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Skills available for Colorado high school math standards

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Colorado Academic Standards: Algebra 1 Colorado Academic Standards: Algebra 1
Colorado Academic Standards: Algebra 2 Colorado Academic Standards: Algebra 2
Colorado Academic Standards: Geometry Colorado Academic Standards: Geometry
Colorado Academic Standards: Mathematics 1 Colorado Academic Standards: Mathematics 1
Colorado Academic Standards: Mathematics 2 Colorado Academic Standards: Mathematics 2
Colorado Academic Standards: Mathematics 3 Colorado Academic Standards: Mathematics 3
Colorado Academic Standards: Precalculus Colorado Academic Standards: Precalculus
Colorado Academic Standards (Common Core) Colorado Academic Standards (Common Core)

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1 Number and Quantity

HS.N-CN.C The Complex Number System: Use complex numbers in polynomial identities and equations.
- HS.N-CN.C.8 Extend polynomial identities to the complex numbers.
- HS.N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  - Fundamental Theorem of Algebra (A2-L.17)

2 Algebra and Functions

HS.A-SSE.A Seeing Structure in Expressions: Interpret the structure of expressions.
- HS.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
  - HS.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
    - Interpret parts of quadratic expressions: word problems (A2-K.)
    - Polynomial vocabulary (A2-L.1)
  - HS.A-SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
    - Interpret parts of quadratic expressions: word problems (A2-K.)
- HS.A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
HS.A-SSE.B Seeing Structure in Expressions: Write expressions in equivalent forms to solve problems.
- HS.A-SSE.B.4 Use the formula for the sum of a finite geometric series (when the common ratio is not 1) to solve problems.
  - Find the sum of a finite geometric series (A2-CC.16)
HS.A-APR.A Arithmetic with Polynomials & Rational Expressions: Perform arithmetic operations on polynomials.
- HS.A-APR.A.1 Explain 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.
  - Add and subtract polynomials (A2-L.2)
  - Multiply polynomials (A2-L.3)
HS.A-APR.B Arithmetic with Polynomials & Rational Expressions: Understand the relationship between zeros and factors of polynomials.
- HS.A-APR.B.2 Know and apply the Remainder Theorem. For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). (Students need not apply the Remainder Theorem to polynomials of degree greater than 4.)
- HS.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.
HS.A-APR.C Arithmetic with Polynomials & Rational Expressions: Use polynomial identities to solve problems.
- HS.A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.
- HS.A-APR.C.5 Know and apply the Binomial Theorem for the expansion of in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
HS.A-APR.D Arithmetic with Polynomials & Rational Expressions: Rewrite rational expressions.
- HS.A-APR.D.6 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, long division, or, for the more complicated examples, a computer algebra system.
- HS.A-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expressions; add, subtract, multiply, and divide rational expressions.
  - Multiply and divide rational expressions (A2-O.5)
  - Add and subtract rational expressions (A2-O.6)
HS.A-CED.A Creating Equations: Create equations that describe numbers or relationships.
- HS.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  - Write exponential functions: word problems (A2-T.)
  - Solve quadratic equations: word problems (PC-C.)
- HS.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales.
- HS.A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
- HS.A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  - Solve multi-variable equations (A2-B.6)
HS.A-REI.A Reasoning with Equations & Inequalities: Understand solving equations as a process of reasoning and explain the reasoning.
- HS.A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  - Solve radical equations (A2-M.14)
  - Solve rational equations (A2-O.7)
HS.A-REI.D Reasoning with Equations & Inequalities: Represent and solve equations and inequalities graphically.
- HS.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. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
HS.F-IF.B Interpreting Functions: Interpret functions that arise in applications in terms of the context.
- HS.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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
- HS.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - Domain and range of polynomials (A2-L.16)
  - Domain and range of radical functions (A2-M.12)
- HS.F-IF.B.6 Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.
  - Average rate of change (A2-D.12)
HS.F-IF.C Interpreting Functions: Analyze functions using different representations.
- HS.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.
  - HS.F-IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
    - Graph an absolute value function (A2-D.13)
    - Graph piecewise-defined functions (A2-D.17)
  - HS.F-IF.C.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
  - HS.F-IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
- HS.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.
  - HS.F-IF.C.8.a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  - HS.F-IF.C.8.b Use the properties of exponents to interpret expressions for exponential functions.
- HS.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
HS.F-BF.A Building Functions: Build a function that models a relationship between two quantities.
- HS.F-BF.A.1 Write a function that describes a relationship between two quantities.
  - HS.F-BF.A.1.b Combine standard function types using arithmetic operations.
HS.F-BF.B Building Functions: Build new functions from existing functions.
- HS.F-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), 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 using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
- HS.F-BF.B.4 Find inverse functions.
  - HS.F-BF.B.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
HS.F-LE.A Linear, Quadratic & Exponential Models: Construct and compare linear, quadratic, and exponential models and solve problems.
- HS.F-LE.A.4 For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
HS.F-TF.A Trigonometric Functions: Extend the domain of trigonometric functions using the unit circle.
- HS.F-TF.A.1 Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
  - Convert between radians and degrees (A2-Y.1)
  - Radians and arc length (A2-Y.2)
- HS.F-TF.A.2 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.
HS.F-TF.B Trigonometric Functions: Model periodic phenomena with trigonometric functions.
- HS.F-TF.B.5 Model periodic phenomena with trigonometric functions with specified amplitude, frequency, and midline.

3 Data, Statistics, and Probability

HS.S-ID.A Interpreting Categorical & Quantitative Data: Summarize, represent, and interpret data on a single count or measurement variable.
- HS.S-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages and identify data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
HS.S-IC.A Making Inferences & Justifying Conclusions: Understand and evaluate random processes underlying statistical experiments.
- HS.S-IC.A.1 Describe statistics as a process for making inferences about population parameters based on a random sample from that population.
  - Identify biased samples (A2-FF.1)
- HS.S-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
HS.S-IC.B Making Inferences & Justifying Conclusions: Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
- HS.S-IC.B.3 Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
  - Experiment design (A2-FF.12)
- HS.S-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
  - Find confidence intervals for population means (A2-FF.9)
  - Find confidence intervals for population proportions (A2-FF.10)
- HS.S-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
  - Analyze the results of an experiment using simulations (A2-FF.13)
- HS.S-IC.B.6 Evaluate reports based on data. Define and explain the meaning of significance, both statistical (using p-values) and practical (using effect size).
  - Interpret confidence intervals for population means (A2-FF.11)
  - Analyze the results of an experiment using simulations (A2-FF.13)
HS.S-MD.B Using Probability to Make Decisions: Use probability to evaluate outcomes of decisions.
- HS.S-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
- HS.S-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
  - Choose the better bet (A2-EE.9)

4 Geometry

HS.G-SRT.D Similarity, Right Triangles, and Trigonometry: Apply trigonometry to general triangles.
- HS.G-SRT.D.9 Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
  - Area of a triangle: sine formula (A2-Z.22)
- HS.G-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
- HS.G-SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
  - Solve a triangle (A2-Z.21)
HS.G-GMD.B Geometric Measurement and Dimension: Visualize relationships between two-dimensional and three-dimensional objects.
- HS.G-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
  - Cross sections of three-dimensional figures (G-H.4)
  - Solids of revolution (G-H.5)
HS.G-MG.A Modeling with Geometry: Apply geometric concepts in modeling situations.
- HS.G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
  - Pythagorean theorem (G-Q.1)
- HS.G-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
  - Calculate density, mass, and volume (G-W.9)
- HS.G-MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Colorado

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1 Number and Quantity

HS.N-CN.C The Complex Number System: Use complex numbers in polynomial identities and equations.

HS.N-CN.C.8 Extend polynomial identities to the complex numbers.

HS.N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

2 Algebra and Functions

HS.A-SSE.A Seeing Structure in Expressions: Interpret the structure of expressions.

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

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

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

HS.A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it.

HS.A-SSE.B Seeing Structure in Expressions: Write expressions in equivalent forms to solve problems.

HS.A-SSE.B.4 Use the formula for the sum of a finite geometric series (when the common ratio is not 1) to solve problems.

HS.A-APR.A Arithmetic with Polynomials & Rational Expressions: Perform arithmetic operations on polynomials.

HS.A-APR.A.1 Explain 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.

HS.A-APR.B Arithmetic with Polynomials & Rational Expressions: Understand the relationship between zeros and factors of polynomials.

HS.A-APR.B.2 Know and apply the Remainder Theorem. For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). (Students need not apply the Remainder Theorem to polynomials of degree greater than 4.)

HS.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.

HS.A-APR.C Arithmetic with Polynomials & Rational Expressions: Use polynomial identities to solve problems.

HS.A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.

HS.A-APR.D Arithmetic with Polynomials & Rational Expressions: Rewrite rational expressions.

HS.A-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expressions; add, subtract, multiply, and divide rational expressions.

HS.A-CED.A Creating Equations: Create equations that describe numbers or relationships.

HS.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

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

HS.A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

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

HS.A-REI.A Reasoning with Equations & Inequalities: Understand solving equations as a process of reasoning and explain the reasoning.

HS.A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

HS.A-REI.D Reasoning with Equations & Inequalities: Represent and solve equations and inequalities graphically.

HS.F-IF.B Interpreting Functions: Interpret functions that arise in applications in terms of the context.

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

HS.F-IF.B.6 Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.

HS.F-IF.C Interpreting Functions: Analyze functions using different representations.

HS.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.

HS.F-IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

HS.F-IF.C.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

HS.F-IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

HS.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.

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

HS.F-IF.C.8.b Use the properties of exponents to interpret expressions for exponential functions.

HS.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

HS.F-BF.A Building Functions: Build a function that models a relationship between two quantities.

HS.F-BF.A.1 Write a function that describes a relationship between two quantities.

HS.F-BF.A.1.b Combine standard function types using arithmetic operations.

HS.F-BF.B Building Functions: Build new functions from existing functions.

HS.F-BF.B.4 Find inverse functions.

HS.F-BF.B.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

HS.F-LE.A Linear, Quadratic & Exponential Models: Construct and compare linear, quadratic, and exponential models and solve problems.

HS.F-LE.A.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

HS.F-TF.A Trigonometric Functions: Extend the domain of trigonometric functions using the unit circle.

HS.F-TF.A.1 Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

HS.F-TF.A.2 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.

HS.F-TF.B Trigonometric Functions: Model periodic phenomena with trigonometric functions.

HS.F-TF.B.5 Model periodic phenomena with trigonometric functions with specified amplitude, frequency, and midline.

3 Data, Statistics, and Probability

HS.S-ID.A Interpreting Categorical & Quantitative Data: Summarize, represent, and interpret data on a single count or measurement variable.

HS.S-ID.A.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages and identify data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

HS.S-IC.A Making Inferences & Justifying Conclusions: Understand and evaluate random processes underlying statistical experiments.

HS.S-IC.A.1 Describe statistics as a process for making inferences about population parameters based on a random sample from that population.

HS.S-IC.A.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

HS.S-IC.B Making Inferences & Justifying Conclusions: Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

HS.S-IC.B.3 Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

HS.S-IC.B.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

HS.S-IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

HS.S-IC.B.6 Evaluate reports based on data. Define and explain the meaning of significance, both statistical (using p-values) and practical (using effect size).

HS.S-MD.B Using Probability to Make Decisions: Use probability to evaluate outcomes of decisions.

HS.S-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

HS.S-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

4 Geometry

HS.G-SRT.D Similarity, Right Triangles, and Trigonometry: Apply trigonometry to general triangles.

HS.G-SRT.D.9 Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

HS.G-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.

HS.G-SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

HS.G-GMD.B Geometric Measurement and Dimension: Visualize relationships between two-dimensional and three-dimensional objects.

HS.G-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

HS.G-MG.A Modeling with Geometry: Apply geometric concepts in modeling situations.

HS.G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

HS.F-LE.A.4 For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.