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#### MAT-09.AR

MAT-09.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-09.AR.01

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

## Progressions

Decompose Numbers and/or Expressions
• MAT-00.AR.OA.03 Decompose numbers less than or equal to 10 into pairs in more than one way using verbal explanations, objects, or drawings.
• MAT-01.AR.OA.03 Decompose numbers less than or equal to 20 in more than one way.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• 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.11 Interpret the parameters in a linear, quadratic, or exponential function in context.
• MAT-12.AR.01 Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.
• MAT-12.AR.03 Interpret expressions that represent a quantity in context.
• MAT-12.AR.06 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, division, or technology for the more complicated examples.
• 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.20 Apply the Binomial Theorem for the expansion of (ax + by)n in powers of x and y for a positive integer n and integers a and b
Factor Pairs/Multiples
• MAT-04.AR.OA.04 Find factor pairs and multiples within the range of 1-36 while classifying numbers as prime or composite.
• MAT-05.AR.OA.04 Find factor pairs and multiples within the range of 1-100 while classifying numbers as prime or composite.
• MAT-06.NO.O.04 Determine 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.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• 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.11 Interpret the parameters in a linear, quadratic, or exponential function 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-12.AR.03 Interpret expressions that represent a quantity in context.
• 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.06 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, division, or technology for the more complicated examples.
• 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.
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.
Patterns
• MAT-00.AR.OA.06 Recognize, duplicate, complete, and extend repeating patterns in a variety of contexts (e.g., shape, color, size, objects, sounds, and movement).
• MAT-01.AR.OA.07 Identify, create, complete, and extend patterns that are repeating, increasing, and decreasing in a variety of contexts.
• MAT-02.AR.OA.06 Identify a group of objects from 0 to 20 as even or odd by showing even numbers as a sum of two equal parts.
• MAT-03.AR.OA.06 Identify arithmetic patterns and explain them using the properties of operations.
• MAT-04.AR.OA.06 Generate a number or shape pattern that follows a given rule while identifying apparent features of the pattern that were not explicit in the rule itself.
• MAT-05.AR.OA.05 Generate two numerical patterns using two given rules and form ordered pairs consisting of corresponding terms from the two patterns. (Graphing on a coordinate plane).
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• MAT-12.AR.2 Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.

#### MAT-09.AR.02

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

## Progressions

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-09.AR.03

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions 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.
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-09.AR.04

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

## Progressions

Exponents

• MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• 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.01 Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.06 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 proper labels and scales.
• 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-0.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.
• MAT-12.NO.01 Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.
• 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.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.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.15 Use properties of logarithms to express the solution to abct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.

Unit Size and Scale

• MAT-06.NO.NS.02 Write, interpret, and explain statements of order for rational numbers on a number line diagram and in authentic contexts.
• MAT-09.NO.03 Choose and interpret the scale and the origin in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• 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-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• 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.NO.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• 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.
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.
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.

#### MAT-09.AR.05

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions 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-09.AR.06

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

## Progressions

Compare Numbers and/or Expressions

• MAT-00.NO.NBT.02 Compare two numbers between 1 and 20 using words greater than, less than, or equal to.
• MAT-01.NO.NBT.02 Compare two two-digit numbers using symbols >, <, and =. Justify comparisons based on the number of tens and ones.
• MAT-02.NO.NBT.02 Compare two three-digit numbers using symbols >, <, and =. Justify comparisons based on the value of thousands, hundreds, tens, and ones.
• MAT-03.NO.NBT.01 Compare two four-digit numbers using symbols, >, <, and =. Justify comparisons based on the value of thousands, hundreds, tens, and ones.
• MAT-04.NO.NBT.02 Compare two numbers to the millions place and decimals to the hundredths place, using symbols >, <, and =. Justify comparisons based on the value of the digits.
• MAT-05.NO.NBT.02 Compare two decimals to thousandths using symbols >, <, and =. Justify comparisons based on the value of the digits.
• MAT-06.NO.NS.01 Explain and show the relationship between non-zero rational numbers and their opposites using horizontal and vertical number lines in authentic problems. Use rational numbers to represent quantities in authentic contexts and explain the meaning of 0 in certain situations.
• MAT-06.NO.NS.02 Write, interpret, and explain statements of order for rational numbers on a number line and in authentic contexts.
• MAT-07.NO.NS.01 Describe the absolute value of a number as its distance from zero on a number line.
• MAT-08.NO.NS.01 Compare and classify real numbers within the real number system.
• MAT-08.NO.NS.02 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them on a number line diagram, and estimate the value of irrational expressions involving one operation.
• MAT-08.NO.NS.03 Use scientific notation to represent very large or very small quantities. Interpret scientific notation generated by technology. Compare and order numbers in scientific and standard notation.
• MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.
• MAT-09.AR.08 Graph the solution set to a two-variable system of linear equations. 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.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.

Exponents

• MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• 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.01 Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.06 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 proper labels and scales.
• 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-0.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.
• MAT-12.NO.01 Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.
• 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.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.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.15 Use properties of logarithms to express the solution to abct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.
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.
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-09.AR.07

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions 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.
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.
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-09.AR.08

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

## Progressions

Compare Numbers and/or Expressions

• MAT-00.NO.NBT.02 Compare two numbers between 1 and 20 using words greater than, less than, or equal to.
• MAT-01.NO.NBT.02 Compare two two-digit numbers using symbols >, <, and =. Justify comparisons based on the number of tens and ones.
• MAT-02.NO.NBT.02 Compare two three-digit numbers using symbols >, <, and =. Justify comparisons based on the value of thousands, hundreds, tens, and ones.
• MAT-03.NO.NBT.01 Compare two four-digit numbers using symbols, >, <, and =. Justify comparisons based on the value of thousands, hundreds, tens, and ones.
• MAT-04.NO.NBT.02 Compare two numbers to the millions place and decimals to the hundredths place, using symbols >, <, and =. Justify comparisons based on the value of the digits.
• MAT-05.NO.NBT.02 Compare two decimals to thousandths using symbols >, <, and =. Justify comparisons based on the value of the digits.
• MAT-06.NO.NS.01 Explain and show the relationship between non-zero rational numbers and their opposites using horizontal and vertical number lines in authentic problems. Use rational numbers to represent quantities in authentic contexts and explain the meaning of 0 in certain situations.
• MAT-06.NO.NS.02 Write, interpret, and explain statements of order for rational numbers on a number line and in authentic contexts.
• MAT-07.NO.NS.01 Describe the absolute value of a number as its distance from zero on a number line.
• MAT-08.NO.NS.01 Compare and classify real numbers within the real number system.
• MAT-08.NO.NS.02 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them on a number line diagram, and estimate the value of irrational expressions involving one operation.
• MAT-08.NO.NS.03 Use scientific notation to represent very large or very small quantities. Interpret scientific notation generated by technology. Compare and order numbers in scientific and standard notation.
• MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.
• MAT-09.AR.08 Graph the solution set to a two-variable system of linear equations. 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.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.
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-09.AR.09

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

## Progressions

Compare Numbers and/or Expressions

• MAT-00.NO.NBT.02 Compare two numbers between 1 and 20 using words greater than, less than, or equal to.
• MAT-01.NO.NBT.02 Compare two two-digit numbers using symbols >, <, and =. Justify comparisons based on the number of tens and ones.
• MAT-02.NO.NBT.02 Compare two three-digit numbers using symbols >, <, and =. Justify comparisons based on the value of thousands, hundreds, tens, and ones.
• MAT-03.NO.NBT.01 Compare two four-digit numbers using symbols, >, <, and =. Justify comparisons based on the value of thousands, hundreds, tens, and ones.
• MAT-04.NO.NBT.02 Compare two numbers to the millions place and decimals to the hundredths place, using symbols >, <, and =. Justify comparisons based on the value of the digits.
• MAT-05.NO.NBT.02 Compare two decimals to thousandths using symbols >, <, and =. Justify comparisons based on the value of the digits.
• MAT-06.NO.NS.01 Explain and show the relationship between non-zero rational numbers and their opposites using horizontal and vertical number lines in authentic problems. Use rational numbers to represent quantities in authentic contexts and explain the meaning of 0 in certain situations.
• MAT-06.NO.NS.02 Write, interpret, and explain statements of order for rational numbers on a number line and in authentic contexts.
• MAT-07.NO.NS.01 Describe the absolute value of a number as its distance from zero on a number line.
• MAT-08.NO.NS.01 Compare and classify real numbers within the real number system.
• MAT-08.NO.NS.02 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them on a number line diagram, and estimate the value of irrational expressions involving one operation.
• MAT-08.NO.NS.03 Use scientific notation to represent very large or very small quantities. Interpret scientific notation generated by technology. Compare and order numbers in scientific and standard notation.
• MAT-09.AR.06 Solve linear equations and inequalities (to include compound inequalities) in one variable.
• MAT-09.AR.08 Graph the solution set to a two-variable system of linear equations. 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.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.
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-09.AR.10

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

## Progressions

• MAT-09.AR.10 Solve quadratic equations in one variable by inspection (e.g., for x2 = 49) taking square roots, the quadratic formula, and factoring, as appropriate to the initial form of the equation.
• MAT-12.NO.09 Apply the Fundamental Theorem of Algebra to determine the number of zeros for polynomial functions. Find all solutions to a polynomial equation.
• 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.10 Derive the quadratic formula from the form 0 = ax2 + bx + c.
• MAT-12.AR.11 Solve quadratic equations with real coefficients that have solutions of the form a + bi and a - bi.
• 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.17 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

#### MAT-09.AR.11

 9th Grade (MAT) Targeted Standard     (AR) Algebraic Reasoning  Learners will look for, generate, and make sense of patterns, relationships, and algebraic symbols to represent mathematical models while adopting approaches and solutions in novel situations.

## Progressions

• MAT-01.NO.NBT.03 Add within 100 using a two-digit number and a one-digit number. Use concrete models, drawings, and strategies that reflect an understanding of place value.
• MAT-02.NO.NBT.03 Add within 100 using place value strategies and/or the relationship between addition and subtraction.
• MAT-01.NO.NBT.04 Subtract multiples of 10 within 100 using concrete models, drawings, and strategies that reflect an understanding of place value.
• MAT-01.NO.NBT.05 Mentally add or subtract 10 to or from a given two-digit number and explain the reasoning used.
• MAT-02.NO.NBT.04 Subtract within 100 using place value strategies and/or the relationship between addition and subtraction.
• MAT-02.NO.NBT.05 Mentally add or subtract 10 or 100 to or from a given number between 100 and 900.
• MAT-03.NO.NBT.03 Add and subtract within 1000 using place value strategies, algorithms, and/or the relationship between addition and subtraction.
• MAT-04.NO.NBT.04 Add and subtract multi-digit whole numbers to the one million place using strategies flexibly, including the algorithm.
• MAT-05.NO.NBT.05 Use concrete models, drawings, place value strategies, properties of operations, and/or relationships to add, subtract, and multiply decimals to hundredths.
• MAT-07.NO.O.01 Add, subtract, multiply, and divide integers using visual models and properties of operations in multi-step authentic and mathematical problems, including authentic problems.
• MAT-07.NO.O.02 Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.
• MAT-07.NO.O.03 Add, subtract, multiply, and divide non-negative decimals to the hundredth place in multi-step problems using strategies or procedures, including authentic problems.
• MAT-08.NO.O.02 Add, subtract, multiply, and divide rational numbers using strategies or procedures.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.11 Add, subtract, and multiply polynomials.
• MAT-12.NO.03 Demonstrate that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.
• MAT-12.AR.13 Add, subtract, and multiply polynomials beyond quadratics. Understand that polynomials form a system comparable to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
• MAT-12.NO.11 Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.
• MAT-12.NO.17 Add and subtract vectors. Represent vector subtraction graphically by connecting the tips of the appropriate order and using the components to perform vector subtraction.
• MAT-12.NO.19 Represent data in a matrix. Perform operations (i.e., addition, subtraction, multiplication) on matrices of appropriate dimensions to solve problems and in context. Know that matrix multiplication is not commutative.

Multiplication and Division

• MAT-03.NO.NBT.04 Multiply one-digit whole numbers by multiples of 10 within 100.
• MAT-04.NO.NBT.05 Multiply a whole number up to four digits by a one-digit whole number and multiply two two digit numbers. Show and justify the calculation using equations, rectangular arrays, and models.
• MAT-05.NO.NBT.04 Multiply multi-digit whole numbers using strategies flexibly, including the algorithm.
• MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
• MAT-04.NO.NBT.06 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors using place value strategies. Show and justify the calculation by using equations, rectangular arrays, and models.
• MAT-05.NO.NBT.05 Use concrete models, drawings, place value strategies, properties of operations, and/or relationships to add, subtract, and multiply decimals to hundredths.
• MAT-05.NO.NBT.06 Find whole-number quotients and remainders with up to four-digit dividends and two-digit divisors using place value strategies. Show and justify the calculation using equations, rectangular arrays, and/or area models.
• MAT-06.NO.O.01 Divide multi-digit whole numbers up to four-digit dividends and two-digit divisors using strategies or procedures.
• MAT-07.NO.O.01 Add, subtract, multiply, and divide integers and positive rational numbers using visual models and properties of operations in multi-step problems, including authentic problems.
• MAT-07.NO.O.02 Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.
• MAT-07.NO.O.03 Add, subtract, multiply, and divide non-negative decimals to the hundredth place in multi-step problems using strategies or procedures, including authentic problems.
• MAT-08.NO.O.01 Evaluate mentally the square roots of perfect squares up to 225 and cube roots of perfect cubes up to 1000.
• MAT-08.NO.O.02 Add, subtract, multiply, and divide rational numbers using strategies or procedures.
• MAT-09.NO.02 Perform basic operations on radicals and simplify radicals to write equivalent expressions.
• MAT-09.AR.11 Add, subtract, and multiply polynomials.
• MAT-12.NO.03 Demonstrate that the sum or product of two rational numbers is rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.
• MAT-12.AR.13 Add, subtract, and multiply polynomials beyond quadratics. Understand that polynomials form a system comparable to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
• MAT-12.NO.11 Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.
• MAT-12.NO.19 Represent data in a matrix. Perform operations (i.e., addition, subtraction, multiplication) on matrices of appropriate dimensions to solve problems and in context. Know that matrix multiplication is not commutative.

#### MAT-09.AR.F.01

 9th 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

#### MAT-09.AR.F.02

 9th 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-09.AR.F.03

 9th 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

#### MAT-09.AR.F.04

 9th 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-09.AR.F.05

 9th 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

Unit Rate
• MAT-06.AR.RP.02 Describe and calculate a unit rate when given a ratio relationship between two quantities using rate language and visual models.
• MAT-06.AR.RP.05 Convert measurement units within and between measurement systems using ratio reasoning given conversion factors.
• MAT-06.AR.RP.04 Calculate a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part, and the percent.
• MAT-07.AR.RP.01 Calculate unit rates associated with ratios of rational numbers, including ratios of lengths, areas, and other quantities measured in like or different units.
• MAT-09.NO.03 Choose and interpret the scale and the units in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.AR.F.05 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-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.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• 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.
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-09.AR.F.06

 9th 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

Exponents

• MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• 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.01 Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.06 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 proper labels and scales.
• 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-0.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.
• MAT-12.NO.01 Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.
• 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.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.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.15 Use properties of logarithms to express the solution to abct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.
Factor Pairs/Multiples
• MAT-04.AR.OA.04 Find factor pairs and multiples within the range of 1-36 while classifying numbers as prime or composite.
• MAT-05.AR.OA.04 Find factor pairs and multiples within the range of 1-100 while classifying numbers as prime or composite.
• MAT-06.NO.O.04 Determine 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.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• 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.11 Interpret the parameters in a linear, quadratic, or exponential function 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-12.AR.03 Interpret expressions that represent a quantity in context.
• 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.06 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, division, or technology for the more complicated examples.
• 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.
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.
Unit Rate
• MAT-06.AR.RP.02 Describe and calculate a unit rate when given a ratio relationship between two quantities using rate language and visual models.
• MAT-06.AR.RP.05 Convert measurement units within and between measurement systems using ratio reasoning given conversion factors.
• MAT-06.AR.RP.04 Calculate a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part, and the percent.
• MAT-07.AR.RP.01 Calculate unit rates associated with ratios of rational numbers, including ratios of lengths, areas, and other quantities measured in like or different units.
• MAT-09.NO.03 Choose and interpret the scale and the units in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.AR.F.05 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-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.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• 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.
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-09.AR.F.07

 9th 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-09.AR.F.08

 9th 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

Exponents

• MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• 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.01 Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.06 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 proper labels and scales.
• 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-0.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.
• MAT-12.NO.01 Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.
• 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.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.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.15 Use properties of logarithms to express the solution to abct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.
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-09.AR.F.09

 9th 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

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

Transformations

• MAT-08.GM.GF.01 Perform single transformations to a figure on or off the coordinate plane and determine whether the figures are congruent or similar.
• MAT-08.GM.GF.02 Describe the characteristics of transformations on the coordinate plane using transformation language.
• MAT-08.GM.GF.03 Name the type of transformation(s) needed to map a pre-image to its image.
• 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 f(x) + k, f(x - h) and af(x), for specific values of a, h, and k (both positive and negative). Find the values of a, h, and k given the graph of the function.
• MAT-10.GM.02 Represent transformations in the plane. Describe transformations as functions that take 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.03 Describe the rotations and reflections of a triangle, rectangle, parallelogram, trapezoid, or regular polygon that map each figure onto itself or another figure.
• MAT-10.GM.04 Develop or verify the characteristics of rotations, reflections, and translations in angles, circles, perpendicular lines, parallel lines, and line segments.
• MAT-10.GM.05 Draw the image of a figure that has undergone a series of transformations [rotation(s),
• reflection(s), or translation(s)] of a geometric figure using a variety of methods (e.g., graph paper, tracing paper, or geometry software).
• MAT-10.GM.06 Predict the effect of a specified rigid motion on a given figure using geometric descriptions of rigid motions. Determine whether two figures are congruent using the definition of congruence in terms of rigid motions.
• 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 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.17 Apply knowledge of congruence and similarity criteria for triangles to solve problems and prove relationships in various geometric figures.
• MAT-12.AR.F.4 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.NO.18 Multiply a vector by a scalar.

#### MAT-09.AR.F.10

 9th 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

#### MAT-09.AR.F.11

 9th 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

Decompose Numbers and/or Expressions
• MAT-00.AR.OA.03 Decompose numbers less than or equal to 10 into pairs in more than one way using verbal explanations, objects, or drawings.
• MAT-01.AR.OA.03 Decompose numbers less than or equal to 20 in more than one way.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• 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.11 Interpret the parameters in a linear, quadratic, or exponential function in context.
• MAT-12.AR.01 Use the structure of an expression (to extend to polynomial and rational expressions) to identify ways to rewrite it.
• MAT-12.AR.03 Interpret expressions that represent a quantity in context.
• MAT-12.AR.06 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, division, or technology for the more complicated examples.
• 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.20 Apply the Binomial Theorem for the expansion of (ax + by)n in powers of x and y for a positive integer n and integers a and b
Factor Pairs/Multiples
• MAT-04.AR.OA.04 Find factor pairs and multiples within the range of 1-36 while classifying numbers as prime or composite.
• MAT-05.AR.OA.04 Find factor pairs and multiples within the range of 1-100 while classifying numbers as prime or composite.
• MAT-06.NO.O.04 Determine 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.
• MAT-09.AR.01 Use the structure of an expression (i.e., quadratic and exponential) to identify ways to rewrite it.
• 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.11 Interpret the parameters in a linear, quadratic, or exponential function 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-12.AR.03 Interpret expressions that represent a quantity in context.
• 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.06 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, division, or technology for the more complicated examples.
• 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.
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-09.AR.F.12

 9th 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

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

#### MAT-09.NO

MAT-09.NO Domain

# (NO) Number and Operations

## Sub-Categories

• (CC) Counting and Cardinality
Learners will understand the relationship between numerical symbols, names, quantities, and counting sequences.
• (NBT) Number Base Ten
Learners will understand the place value structure of the base-ten number system and represent, compare, and perform operations with multi-digit whole numbers and decimals.
• (NF) Number Fractions
Learners will understand fractions and equivalency to represent, compare, and perform operations of fractions and decimals.
• (NS) Number System
Learners will expand their knowledge of the number system to create connections and solve problems within and across concepts.
• (O) Operations
Learners will expand their computational fluency to create connections and solve problems within and across concepts.

## 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-09.NO.01

 9th Grade (MAT) Targeted Standard    (NO) Number and Operations  Learners will develop a foundational understanding of the number system, operations, and computational fluency to create connections and solve problems within and across concepts.

## Progressions

Exponents

• MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• 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.01 Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.06 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 proper labels and scales.
• 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-0.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.
• MAT-12.NO.01 Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.
• 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.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.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.15 Use properties of logarithms to express the solution to abct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.

#### MAT-09.NO.02

 9th Grade (MAT) Targeted Standard    (NO) Number and Operations  Learners will develop a foundational understanding of the number system, operations, and computational fluency to create connections and solve problems within and across concepts.

## Progressions

• MAT-01.NO.NBT.03 Add within 100 using a two-digit number and a one-digit number. Use concrete models, drawings, and strategies that reflect an understanding of place value.
• MAT-02.NO.NBT.03 Add within 100 using place value strategies and/or the relationship between addition and subtraction.
• MAT-01.NO.NBT.04 Subtract multiples of 10 within 100 using concrete models, drawings, and strategies that reflect an understanding of place value.
• MAT-01.NO.NBT.05 Mentally add or subtract 10 to or from a given two-digit number and explain the reasoning used.
• MAT-02.NO.NBT.04 Subtract within 100 using place value strategies and/or the relationship between addition and subtraction.
• MAT-02.NO.NBT.05 Mentally add or subtract 10 or 100 to or from a given number between 100 and 900.
• MAT-03.NO.NBT.03 Add and subtract within 1000 using place value strategies, algorithms, and/or the relationship between addition and subtraction.
• MAT-04.NO.NBT.04 Add and subtract multi-digit whole numbers to the one million place using strategies flexibly, including the algorithm.
• MAT-05.NO.NBT.05 Use concrete models, drawings, place value strategies, properties of operations, and/or relationships to add, subtract, and multiply decimals to hundredths.
• MAT-07.NO.O.01 Add, subtract, multiply, and divide integers using visual models and properties of operations in multi-step authentic and mathematical problems, including authentic problems.
• MAT-07.NO.O.02 Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.
• MAT-07.NO.O.03 Add, subtract, multiply, and divide non-negative decimals to the hundredth place in multi-step problems using strategies or procedures, including authentic problems.
• MAT-08.NO.O.02 Add, subtract, multiply, and divide rational numbers using strategies or procedures.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.11 Add, subtract, and multiply polynomials.
• MAT-12.NO.03 Demonstrate that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.
• MAT-12.AR.13 Add, subtract, and multiply polynomials beyond quadratics. Understand that polynomials form a system comparable to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
• MAT-12.NO.11 Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.
• MAT-12.NO.17 Add and subtract vectors. Represent vector subtraction graphically by connecting the tips of the appropriate order and using the components to perform vector subtraction.
• MAT-12.NO.19 Represent data in a matrix. Perform operations (i.e., addition, subtraction, multiplication) on matrices of appropriate dimensions to solve problems and in context. Know that matrix multiplication is not commutative.

Multiplication and Division

• MAT-03.NO.NBT.04 Multiply one-digit whole numbers by multiples of 10 within 100.
• MAT-04.NO.NBT.05 Multiply a whole number up to four digits by a one-digit whole number and multiply two two digit numbers. Show and justify the calculation using equations, rectangular arrays, and models.
• MAT-05.NO.NBT.04 Multiply multi-digit whole numbers using strategies flexibly, including the algorithm.
• MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
• MAT-04.NO.NBT.06 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors using place value strategies. Show and justify the calculation by using equations, rectangular arrays, and models.
• MAT-05.NO.NBT.05 Use concrete models, drawings, place value strategies, properties of operations, and/or relationships to add, subtract, and multiply decimals to hundredths.
• MAT-05.NO.NBT.06 Find whole-number quotients and remainders with up to four-digit dividends and two-digit divisors using place value strategies. Show and justify the calculation using equations, rectangular arrays, and/or area models.
• MAT-06.NO.O.01 Divide multi-digit whole numbers up to four-digit dividends and two-digit divisors using strategies or procedures.
• MAT-07.NO.O.01 Add, subtract, multiply, and divide integers and positive rational numbers using visual models and properties of operations in multi-step problems, including authentic problems.
• MAT-07.NO.O.02 Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.
• MAT-07.NO.O.03 Add, subtract, multiply, and divide non-negative decimals to the hundredth place in multi-step problems using strategies or procedures, including authentic problems.
• MAT-08.NO.O.01 Evaluate mentally the square roots of perfect squares up to 225 and cube roots of perfect cubes up to 1000.
• MAT-08.NO.O.02 Add, subtract, multiply, and divide rational numbers using strategies or procedures.
• MAT-09.NO.02 Perform basic operations on radicals and simplify radicals to write equivalent expressions.
• MAT-09.AR.11 Add, subtract, and multiply polynomials.
• MAT-12.NO.03 Demonstrate that the sum or product of two rational numbers is rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.
• MAT-12.AR.13 Add, subtract, and multiply polynomials beyond quadratics. Understand that polynomials form a system comparable to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
• MAT-12.NO.11 Represent addition, subtraction, multiplication, conjugation, powers, and roots of complex numbers geometrically on the complex and/or polar plane; use properties of this representation for computation.
• MAT-12.NO.19 Represent data in a matrix. Perform operations (i.e., addition, subtraction, multiplication) on matrices of appropriate dimensions to solve problems and in context. Know that matrix multiplication is not commutative.

Multiplying and Dividing Fractions

• MAT-04.NO.NF.07 Solve problems by multiplying fractions and whole numbers using visual fraction models (proper and improper fractions limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100).
• MAT-05.NO.NF.02 Explain why multiplying a given number by a fraction greater than one results in a product greater than the given number and explain why multiplying a given number by a fraction less than one results in a product smaller than the given number.
• MAT-05.NO.NF.04 Solve authentic word problems by multiplying fractions and mixed numbers using visual fraction models and equations.
• MAT-06.NO.O.03 Apply multiplication and division of fractions and decimals to solve and interpret problems using visual models, including authentic problems.
• MAT-07.NO.O.02 Add, subtract, multiply, and divide non-negative fractions in multi-step problems, including authentic problems.
• MAT-08.NO.O.02 Add, subtract, multiply, and divide rational numbers using strategies or procedures.
• MAT-09.NO.02 Perform basic operations of simple radical expressions to write a simplified equivalent expression.
• MAT-12.NO.03 Demonstrate that the sum or product of two rational numbers is rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational.
• MAT-12.AR.5 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.NO.06 Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Understand the hierarchal relationships among subsets of the complex number system.

Exponents

• MAT-05.NO.NBT.07 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
• MAT-06.AR.EE.01 Write, read, and evaluate numerical expressions, including expressions with whole number exponents and grouping symbols.
• 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.01 Explain how the definition of rational exponents follows from extending the properties of integer exponents; rewrite simple expressions involving radicals and rational exponents using the properties of exponents.
• MAT-09.NO.02 Perform basic operations on simple radical expressions to write a simplified equivalent expression.
• MAT-09.AR.06 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 proper labels and scales.
• 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-0.AR.F.08 Identify situations that can be modeled with linear, quadratic, and exponential functions. Justify the most appropriate model for a situation based on the rate of change over equal intervals. Include situations in which a quantity grows or decays.
• MAT-12.NO.01 Rewrite complex expressions involving radicals and rational exponents using the properties of exponents.
• 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.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.06 Apply the inverse relationship between exponents and logarithms to solve problems.
• MAT-12.AR.F.15 Use properties of logarithms to express the solution to abct = d where a, c, and d are real numbers and b is a positive real number. Evaluate the logarithm using technology when appropriate.

#### MAT-09.NO.03

 9th Grade (MAT) Targeted Standard    (NO) Number and Operations  Learners will develop a foundational understanding of the number system, operations, and computational fluency to create connections and solve problems within and across concepts.

## Progressions

Unit Size and Scale

• MAT-06.NO.NS.02 Write, interpret, and explain statements of order for rational numbers on a number line diagram and in authentic contexts.
• MAT-09.NO.03 Choose and interpret the scale and the origin in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• 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-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• 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.NO.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• 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.

Unit Rate

• MAT-06.AR.RP.02 Describe and calculate a unit rate when given a ratio relationship between two quantities using rate language and visual models.
• MAT-06.AR.RP.05 Convert measurement units within and between measurement systems using ratio reasoning given conversion factors.
• MAT-06.AR.RP.04 Calculate a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part, and the percent.
• MAT-07.AR.RP.01 Calculate unit rates associated with ratios of rational numbers, including ratios of lengths, areas, and other quantities measured in like or different units.
• MAT-09.NO.03 Choose and interpret the scale and the units in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.AR.F.05 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-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.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• 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.

Units of Measurement

• MAT-03.GM.M.02 Measure and estimate liquid volumes and masses of objects using standard units. Solve one step authentic word problems involving masses or volumes given in the same units.
• MAT-04.GM.M.01 Know the relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec. Record measurement equivalents in a two-column table.
• MAT-04.GM.M.03 Identify and use the appropriate tools, operations, and units of measurement, both customary and metric to solve problems involving time, length, weight, mass, and capacity.
• MAT-04.GM.M.02 Generate simple conversions from a larger unit to a smaller unit to solve authentic problems within a single system of measurement, both customary and metric systems.
• MAT-05.GM.M.01 Generate conversions among different-sized standard measurement units within a given measurement system, both customary and metric systems. Use these conversions in solving multi-step, authentic word problems.
• MAT-06.AR.RP.05 Convert measurement units within and between measurement systems using ratio reasoning given conversion factors.
• 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-09.NO.03 Choose and interpret the scale and the origin in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• MAT-12.NO.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.

Displaying Data

• MAT-01.DPS.D.01 Collect, organize and represent data with up to three categories using picture and bar graphs.
• MAT-02.DPS.D.01 Formulate questions and collect, organize, and represent data, with up to four categories using single unit scaled pictures and bar graphs.
• MAT-03.DPS.D.01 Formulate questions to collect, organize, and represent data with more than four categories using scaled pictures and bar graphs.
• MAT-04.DPS.D.01 Formulate questions to collect, organize, and represent data to reason with math and across disciplines.
• MAT-02.DPS.D.02 Generate data and create line plots marked in whole number units.
• MAT-03.DPS.D.02 Generate data and create line plots marked in whole numbers, halves, and fourths of a unit.
• MAT-04.DPS.D.02 Generate data and create line plots to display a data set of fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.
• MAT-05.DPS.D.01 Generate data and create line plots to display a data set of fractions of a unit (1/2, 1/4, 1/8). Use grade-level operations for fractions to solve problems involving information presented in line plots.
• MAT-06.DPS.DA.04 Display numerical data in plots on a number line, including dot plots and histograms. Describe any overall patterns in data, such as gaps, clusters, and skews.
• MAT-09.NO.03 Choose and interpret the scale and the units in graphs and data displays.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• MAT-10.DPS.01 Represent data with plots on the real number line (dot plots, histograms, and box plots).
• MAT-10.DPS.03 Represent data on two quantitative variables on a scatter plot and describe how the variables are related.
• MAT-10.DPS.10 Construct and interpret two-way frequency tables of data for two categorical variables. Use the two-way table as a sample space to decide if events are independent and approximate conditional probabilities.
• MAT-12.NO.04 Use units as a way to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.DPS.04 Represent data on a scatter plot for two quantitative variables and describe how the variables are related.

#### MAT-09.NO.04

 9th Grade (MAT) Targeted Standard    (NO) Number and Operations  Learners will develop a foundational understanding of the number system, operations, and computational fluency to create connections and solve problems within and across concepts.

## Progressions

Unit Size and Scale

• MAT-06.NO.NS.02 Write, interpret, and explain statements of order for rational numbers on a number line diagram and in authentic contexts.
• MAT-09.NO.03 Choose and interpret the scale and the origin in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• 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-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• 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.NO.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• 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.

Unit Rate

• MAT-06.AR.RP.02 Describe and calculate a unit rate when given a ratio relationship between two quantities using rate language and visual models.
• MAT-06.AR.RP.05 Convert measurement units within and between measurement systems using ratio reasoning given conversion factors.
• MAT-06.AR.RP.04 Calculate a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part, and the percent.
• MAT-07.AR.RP.01 Calculate unit rates associated with ratios of rational numbers, including ratios of lengths, areas, and other quantities measured in like or different units.
• MAT-09.NO.03 Choose and interpret the scale and the units in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.AR.F.05 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-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.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• 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.

Units of Measurement

• MAT-03.GM.M.02 Measure and estimate liquid volumes and masses of objects using standard units. Solve one step authentic word problems involving masses or volumes given in the same units.
• MAT-04.GM.M.01 Know the relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec. Record measurement equivalents in a two-column table.
• MAT-04.GM.M.03 Identify and use the appropriate tools, operations, and units of measurement, both customary and metric to solve problems involving time, length, weight, mass, and capacity.
• MAT-04.GM.M.02 Generate simple conversions from a larger unit to a smaller unit to solve authentic problems within a single system of measurement, both customary and metric systems.
• MAT-05.GM.M.01 Generate conversions among different-sized standard measurement units within a given measurement system, both customary and metric systems. Use these conversions in solving multi-step, authentic word problems.
• MAT-06.AR.RP.05 Convert measurement units within and between measurement systems using ratio reasoning given conversion factors.
• 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-09.NO.03 Choose and interpret the scale and the origin in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• MAT-12.NO.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.

#### MAT-09.NO.05

 9th Grade (MAT) Targeted Standard    (NO) Number and Operations  Learners will develop a foundational understanding of the number system, operations, and computational fluency to create connections and solve problems within and across concepts.

## Progressions

Unit Size and Scale

• MAT-06.NO.NS.02 Write, interpret, and explain statements of order for rational numbers on a number line diagram and in authentic contexts.
• MAT-09.NO.03 Choose and interpret the scale and the origin in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• 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-10.GM.14 Verify experimentally and justify the properties of dilations given by a center and a scale factor.
• 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.NO.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• 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.

Units of Measurement

• MAT-03.GM.M.02 Measure and estimate liquid volumes and masses of objects using standard units. Solve one step authentic word problems involving masses or volumes given in the same units.
• MAT-04.GM.M.01 Know the relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb., oz.; l, ml; hr., min., sec. Record measurement equivalents in a two-column table.
• MAT-04.GM.M.03 Identify and use the appropriate tools, operations, and units of measurement, both customary and metric to solve problems involving time, length, weight, mass, and capacity.
• MAT-04.GM.M.02 Generate simple conversions from a larger unit to a smaller unit to solve authentic problems within a single system of measurement, both customary and metric systems.
• MAT-05.GM.M.01 Generate conversions among different-sized standard measurement units within a given measurement system, both customary and metric systems. Use these conversions in solving multi-step, authentic word problems.
• MAT-06.AR.RP.05 Convert measurement units within and between measurement systems using ratio reasoning given conversion factors.
• 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-09.NO.03 Choose and interpret the scale and the origin in graphs and data displays.
• MAT-09.NO.04 Define appropriate quantities and units for the purpose of descriptive modeling.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• MAT-12.NO.04 Use units to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.

Displaying Data

• MAT-01.DPS.D.01 Collect, organize and represent data with up to three categories using picture and bar graphs.
• MAT-02.DPS.D.01 Formulate questions and collect, organize, and represent data, with up to four categories using single unit scaled pictures and bar graphs.
• MAT-03.DPS.D.01 Formulate questions to collect, organize, and represent data with more than four categories using scaled pictures and bar graphs.
• MAT-04.DPS.D.01 Formulate questions to collect, organize, and represent data to reason with math and across disciplines.
• MAT-02.DPS.D.02 Generate data and create line plots marked in whole number units.
• MAT-03.DPS.D.02 Generate data and create line plots marked in whole numbers, halves, and fourths of a unit.
• MAT-04.DPS.D.02 Generate data and create line plots to display a data set of fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.
• MAT-05.DPS.D.01 Generate data and create line plots to display a data set of fractions of a unit (1/2, 1/4, 1/8). Use grade-level operations for fractions to solve problems involving information presented in line plots.
• MAT-06.DPS.DA.04 Display numerical data in plots on a number line, including dot plots and histograms. Describe any overall patterns in data, such as gaps, clusters, and skews.
• MAT-09.NO.03 Choose and interpret the scale and the units in graphs and data displays.
• MAT-09.NO.05 Choose a level of accuracy or precision appropriate to limitations on measurement when reporting quantities.
• MAT-10.DPS.01 Represent data with plots on the real number line (dot plots, histograms, and box plots).
• MAT-10.DPS.03 Represent data on two quantitative variables on a scatter plot and describe how the variables are related.
• MAT-10.DPS.10 Construct and interpret two-way frequency tables of data for two categorical variables. Use the two-way table as a sample space to decide if events are independent and approximate conditional probabilities.
• MAT-12.NO.04 Use units as a way to understand problems and to guide the solution of multi-step problems (e.g., unit analysis). Choose and interpret units consistently in formulas. Choose and interpret the scale and the units in graphs and data displays.
• MAT-12.DPS.04 Represent data on a scatter plot for two quantitative variables and describe how the variables are related.

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