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

• 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. (CCSS: HS.A-REI.C.5)

• Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (CCSS: HS.A-REI.C.6)

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

• 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. (CCSS: HS.A-REI.D.12)

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

• Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.⭑ (CCSS: HS.F-BF.A.2)

HS.S-CP.A. Conditional Probability & the Rules of Probability: Understand independence and conditional probability and use them to interpret data.

• Explain 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. (CCSS: HS.S-CP.A.2)

HS.S-CP.B. Conditional Probability & the Rules of Probability: Use the rules of probability to compute probabilities of compound events in a uniform probability model.

• Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. (CCSS: HS.S-CP.B.7)

HS.G-SRT.C. Similarity, Right Triangles, and Trigonometry: Define trigonometric ratios and solve problems involving right triangles.

• Use trigonometric ratios to solve right triangles in applied problems.⭑ (CCSS: HS.G-SRT.C.8)

A star symbol (⭑) represents grade level expectations and evidence outcomes that make up a mathematical modeling standards category.

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

• (+) 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). ((CCSS: HS.N-VM.A.1)

• (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. (CCSS: HS.N-VM.A.2)

• (+) Solve problems involving velocity and other quantities that can be represented by vectors. (CCSS: HS.N-VM.A.3)

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

• Add and subtract vectors. (HS.N-VM.B.4)

  • (+) 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. (CCSS: HS.N-VM.B.4.a)

  • (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. (CCSS: HS.N-VM.B.4.b)

  • (+) 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. (CCSS: HS.N-VM.B.4.c)

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

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

  • (+) 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). (CCSS: HS.N-VM.B.5.b)

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

• (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. (HS.N-VM.C.6)

• (+) 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.7)

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

• (+) Understand that, unlike 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.9)

• (+) 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.10)

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

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

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

• Use the formula for the sum of a finite geometric series (when the common ratio is not 1) to solve problems. For example, calculate mortgage payments.⭑ (CCSS: HS.A-SSE.B.4)

  • (+) Derive the formula for the sum of a finite geometric series (when the common ratio is not 1). (CCSS: HS.A-SSE.B.4) 

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

• (+) Know and apply the Binomial Theorem for the expansion of (x + y)n 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.) (CCSS: HS.A-APR.C.5)

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

• (+) Represent a system of linear equations as a single matrix equation in a vector variable. (CCSS: HS.A-REI.C.8)

• (+) 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). (CCSS: HS.A-REI.C.9)

HS.S-CP.B. Conditional Probability & the Rules of Probability: Use the rules of probability to compute probabilities of compound events in a uniform probability model.

• (+) Use permutations and combinations to compute probabilities of compound events and solve problems. (CCSS: HS.S-CP.B.9)

A star symbol (⭑) represents grade level expectations and evidence outcomes that make up a mathematical modeling standards category.

Please visit the complete 2020 Colorado Academic Standards for High School Mathematics to view the following:

• Colorado Essential Skills and Mathematical Practices connections

• Inquiry Questions

• Coherence Connections