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

• Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (CCSS: HS.N-CN.A.2)

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

• 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.) (CCSS: HS.A-APR.B.2)

• 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. (CCSS: HS.A-APR.B.3)

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

• 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. (CCSS: HS.F-IF.B.4)

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

• Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.⭑ (CCSS: HS.F-IF.C.7)

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

  • Graph trigonometric functions, showing period, midline, and amplitude. (CCSS: HS.F-IF.C.7.e)

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

• 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. (CCSS: HS.F-BF.B.3)

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

• Explain and use the relationship between the sine and cosine of complementary angles. (CCSS: HS.G-SRT.C.7)

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

• (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. (CCSS: HS.N-CN.A.3)

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

• (+) 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. (CCSS: HS.N-CN.B.4)

• (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1 + √3i)3 = 8 because (–1 + √3i) has modulus 2 and argument 120°. (CCSS: HS.N- CN.B.5)

• (+) 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. (CCSS: HS.N-CN.B.6)

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

• (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). (CCSS: HS.N-CN.C.8)

• (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (CCSS: HS.N-CN.C.9)

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

• (+) Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 − y2)2 + (2xy)2 can be used to generate Pythagorean triples. (CCSS: HS.A-APR.C.4)

• (+) 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-APR.D. Arithmetic with Polynomials & Rational Expressions: Rewrite rational expressions.

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

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

• Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.⭑ (CCSS: HS.F-IF.C.7)

  • (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (CCSS: HS.F-IF.C.7.d)

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

• Write a function that describes a relationship between two quantities.⭑ (CCSS: HS.F-BF.A.1)

  • (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. (CCSS: HS.F- BF.A.1.c)

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

• Find inverse functions. (CCSS: HS.F-BF.B.4)

  • (+) Verify by composition that one function is the inverse of another. (CCSS: HS.F-BF.B.4.b)

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

  • (+) Produce an invertible function from a non-invertible function by restricting the domain. (CCSS: HS.F-BF.B.4.d)

• (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (CCSS: HS.F-BF.B.5)

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

• (+) Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle. (CCSS: HS.F-TF.A.1)

• (+) 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. (CCSS: HS.F-TF.A.2)

• (+) 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. (CCSS: HS.F-TF.A.3)

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

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

• (+) Model periodic phenomena with trigonometric functions with specified amplitude, frequency, and midline.⭑ (CCSS: HS.F-TF.B.5)

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

• (+) 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.⭑ (CCSS: HS.F-TF.B.7)

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

• (+) Prove the Pythagorean identity sin2(𝜃) + cos2(𝜃) = 1 and use it to find sin(𝜃), cos(𝜃), or tan(𝜃) given sin(𝜃), cos(𝜃), or tan(𝜃) and the quadrant of the angle. (CCSS: HS.F-TF.C.8)

• (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. (CCSS: HS.F-TF.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 solve problems. (CCSS: HS.S-CP.B.9) (Binomial Theorem)

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

• (+) Derive the formula A = 1/2absin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. (CCSS: HS.G-SRT.D.9)

• (+) Prove the Laws of Sines and Cosines and use them to solve problems. (CCSS: HS.G-SRT.D.10)

• (+) 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). (CCSS: HS.G-SRT.D.11)

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

• (+) Derive the equation of a parabola given a focus and directrix. (CCSS: HS.G-GPE.A.2)

• (+) 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. (CCSS: HS.G-GPE.A.3)

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

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.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) (Reinforce from Algebra 1)