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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.A The Complex Number System: Perform arithmetic operations with complex numbers.
- HS.N-CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
HS.N-CN.B The Complex Number System: Represent complex numbers and their operations on the complex plane.
- HS.N-CN.B.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
- HS.N-CN.B.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
- HS.N-CN.B.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
  - Midpoints in the complex plane (PC-S.7)
  - Distance in the complex plane (PC-S.8)
HS.N-VM.A Vector & Matrix Quantities: Represent and model with vector quantities.
- HS.N-VM.A.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
- HS.N-VM.A.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
  - Find the component form of a vector (PC-U.2)
  - Find the component form of a three-dimensional vector (PC-V.2)
- HS.N-VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.
HS.N-VM.B Vector & Matrix Quantities: Perform operations on vectors.
- HS.N-VM.B.4 Add and subtract vectors.
  - HS.N-VM.B.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
  - HS.N-VM.B.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
    - Find the magnitude and direction of a vector sum (PC-U.9)
  - HS.N-VM.B.4.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
- HS.N-VM.B.5 Multiply a vector by a scalar.
  - HS.N-VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v _y) = (cv_x, cv_y).
    - Multiply a vector by a scalar (PC-U.10)
    - Scalar multiples of three-dimensional vectors (PC-V.4)
  - HS.N-VM.B.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
    - Find the magnitude of a vector scalar multiple (PC-U.11)
    - Determine the direction of a vector scalar multiple (PC-U.12)
HS.N-VM.C Vector & Matrix Quantities: Perform operations on matrices and use matrices in applications.
- HS.N-VM.C.6 Use matrices to represent and manipulate data, e.g., as when all of the payoffs or incidence relationships in a network.
- HS.N-VM.C.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
  - Multiply a matrix by a scalar (PC-L.4)
- HS.N-VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.
- HS.N-VM.C.9 Understand that, unlike the multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
  - Properties of matrices (PC-L.8)
- HS.N-VM.C.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
- HS.N-VM.C.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimension to produce another vector. Work with matrices as transformations of vectors.
- HS.N-VM.C.12 Work with 2 × 2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.

2 Algebra and Functions

HS.A-REI.C Reasoning with Equations & Inequalities: Solve systems of equations.
- HS.A-REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.
  - Solve a system of equations using augmented matrices (PC-I.8)
  - Solve a system of equations using augmented matrices: word problems (PC-I.9)
- HS.A-REI.C.9 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).
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.d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
    - Rational functions: asymptotes and excluded values (PC-E.1)
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.c Compose functions.
    - Composition of functions (PC-A.11)
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.b Verify by composition that one function is the inverse of another.
    - Identify inverse functions (PC-A.12)
    - Check whether two rational functions are inverses (PC-E.3)
  - HS.F-BF.B.4.c Read values of an inverse function from a graph or table, given that the function has an inverse.
    - Find values of inverse functions from tables (PC-A.13)
    - Find values of inverse functions from graphs (PC-A.14)
  - HS.F-BF.B.4.d Produce an invertible function from a non-invertible function by restricting the domain.
- HS.F-BF.B.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
HS.F-TF.A Trigonometric Functions: Extend the domain of trigonometric functions using the unit circle.
- HS.F-TF.A.3 Use special triangles to determine geometrically the values to sine, cosine, tangent for π/3, π/4, and π/6 and use the unit circle to express the values sine, cosine, and tangent for x, π + x, and 2π - x and in terms of their values for x where x is any real number.
  - Find trigonometric ratios using right triangles (PC-M.5)
  - Find trigonometric ratios of special angles (PC-M.7)
- HS.F-TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
  - Symmetry and periodicity of trigonometric functions (PC-O.2)
HS.F-TF.B Trigonometric Functions: Model periodic phenomena with trigonometric functions.
- HS.F-TF.B.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
  - Inverses of trigonometric functions (PC-M.9)
  - Inverses of trigonometric functions using a calculator (PC-M.10)
- HS.F-TF.B.7 Use inverse function to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
  - Solve trigonometric equations (PC-M.11)
HS.F-TF.C Trigonometric Functions: Prove and apply trigonometric identities.
- HS.F-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

3 Data, Statistics, and Probability

HS.S-MD.A Using Probability to Make Decisions: Calculate expected values and use them to solve problems.
- HS.S-MD.A.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
  - Write a discrete probability distribution (PC-Y.2)
  - Graph a discrete probability distribution (PC-Y.3)
- HS.S-MD.A.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
  - Expected values of random variables (PC-Y.4)
- HS.S-MD.A.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
  - Write the probability distribution for a game of chance (PC-Y.7)
  - Expected values for a game of chance (PC-Y.8)
- HS.S-MD.A.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
HS.S-MD.B Using Probability to Make Decisions: Use probability to evaluate outcomes of decisions.
- HS.S-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
  - HS.S-MD.B.5.a Find the expected payoff for a game of chance.
    - Expected values for a game of chance (PC-Y.8)
  - HS.S-MD.B.5.b Evaluate and compare strategies on the basis of expected values.
    - Choose the better bet (PC-Y.9)

4 Geometry

HS.G-GPE.A Expressing Geometric Properties with Equations: Translate between the geometric description and the equation for a conic section.
- HS.G-GPE.A.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
  - Write equations of ellipses in standard form (PC-P.9)
  - Write equations of hyperbolas in standard form (PC-P.12)
HS.G-GMD.A Geometric Measurement and Dimension: Explain volume formulas and use them to solve problems.
- HS.G-GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

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1 Number and Quantity

HS.N-CN.A The Complex Number System: Perform arithmetic operations with complex numbers.

HS.N-CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

HS.N-CN.B The Complex Number System: Represent complex numbers and their operations on the complex plane.

HS.N-CN.B.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

HS.N-CN.B.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

HS.N-CN.B.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

HS.N-VM.A Vector & Matrix Quantities: Represent and model with vector quantities.

HS.N-VM.A.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

HS.N-VM.A.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

HS.N-VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.

HS.N-VM.B Vector & Matrix Quantities: Perform operations on vectors.

HS.N-VM.B.4 Add and subtract vectors.

HS.N-VM.B.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

HS.N-VM.B.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

HS.N-VM.B.5 Multiply a vector by a scalar.

HS.N-VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, v y) = (cvx, cvy).

HS.N-VM.B.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

HS.N-VM.C Vector & Matrix Quantities: Perform operations on matrices and use matrices in applications.

HS.N-VM.C.6 Use matrices to represent and manipulate data, e.g., as when all of the payoffs or incidence relationships in a network.

HS.N-VM.C.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

HS.N-VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.

HS.N-VM.C.9 Understand that, unlike the multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

HS.N-VM.C.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

HS.N-VM.C.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimension to produce another vector. Work with matrices as transformations of vectors.

HS.N-VM.C.12 Work with 2 × 2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.

2 Algebra and Functions

HS.A-REI.C Reasoning with Equations & Inequalities: Solve systems of equations.

HS.A-REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.

HS.A-REI.C.9 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).

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.d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

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.c Compose functions.

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.b Verify by composition that one function is the inverse of another.

HS.F-BF.B.4.c Read values of an inverse function from a graph or table, given that the function has an inverse.

HS.F-BF.B.4.d Produce an invertible function from a non-invertible function by restricting the domain.

HS.F-BF.B.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

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

HS.F-TF.A.3 Use special triangles to determine geometrically the values to sine, cosine, tangent for π/3, π/4, and π/6 and use the unit circle to express the values sine, cosine, and tangent for x, π + x, and 2π - x and in terms of their values for x where x is any real number.

HS.F-TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

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

HS.F-TF.B.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

HS.F-TF.B.7 Use inverse function to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

HS.F-TF.C Trigonometric Functions: Prove and apply trigonometric identities.

HS.F-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

3 Data, Statistics, and Probability

HS.S-MD.A Using Probability to Make Decisions: Calculate expected values and use them to solve problems.

HS.S-MD.A.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

HS.S-MD.A.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

HS.S-MD.A.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

HS.S-MD.A.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

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

HS.S-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

HS.S-MD.B.5.a Find the expected payoff for a game of chance.

HS.S-MD.B.5.b Evaluate and compare strategies on the basis of expected values.

4 Geometry

HS.G-GPE.A Expressing Geometric Properties with Equations: Translate between the geometric description and the equation for a conic section.

HS.G-GPE.A.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

HS.G-GMD.A Geometric Measurement and Dimension: Explain volume formulas and use them to solve problems.

HS.G-GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

HS.N-VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v _y) = (cv_x, cv_y).