• Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and exponential functions. (CCSS: HS.A-CED.A.1)

• Explain that a function is a correspondence from one set (called the domain) to another set (called the range) that 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). (CCSS: HS.F-IF.A.1)

• Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (CCSS: HS.F-IF.A.2)

• 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)

• 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. (CCSS: HS.F-BF.B.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).⭑ (CCSS: HS.G-MG.A.3)

• Model data in context with plots on the real number line (dot plots, histograms, and box plots). (CCSS: HS.S-ID.A.1)

• Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. (CCSS: HS.S-ID.B.6.a)

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

• 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)

• Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in 10th grade. Do the same for other subjects and compare the results. (CCSS: HS.S-CP.A.4)

• 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)

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

Instructional Resources:

OpenStax Prealgebra 2e (Lessons 9.5, 9.6)

OpenStax Introductory Statistics (Chapters 1-3)

Khan Academy High School Statistics (Units 1, 3, 4, 5, 6)

• Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. (CCSS: HS.A-CED.A.4)

• Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (CCSS: HS.A-REI.B.3)

• 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)

• Graph linear functions and show intercepts. (CCSS: HS.F-IF.C.7.a)

• Graph absolute value functions. (CCSS: HS.F-IF.C.7.b)

• Construct linear functions, including arithmetic sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (CCSS: HS.F-LE.A.2)

• Interpret the parameters in a linear or exponential function in terms of a context. (CCSS: HS.F-LE.B.5)

• Fit a linear function for a scatter plot that suggests a linear association. (CCSS: HS.S-ID.B.6.c)

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

Instructional Resources:

OpenStax Elementary Algebra 2e (Chapters 2, 4, 5)

OpenStax Intermediate Algebra 2e (Chapters 2, 3, Lessons 4.1, 4.2, 4.7)

Khan Academy Algebra 1 (Units 2, 4, 5, 6, 7, 8, 10 [absolute value functions only])

• Factor a quadratic expression to reveal the zeros of the function it defines. (CCSS: HS.A-SSE.B.3.a)

• Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (CCSS: HS.A-SSE.B.3.b)

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

• Solve quadratic equations (e.g., for x2 = 49) by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. (CCSS: HS.A-REI.B.4.b)

• Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. (CCSS: HS.A-REI.C.7)

• Graph quadratic functions and show intercepts, maxima, and minima. (CCSS: HS.F-IF.C.7.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. (CCSS: HS.F-IF.C.8.a)

• Use the Pythagorean Theorem to solve right triangles in applied problems. (CCSS: HS.G-SRT.C.8)

• Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. (CCSS: HS.G-GPE.A.1)

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

Instructional Resources:

OpenStax Elementary Algebra 2e (Chapters 6, 7, 10)

OpenStax Intermediate Algebra 2e (Lesson 5.3, Chapters 6, 9)

Khan Academy Algebra 1 (Units 13, 14)

• Graph exponential functions, showing intercepts and end behavior. (CCSS: HS.F-IF.C.7.e)

• Construct exponential functions, including geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (CCSS: HS.F-LE.A.2)

• 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. (CCSS: HS.F-LE.A.4) (limit to base 2 and 10 for Intermediate Algebra)

• Interpret the parameters in an exponential function in terms of a context. (CCSS: HS.F-LE.B.5)

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

Instructional Resources:

OpenStax Intermediate Algebra 2e (Lesson 2.7, Chapter 10)

Khan Academy Algebra 1 (Units 11 [properties of exponents only], 12)

• For a rational 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)

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

Instructional Resources:

OpenStax College Algebra 2e (Lesson 5.6)

Khan Academy Precalculus (Unit 4 [graphing rational functions only])

The following concepts and skills should also be reinforced through daily warm-ups to prepare students for the SAT, focusing primarily on the Heart of Algebra (33%) and Problem Solving and Data Analysis (29%) subscore areas:

Heart of Algebra

• Create, solve, or interpret a linear expression or equation in one variable 

• Create, solve, or interpret linear inequalities in one variable

• Build a linear function that models a linear relationship between two quantities

• Create, solve, and interpret systems of linear inequalities in two variables

• Create, solve, and interpret systems of two linear equations in two variables

• Algebraically solve linear equations (or inequalities) in one variable

• Algebraically solve systems of two linear equations in two variables

• Interpret the variables and constants in expressions for linear functions within the context presented

• Understand connections between algebraic and graphical representations

Problem Solving and Data Analysis

• Use ratios, rates, proportional relationships, and scale drawings to solve single- and multistep problems

• Solve single- and multistep problems involving percentages

• Solve single- and multistep problems involving measurement quantities, units, and unit conversion

• Given a scatterplot, use linear, quadratic, or exponential models to describe how the variables are related

• Use the relationship between two variables to investigate key features of the graph

• Compare linear growth with exponential growth

• Use two-way tables to summarize categorical data and relative frequencies, and calculate conditional probability

• Make inferences about population parameters based on sample data

• Use statistics to investigate measures of center of data and analyze shape, center, and spread

• Evaluate reports to make inferences, justify conclusions, and determine appropriateness of data collection methods

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