text: 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.
explanation: The provided text is a list of educational standards and learning objectives for mathematics, specifically focusing on algebraic reasoning and functions. It outlines various skills and knowledge that students are expected to develop at different grade levels. The text does not contain a specific question or a question and answer.
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distilabel_metadata.content: 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a list of educational standards and learning objectives for mathematics, specifically focusing on algebraic reasoning and functions. It outlines various skills and knowledge that students are expected to develop at different grade levels. The text does not contain a specific question or a question and answer.
  
  Label: no-question