetic and geometric sequences both recursively and with an explicit formula and convert between the two forms. Use sequences to model situations.
• MAT-12.AR.F.16 Extend right triangle trigonometry and apply knowledge of the unit circle to determine values of sine, cosine, and tangent for multiples of π /3, π/4 and π/6.
• MAT-12.AR.F.17 Use the Pythagorean Identity sin²(θ) + cos²(θ) = 1 to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
• MAT-12.AR.F.18 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
• MAT-12.AR.F.19 Use the unit circle to express the values of sine, cosine, and tangent for π - x, π + x, and 2π - x in terms of their values for x, where x is any real number.
• MAT-12.AR.F.20 Use the unit circle to explain the symmetry (odd and even) and the periodicity of trigonometric functions.
• MAT-12.AR.F.21 Create a trigonometric function to model periodic phenomena.
• MAT-12.AR.F.22 Restrict the domain of a trigonometric function to construct its inverse.
• MAT-12.AR.F.23 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions and interpret them in context.
• MAT-12.AR.F.24 Know and apply the addition and subtraction formulas for sine, cosine, and tangent to solve problems.

Graphing Functions

• MAT-08.AR.F.05 Describe qualitatively the functional relationship between two quantities by analyzing a graph, including where the function is constant, increasing, or decreasing; linear or nonlinear; and discrete or continuous. Create a graph that exhibits the qualitative features of a function described.
• MAT-09.AR.F.01 Determine whether a relationship is a function given a table, graph, or words, identifying x as an element of the domain and f(x) as an element in the range. Determine the domain and range of a function in context.
• MAT-09.AR.F.03 Sketch key features (to include intercepts, maximums, minimums, and lines of symmetry, where applicable) of linear, exponential, and quadratic functions modeling the relationship between two quantities using tables, graphs, written descriptions, and equations.
• MAT-09.AR.F.09 Identify the effect of transformations on the graph of a linear, absolute value, or quadratic function by replacing f(x) with af(x), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the value of a, h, and k given the graph of the function.
• MAT-09.AR.F.10 Find the inverse of a linear function and describe the relationship between the domain, range, and graph of the function and its inverse. Graph the inverse of a linear function.
• MAT-09.AR.F.12 Identify, using graphs or tables, the solution(s) to linear or exponential functions f(x) = g(x) as xvalues that result in equivalent y-values.
• MAT-12.AR.14 Identify zeros of polynomials when suitable factorizations are available. Use the zeros to construct a rough graph of the function defined by the polynomial.
• MAT-12.AR.16 Identify, using graphs, technology, tables, or successive approximations, that the solution(s) to the equation f(x) = g(x) is the x-value(s) that result in the y-values of f(x) and g(x) being the same.
• MAT-12.AR.F.04 Identify the effect of transformations on the graph of a function by replacing f(x) with af(x), f(bx), f(x - h), and f(x) + k, for specific values of a, h, and k (both positive and negative). Find the values of a, b, h, and k given the graph of the function. Recognize even and odd functions from their graphs and equations.
• MAT-12.AR.F.08 Use tables, graphs, verbal descriptions, and equations to interpret and sketch the key features of a function modeling the relationship between two quantities.
• MAT-12.AR.F.09 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
• MAT-12.AR.F.10 Graph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases.
• MAT-12.AR.F.11 Analyze and graph functions expressed symbolically (by hand in simple cases and using technology for more complicated cases), identifying key features of the graph.
• MAT-12.AR.F.12 Compare the end behavior of linear, quadratic, and exponential functions using graphs and/or tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing as a linear or quadratic function

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