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Skills available for Arizona high school math standards

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Arizona Mathematics Standards: Algebra 1 Arizona Mathematics Standards: Algebra 1
Arizona Mathematics Standards: Algebra 2 Arizona Mathematics Standards: Algebra 2
Arizona Mathematics Standards: Geometry Arizona Mathematics Standards: Geometry
Arizona Mathematics Standards: Plus Standards Arizona Mathematics Standards: Plus Standards
Arizona's College and Career Ready Standards Arizona's College and Career Ready Standards

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A1.N Number and Quantity

A1.N-RN The Real Number System
- A1.N-RN.B Use properties of rational and irrational numbers.
  - A1.N-RN.B.3 Explain why 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.
A1.N-Q Quantities
- A1.N-Q.A Reason quantitatively and use units to solve problems.
  - A1.N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.
  - A1.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.
  - A1.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.
  - Checkpoint opportunity
    - Checkpoint: Units and quantities (A1-O.6)

A1.A Algebra

A1.A-SSE Seeing Structure in Expressions
- A1.A-SSE.A Interpret the structure of expressions.
  - A1.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
    - A1.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
    - A1.A-SSE.A.1.b Interpret expressions by viewing one or more of their parts as a single entity.
      - Interpret parts of quadratic expressions: word problems (A1-CC.)
  - A1.A-SSE.A.2 Use structure to identify ways to rewrite numerical and polynomial expressions. Focus on polynomial multiplication and factoring patterns.
- A1.A-SSE.B Write expressions in equivalent forms to solve problems.
  - A1.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
    - A1.A-SSE.B.3.a Factor a quadratic expression to reveal the zeros of the function it defines.
    - A1.A-SSE.B.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  - Checkpoint opportunity
    - Checkpoint: Write and interpret equivalent expressions (A1)
A1.A-APR Arithmetic with Polynomials and Rational Expressions
- A1.A-APR.A Perform arithmetic operations on polynomials.
  - A1.A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
- A1.A-APR.B Understand the relationship between zeros and factors of polynomials.
  - A1.A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Focus on quadratic and cubic polynomials in which linear and quadratic factors are available.
    - Solve a quadratic equation using the zero product property (A1-CC.7)
    - Match quadratic functions and graphs (A1-CC.15)
  - Checkpoint opportunity
    - Checkpoint: Polynomial operations (A1-AA.13)
A1.A-CED Creating Equations
- A1.A-CED.A Create equations that describe numbers or relationships.
  - A1.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on equations and inequalities that are linear, quadratic, or exponential.
  - A1.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  - A1.A-CED.A.3 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.
  - A1.A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
    - Rearrange multi-variable equations (A1-I.11)
  - Checkpoint opportunity
    - Checkpoint: Represent constraints (A1-V.17)
    - Checkpoint: Problem solving with equations and inequalities (A1-DD.13)
A1.A-REI Reasoning with Equations and Inequalities
- A1.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.
  - A1.A-REI.A.1 Explain each step in solving linear and quadratic equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- A1.REI.B Solve equations and inequalities in one variable.
  - A1.A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  - A1.A-REI.B.4 Solve quadratic equations in one variable.
    - A1.A-REI.B.4.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x + k)² = q that has the same solutions. Derive the quadratic formula from this form.
      - Complete the square (A1-CC.9)
    - A1.A-REI.B.4.b Solve quadratic equations by inspection (e.g., x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Focus on solutions for quadratic equations that have real roots. Include cases that recognize when a quadratic equation has no real solutions.
  - Checkpoint opportunity
    - Checkpoint: Solve linear equations and inequalities (A1-K.16)
    - Checkpoint: Quadratic equations (A1-CC.20)
- A1.A-REI.C Solve systems of equations.
  - A1.A-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
    - Solve a system of equations using elimination (A1-V.10)
    - Solve a system of equations using elimination: word problems (A1-V.11)
  - A1.A-REI.C.6 Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables. Include problem solving opportunities utilizing real-world context.
- A1.A-REI.D Represent and solve equations and inequalities graphically.
  - A1.A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve, which could be a line.
  - A1.A-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately (e.g., using technology to graph the functions, make tables of values, or find successive approximations). Focus on cases where f(x) and/or g(x) are linear, absolute value, quadratic, and exponential functions.
  - A1.A-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary in the case of a strict inequality, and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
  - Checkpoint opportunity
    - Checkpoint: Solve equations using graphs and tables (A1-Q.22)
    - Checkpoint: Systems of equations and inequalities (A1-V.16)

A1.F Functions

A1.F-IF Interpreting Functions
- A1.F-IF.A Understand the concept of a function and use function notation.
  - A1.F-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  - A1.F-IF.A.2 Evaluate a function for inputs in the domain, and interpret statements that use function notation in terms of a context.
  - A1.F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
- A1.F-IF.B Interpret functions that arise in applications in terms of the context.
  - A1.F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Include problem-solving opportunities utilizing real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums. Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).
  - A1.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - A1.F-IF.B.6 Calculate and interpret the average rate of change of a continuous function (presented symbolically or as a table) on a closed interval. Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context. Focus on linear, absolute value, quadratic, and exponential functions.
  - Checkpoint opportunity
    - Checkpoint: Function concepts (A1-Q.20)
    - Checkpoint: Average rate of change (A1-Q.21)
- A1.F-IF.C Analyze functions using different representations.
  - A1.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Functions include linear, exponential, quadratic, and piecewise-defined functions (limited to the aforementioned functions).
  - A1.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - A1.F-IF.C.8.a Use the process of factoring and completing the square of a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  - A1.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).
    - Compare linear functions: graphs and equations (A1-T.15)
    - Compare linear functions: tables, graphs, and equations (A1-T.16)
  - Checkpoint opportunity
    - Checkpoint: Features of functions (A1-DD.14)
    - Checkpoint: Graph and analyze functions (A1)
A1.F-BF Building Functions
- A1.F-BF.A Build a function that models a relationship between two quantities.
  - A1.F-BF.A.1 Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from real-world context. Focus on linear, absolute value, quadratic, exponential, and piecewise-defined functions (limited to the aforementioned functions).
- A1.F-BF.B Build new functions from existing functions.
  - A1.F-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph. Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).
  - Checkpoint opportunity
    - Checkpoint: Function transformations (A1-EE.14)
    - Checkpoint: Build functions (A1)
A1.F-LE Linear, Quadratic, and Exponential Models
- A1.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.
  - A1.F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
    - A1.F-LE.A.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    - A1.F-LE.A.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
    - A1.F-LE.A.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
  - A1.F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or input/output pairs.
  - A1.F-LE.A.3 Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.
    - Compare linear and exponential growth (A1-DD.11)
    - Compare linear, exponential, and quadratic growth (A1-DD.12)
- A1.F-LE.B Interpret expressions for functions in terms of the situation they model.
  - A1.F-LE.B.5 Interpret the parameters in a linear or exponential function with integer exponents utilizing real world context.
  - Checkpoint opportunity
    - Checkpoint: Compare linear and exponential functions (A1-DD.15)

A1.S Statistics and Probability

A1.S-ID Summarize, represent, and interpret data on a single count or measurement variable.
- A1.S-ID.A Summarize, represent, and interpret data on a single count or measurement variable.
  - A1.S-ID.A.1 Represent real-value data with plots for the purpose of comparing two or more data sets.
  - A1.S-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
  - A1.S-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of outliers if present.
    - Box plots (A1-N.9)
    - Identify an outlier and describe the effect of removing it (A1-MM.5)
  - Checkpoint opportunity
    - Checkpoint: Line plots, histograms, and box plots (A1-N.10)
    - Checkpoint: Compare data sets (A1-MM.17)
- A1.S-ID.B Summarize, represent, and interpret data on two categorical and quantitative variables.
  - A1.S-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data, including joint, marginal, and conditional relative frequencies. Recognize possible associations and trends in the data.
    - Find probabilities using two-way frequency tables (A1-LL.3)
    - Find conditional probabilities using two-way frequency tables (A1-LL.4)
  - A1.S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the quantities are related.
    - A1.S-ID.B.6.a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Focus on linear models.
    - A1.S-ID.B.6.b Informally assess the fit of a function by plotting and analyzing residuals.
      - Interpret a scatter plot (A1-MM.8)
  - Checkpoint opportunity
    - Checkpoint: Two-way frequency tables (A1-LL.11)
- A1.S-ID.C Interpret linear models.
  - A1.S-ID.C.7 Interpret the slope as a rate of change and the constant term of a linear model in the context of the data.
    - Interpret regression lines (A1-MM.14)
    - Analyze a regression line of a data set (A1-MM.15)
  - A1.S-ID.C.8 Compute and interpret the correlation coefficient of a linear relationship.
    - Match correlation coefficients to scatter plots (A1-MM.10)
    - Calculate correlation coefficients (A1-MM.11)
  - A1.S-ID.C.9 Distinguish between correlation and causation.
    - Correlation and causation (A1-MM.16)
  - Checkpoint opportunity
    - Checkpoint: Linear modeling (A1-MM.18)
A1.S-CP Conditional Probability and the Rules of Probability
- A1.S-CP.A Understand independence and conditional probability and use them to interpret data.
  - A1.S-CP.A.1 Describe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections, or complements of other events.
    - Theoretical probability (A1-LL.1)
    - Outcomes of compound events (A1-LL.5)
  - A1.S-CP.A.2 Use the Multiplication Rule for independent events to understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
    - Identify independent and dependent events (A1-LL.6)
    - Probability of independent and dependent events (A1-LL.7)

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A1.N Number and Quantity

A1.N-RN The Real Number System

A1.N-RN.B Use properties of rational and irrational numbers.

A1.N-RN.B.3 Explain why 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.

A1.N-Q Quantities

A1.N-Q.A Reason quantitatively and use units to solve problems.

A1.N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays, include utilizing real-world context.

A1.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. Include problem-solving opportunities utilizing real-world context.

A1.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities utilizing real-world context.

Checkpoint opportunity

A1.A Algebra

A1.A-SSE Seeing Structure in Expressions

A1.A-SSE.A Interpret the structure of expressions.

A1.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

A1.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.

A1.A-SSE.A.1.b Interpret expressions by viewing one or more of their parts as a single entity.

A1.A-SSE.A.2 Use structure to identify ways to rewrite numerical and polynomial expressions. Focus on polynomial multiplication and factoring patterns.

A1.A-SSE.B Write expressions in equivalent forms to solve problems.

A1.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

A1.A-SSE.B.3.a Factor a quadratic expression to reveal the zeros of the function it defines.

A1.A-SSE.B.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Checkpoint opportunity

A1.A-APR Arithmetic with Polynomials and Rational Expressions

A1.A-APR.A Perform arithmetic operations on polynomials.

A1.A-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

A1.A-APR.B Understand the relationship between zeros and factors of polynomials.

A1.A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Focus on quadratic and cubic polynomials in which linear and quadratic factors are available.

Checkpoint opportunity

A1.A-CED Creating Equations

A1.A-CED.A Create equations that describe numbers or relationships.

A1.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include problem-solving opportunities utilizing real-world context. Focus on equations and inequalities that are linear, quadratic, or exponential.

A1.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A1.A-CED.A.3 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.

A1.A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Checkpoint opportunity

A1.A-REI Reasoning with Equations and Inequalities

A1.A-REI.A Understand solving equations as a process of reasoning and explain the reasoning.

A1.A-REI.A.1 Explain each step in solving linear and quadratic equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A1.REI.B Solve equations and inequalities in one variable.

A1.A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A1.A-REI.B.4 Solve quadratic equations in one variable.

A1.A-REI.B.4.a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x + k)² = q that has the same solutions. Derive the quadratic formula from this form.

Checkpoint opportunity

A1.A-REI.C Solve systems of equations.

A1.A-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

A1.A-REI.C.6 Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables. Include problem solving opportunities utilizing real-world context.

A1.A-REI.D Represent and solve equations and inequalities graphically.

A1.A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve, which could be a line.

A1.A-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary in the case of a strict inequality, and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Checkpoint opportunity

A1.F Functions

A1.F-IF Interpreting Functions

A1.F-IF.A Understand the concept of a function and use function notation.

A1.F-IF.A.2 Evaluate a function for inputs in the domain, and interpret statements that use function notation in terms of a context.

A1.F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

A1.F-IF.B Interpret functions that arise in applications in terms of the context.

A1.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Checkpoint opportunity

A1.F-IF.C Analyze functions using different representations.

A1.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Functions include linear, exponential, quadratic, and piecewise-defined functions (limited to the aforementioned functions).

A1.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

A1.F-IF.C.8.a Use the process of factoring and completing the square of a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

A1.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Focus on linear, absolute value, quadratic, exponential and piecewise-defined functions (limited to the aforementioned functions).

Checkpoint opportunity

A1.F-BF Building Functions

A1.F-BF.A Build a function that models a relationship between two quantities.

A1.F-BF.B Build new functions from existing functions.

Checkpoint opportunity

A1.F-LE Linear, Quadratic, and Exponential Models

A1.F-LE.A Construct and compare linear, quadratic, and exponential models and solve problems.

A1.F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

A1.F-LE.A.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

A1.F-LE.A.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

A1.F-LE.A.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

A1.F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or input/output pairs.

A1.F-LE.A.3 Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

A1.F-LE.B Interpret expressions for functions in terms of the situation they model.