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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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9-12.1 Number Sense, Properties, and Operations

9-12.1.1 The complex number system includes real numbers and imaginary numbers
- 9-12.1.1.a Extend the properties of exponents to rational exponents.
  - 9-12.1.1.a.i Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
  - 9-12.1.1.a.ii Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- 9-12.1.1.b Use properties of rational and irrational numbers.
  - 9-12.1.1.b.i Explain why the sum or product of two rational numbers is rational.
  - 9-12.1.1.b.ii Explain why the sum of a rational number and an irrational number is irrational.
  - 9-12.1.1.b.iii Explain why the product of a nonzero rational number and an irrational number is irrational.
- 9-12.1.1.c Perform arithmetic operations with complex numbers.
  - 9-12.1.1.c.i Define the complex number i such that i² = -1, and show that every complex number has the form a + bi where a and b are real numbers.
    - Introduction to complex numbers (A2-I.1)
    - Introduction to the complex plane (PC-S.1)
  - 9-12.1.1.c.ii Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- 9-12.1.1.d Use complex numbers in polynomial identities and equations.
  - 9-12.1.1.d.i Solve quadratic equations with real coefficients that have complex solutions.
    - Complex conjugate theorem (A2-L.12)
    - Complex conjugate theorem (PC-D.7)
9-12.1.2 Quantitative reasoning is used to make sense of quantities and their relationships in problem situations
- 9-12.1.2.a Reason quantitatively and use units to solve problems.
  - 9-12.1.2.a.i Use units as a way to understand problems and to guide the solution of multi-step problems.
    - 9-12.1.2.a.i.1 Choose and interpret units consistently in formulas.
    - 9-12.1.2.a.i.2 Choose and interpret the scale and the origin in graphs and data displays.
  - 9-12.1.2.a.ii Define appropriate quantities for the purpose of descriptive modeling.
    - Interpret bar graphs, line graphs, and histograms (A1-N.5)
    - Create bar graphs, line graphs, and histograms (A1-N.6)
  - 9-12.1.2.a.iii Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
  - 9-12.1.2.a.iv Describe factors affecting take-home pay and calculate the impact
  - 9-12.1.2.a.v Design and use a budget, including income (net take-home pay) and expenses (mortgage, car loans, and living expenses) to demonstrate how living within your means is essential for a secure financial future

9-12.2 Patterns, Functions, and Algebraic Structures

9-12.2.1 Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables
- 9-12.2.1.a Formulate the concept of a function and use function notation.
  - 9-12.2.1.a.i 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.
  - 9-12.2.1.a.ii Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  - 9-12.2.1.a.iii Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
- 9-12.2.1.b Interpret functions that arise in applications in terms of the context
  - 9-12.2.1.b.i 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 given a verbal description of the relationship.
  - 9-12.2.1.b.ii Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - 9-12.2.1.b.iii Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.
- 9-12.2.1.c Analyze functions using different representations
  - 9-12.2.1.c.i Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
  - 9-12.2.1.c.ii Graph linear and quadratic functions and show intercepts, maxima, and minima.
  - 9-12.2.1.c.iii Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  - 9-12.2.1.c.iv Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
    - Match polynomials and graphs (A2-L.15)
    - Match polynomials and graphs (PC-D.11)
  - 9-12.2.1.c.v Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
  - 9-12.2.1.c.vi Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - 9-12.2.1.c.vi.1 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.
    - 9-12.2.1.c.vi.2 Use the properties of exponents to interpret expressions for exponential functions.
    - 9-12.2.1.c.vi.3 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
- 9-12.2.1.d Build a function that models a relationship between two quantities
  - 9-12.2.1.d.i Write a function that describes a relationship between two quantities.
    - 9-12.2.1.d.i.1 Determine an explicit expression, a recursive process, or steps for calculation from a context.
    - 9-12.2.1.d.i.2 Combine standard function types using arithmetic operations.
  - 9-12.2.1.d.ii Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
- 9-12.2.1.e Build new functions from existing functions
  - 9-12.2.1.e.i Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k, and find the value of k given the graphs.
  - 9-12.2.1.e.ii Experiment with cases and illustrate an explanation of the effects on the graph using technology.
    - Describe function transformations (A2-Q.6)
    - Describe function transformations (PC-B.6)
  - 9-12.2.1.e.iii Find inverse functions.
- 9-12.2.1.f Extend the domain of trigonometric functions using the unit circle
  - 9-12.2.1.f.i Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
    - Radians and arc length (A2-Y.2)
    - Radians and arc length (PC-M.2)
  - 9-12.2.1.f.ii 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.
    - Find trigonometric ratios using the unit circle (A2-Z.6)
    - Find trigonometric ratios using the unit circle (PC-M.6)
9-12.2.2 Quantitative relationships in the real world can be modeled and solved using functions
- 9-12.2.2.a Construct and compare linear, quadratic, and exponential models and solve problems
  - 9-12.2.2.a.i Distinguish between situations that can be modeled with linear functions and with exponential functions.
    - 9-12.2.2.a.i.1 Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    - 9-12.2.2.a.i.2 Identify situations in which one quantity changes at a constant rate per unit interval relative to another.
    - 9-12.2.2.a.i.3 Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
  - 9-12.2.2.a.ii Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.
  - 9-12.2.2.a.iii Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
  - 9-12.2.2.a.iv For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
- 9-12.2.2.b Interpret expressions for functions in terms of the situation they model
  - 9-12.2.2.b.i Interpret the parameters in a linear or exponential function in terms of a context.
- 9-12.2.2.c Model periodic phenomena with trigonometric functions.
  - 9-12.2.2.c.i Choose the trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
    - Symmetry and periodicity of trigonometric functions (A2-BB.2)
    - Symmetry and periodicity of trigonometric functions (PC-O.2)
- 9-12.2.2.d Model personal financial situations
  - 9-12.2.2.d.i Analyze the impact of interest rates on a personal financial plan
  - 9-12.2.2.d.ii Evaluate the costs and benefits of credit
  - 9-12.2.2.d.iii Analyze various lending sources, services, and financial institutions
9-12.2.3 Expressions can be represented in multiple, equivalent forms
- 9-12.2.3.a Interpret the structure of expressions
  - 9-12.2.3.a.i Interpret expressions that represent a quantity in terms of its context.
    - 9-12.2.3.a.i.1 Interpret parts of an expression, such as terms, factors, and coefficients.
      - Polynomial vocabulary (A1-AA.1)
      - Polynomial vocabulary (A2-L.1)
    - 9-12.2.3.a.i.2 Interpret complicated expressions by viewing one or more of their parts as a single entity.
  - 9-12.2.3.a.ii Use the structure of an expression to identify ways to rewrite it.
- 9-12.2.3.b Write expressions in equivalent forms to solve problems
  - 9-12.2.3.b.i Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
    - 9-12.2.3.b.i.1 Factor a quadratic expression to reveal the zeros of the function it defines.
    - 9-12.2.3.b.i.2 Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
    - 9-12.2.3.b.i.3 Use the properties of exponents to transform expressions for exponential functions.
  - 9-12.2.3.b.ii Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
- 9-12.2.3.c Perform arithmetic operations on polynomials
  - 9-12.2.3.c.i 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.
- 9-12.2.3.d Understand the relationship between zeros and factors of polynomials
  - 9-12.2.3.d.i State and apply the Remainder Theorem.
  - 9-12.2.3.d.ii 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.
- 9-12.2.3.e Use polynomial identities to solve problems
  - 9-12.2.3.e.i Prove polynomial identities and use them to describe numerical relationships.
- 9-12.2.3.f Rewrite rational expressions
- 9-12.2.3.g Rewrite simple rational expressions in different forms.
9-12.2.4 Solutions to equations, inequalities and systems of equations are found using a variety of tools
- 9-12.2.4.a Create equations that describe numbers or relationships
- 9-12.2.4.b Understand solving equations as a process of reasoning and explain the reasoning
  - 9-12.2.4.b.i Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution.
    - Properties of equality (A1-H.4)
    - Weighted averages: word problems (A1-O.5)
  - 9-12.2.4.b.ii Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
- 9-12.2.4.c Solve equations and inequalities in one variable
  - 9-12.2.4.c.i Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  - 9-12.2.4.c.ii Solve quadratic equations in one variable.
    - 9-12.2.4.c.ii.1 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.
    - 9-12.2.4.c.ii.2 Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation.
    - 9-12.2.4.c.ii.3 Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
- 9-12.2.4.d Solve systems of equations
  - 9-12.2.4.d.i 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.
  - 9-12.2.4.d.ii Solve systems of linear equations exactly and approximately, focusing on pairs of linear equations in two variables.
  - 9-12.2.4.d.iii Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
    - Solve a nonlinear system of equations (A2-E.18)
- 9-12.2.4.e Represent and solve equations and inequalities graphically
  - 9-12.2.4.e.i Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve.
  - 9-12.2.4.e.ii 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.
  - 9-12.2.4.e.iii 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.

9-12.3 Data Analysis, Statistics, and Probability

9-12.3.1 Visual displays and summary statistics condense the information in data sets into usable knowledge
- 9-12.3.1.a Summarize, represent, and interpret data on a single count or measurement variable
  - 9-12.3.1.a.i Represent data with plots on the real number line (dot plots, histograms, and box plots).
    - Create bar graphs, line graphs, and histograms (A1-N.6)
  - 9-12.3.1.a.ii Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
  - 9-12.3.1.a.iii Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
  - 9-12.3.1.a.iv 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.
    - Find probabilities using the normal distribution I (PC-Y.12)
    - Use normal distributions to approximate binomial distributions (PC-Y.18)
  - 9-12.3.1.a.v Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
- 9-12.3.1.b Summarize, represent, and interpret data on two categorical and quantitative variables
  - 9-12.3.1.b.i Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
    - Find probabilities using two-way frequency tables (PC-X.5)
    - Find conditional probabilities using two-way frequency tables (PC-X.9)
  - 9-12.3.1.b.ii Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
    - 9-12.3.1.b.ii.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.
      - Analyze a regression line of a data set (PC-Z.10)
      - Analyze a regression line using statistics of a data set (PC-Z.11)
    - 9-12.3.1.b.ii.2 Informally assess the fit of a function by plotting and analyzing residuals.
      - Interpret a scatter plot (A1-MM.8)
    - 9-12.3.1.b.ii.3 Fit a linear function for a scatter plot that suggests a linear association.
      - Write equations for lines of best fit (A1-MM.12)
      - Find the equation of a regression line (PC-Z.8)
- 9-12.3.1.c Interpret linear models
  - 9-12.3.1.c.i Interpret the slope and the intercept of a linear model in the context of the data.
    - Find the equation of a regression line (PC-Z.8)
    - Interpret regression lines (PC-Z.9)
  - 9-12.3.1.c.ii Using technology, compute and interpret the correlation coefficient of a linear fit.
    - Calculate correlation coefficients (PC-Z.7)
    - Find the equation of a regression line (PC-Z.8)
  - 9-12.3.1.c.iii Distinguish between correlation and causation.
9-12.3.2 Statistical methods take variability into account supporting informed decisions making through quantitative studies designed to answer specific questions
- 9-12.3.2.a Understand and evaluate random processes underlying statistical experiments
  - 9-12.3.2.a.i Describe statistics as a process for making inferences about population parameters based on a random sample from that population.
    - Identify biased samples (A1-MM.1)
  - 9-12.3.2.a.ii Decide if a specified model is consistent with results from a given data-generating process.
- 9-12.3.2.b Make inferences and justify conclusions from sample surveys, experiments, and observational studies
  - 9-12.3.2.b.i Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
    - Identify biased samples (A2-FF.1)
    - Identify biased samples (PC-Z.1)
  - 9-12.3.2.b.ii Use data from a sample survey to estimate a population mean or proportion.
    - Experimental probability (A1-LL.2)
  - 9-12.3.2.b.iii Develop a margin of error through the use of simulation models for random sampling.
  - 9-12.3.2.b.iv 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 (PC-Z.16)
  - 9-12.3.2.b.v Define and explain the meaning of significance, both statistical (using p-values) and practical (using effect size).
  - 9-12.3.2.b.vi Evaluate reports based on data.
    - Experiment design (A2-FF.12)
    - Experiment design (PC-Z.15)
9-12.3.3 Probability models outcomes for situations in which there is inherent randomness
- 9-12.3.3.a Understand independence and conditional probability and use them to interpret data
  - 9-12.3.3.a.i Describe events as subsets of a sample space using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events.
  - 9-12.3.3.a.ii 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.
  - 9-12.3.3.a.iii Using the conditional probability of A given B as P(A and B)/P(B), interpret the independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
    - Independence and conditional probability (A2-DD.10)
    - Independence and conditional probability (PC-X.8)
  - 9-12.3.3.a.iv 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.
  - 9-12.3.3.a.v Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
    - Independence and conditional probability (A2-DD.10)
    - Independence and conditional probability (PC-X.8)
- 9-12.3.3.b Use the rules of probability to compute probabilities of compound events in a uniform probability model
  - 9-12.3.3.b.i Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
  - 9-12.3.3.b.ii 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.
    - Find probabilities using the addition rule (A2-DD.12)
    - Find probabilities using the addition rule (PC-X.10)
- 9-12.3.3.c Analyze the cost of insurance as a method to offset the risk of a situation

9-12.4 Shape, Dimension, and Geometric Relationships

9-12.4.1 Objects in the plane can be transformed, and those transformations can be described and analyzed mathematically
- 9-12.4.1.a Experiment with transformations in the plane
  - 9-12.4.1.a.i State precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
    - Lines, line segments, and rays (G-B.1)
    - Angle vocabulary (G-C.1)
  - 9-12.4.1.a.ii Represent transformations in the plane using appropriate tools.
  - 9-12.4.1.a.iii Describe transformations as functions that take points in the plane as inputs and give other points as outputs.
  - 9-12.4.1.a.iv Compare transformations that preserve distance and angle to those that do not.
  - 9-12.4.1.a.v Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
    - Rotate polygons about a point (G-L.7)
    - Transformations that carry a polygon onto itself (G-L.13)
  - 9-12.4.1.a.vi Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  - 9-12.4.1.a.vii Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools.
  - 9-12.4.1.a.viii Specify a sequence of transformations that will carry a given figure onto another.
    - Sequences of congruence transformations: graph the image (G-L.11)
- 9-12.4.1.b Understand congruence in terms of rigid motions
  - 9-12.4.1.b.i Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure.
    - Solids of revolution (G-H.5)
  - 9-12.4.1.b.ii Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
  - 9-12.4.1.b.iii Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
  - 9-12.4.1.b.iv Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
- 9-12.4.1.c Prove geometric theorems
  - 9-12.4.1.c.i Prove theorems about lines and angles.
  - 9-12.4.1.c.ii Prove theorems about triangles.
  - 9-12.4.1.c.iii Prove theorems about parallelograms.
- 9-12.4.1.d Make geometric constructions
  - 9-12.4.1.d.i Make formal geometric constructions with a variety of tools and methods.
  - 9-12.4.1.d.ii Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
9-12.4.2 Concepts of similarity are foundational to geometry and its applications
- 9-12.4.2.a Understand similarity in terms of similarity transformations
  - 9-12.4.2.a.i Verify experimentally the properties of dilations given by a center and a scale factor:
    - 9-12.4.2.a.i.1 Show that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
    - 9-12.4.2.a.i.2 Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
      - Dilations and scale factors (A1-F.8)
      - Dilations: find the scale factor (G-L.18)
  - 9-12.4.2.a.ii Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.
    - Similar figures: side lengths and angle measures (A1-F.6)
    - Identify similar figures (G-P.3)
  - 9-12.4.2.a.iii Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
    - Similar figures: side lengths and angle measures (A1-F.6)
  - 9-12.4.2.a.iv Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
    - Similar triangles and indirect measurement (A1-F.7)
- 9-12.4.2.b Prove theorems involving similarity
  - 9-12.4.2.b.i Prove theorems about triangles.
  - 9-12.4.2.b.ii Prove that all circles are similar.
  - 9-12.4.2.b.iii Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- 9-12.4.2.c Define trigonometric ratios and solve problems involving right triangles
  - 9-12.4.2.c.i Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
  - 9-12.4.2.c.ii Explain and use the relationship between the sine and cosine of complementary angles.
    - Complementary angle identities (PC-O.1)
  - 9-12.4.2.c.iii Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
- 9-12.4.2.d Prove and apply trigonometric identities
  - 9-12.4.2.d.i Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1.
  - 9-12.4.2.d.ii Use the Pythagorean identity to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
    - Find trigonometric ratios using the unit circle (PC-M.6)
- 9-12.4.2.e Understand and apply theorems about circles.
  - 9-12.4.2.e.i Identify and describe relationships among inscribed angles, radii, and chords.
    - Arcs and chords (G-U.9)
    - Inscribed angles (G-U.12)
  - 9-12.4.2.e.ii Construct the inscribed and circumscribed circles of a triangle.
    - Construct an equilateral triangle inscribed in a circle (G-U.19)
    - Construct the inscribed or circumscribed circle of a triangle (G-U.22)
  - 9-12.4.2.e.iii Prove properties of angles for a quadrilateral inscribed in a circle.
    - Angles in inscribed quadrilaterals I (G-U.14)
- 9-12.4.2.f Find arc lengths and areas of sectors of circles.
  - 9-12.4.2.f.i Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality.
    - Radians and arc length (A2-Y.2)
    - Radians and arc length (PC-M.2)
  - 9-12.4.2.f.ii Derive the formula for the area of a sector.
    - Area of sectors (G-U.7)
9-12.4.3 Objects in the plane can be described and analyzed algebraically
- 9-12.4.3.a Express Geometric Properties with Equations.
  - 9-12.4.3.a.i Translate between the geometric description and the equation for a conic section
    - 9-12.4.3.a.i.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem.
    - 9-12.4.3.a.i.2 Complete the square to find the center and radius of a circle given by an equation.

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9-12.1 Number Sense, Properties, and Operations

9-12.1.1 The complex number system includes real numbers and imaginary numbers

9-12.1.1.a Extend the properties of exponents to rational exponents.

9-12.1.1.a.i Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

9-12.1.1.a.ii Rewrite expressions involving radicals and rational exponents using the properties of exponents.

9-12.1.1.b Use properties of rational and irrational numbers.

9-12.1.1.b.i Explain why the sum or product of two rational numbers is rational.

9-12.1.1.b.ii Explain why the sum of a rational number and an irrational number is irrational.

9-12.1.1.b.iii Explain why the product of a nonzero rational number and an irrational number is irrational.

9-12.1.1.c Perform arithmetic operations with complex numbers.

9-12.1.1.c.i Define the complex number i such that i2 = -1, and show that every complex number has the form a + bi where a and b are real numbers.

9-12.1.1.c.ii Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

9-12.1.1.d Use complex numbers in polynomial identities and equations.

9-12.1.1.d.i Solve quadratic equations with real coefficients that have complex solutions.

9-12.1.2 Quantitative reasoning is used to make sense of quantities and their relationships in problem situations

9-12.1.2.a Reason quantitatively and use units to solve problems.

9-12.1.2.a.i Use units as a way to understand problems and to guide the solution of multi-step problems.

9-12.1.2.a.i.1 Choose and interpret units consistently in formulas.

9-12.1.2.a.i.2 Choose and interpret the scale and the origin in graphs and data displays.

9-12.1.2.a.ii Define appropriate quantities for the purpose of descriptive modeling.

9-12.1.2.a.iii Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

9-12.1.2.a.iv Describe factors affecting take-home pay and calculate the impact

9-12.1.2.a.v Design and use a budget, including income (net take-home pay) and expenses (mortgage, car loans, and living expenses) to demonstrate how living within your means is essential for a secure financial future

9-12.2 Patterns, Functions, and Algebraic Structures

9-12.2.1 Functions model situations where one quantity determines another and can be represented algebraically, graphically, and using tables

9-12.2.1.a Formulate the concept of a function and use function notation.

9-12.2.1.a.i 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.

9-12.2.1.a.ii Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

9-12.2.1.a.iii Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

9-12.2.1.b Interpret functions that arise in applications in terms of the context

9-12.2.1.b.i 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 given a verbal description of the relationship.

9-12.2.1.b.ii Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

9-12.2.1.b.iii Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.

9-12.2.1.c Analyze functions using different representations

9-12.2.1.c.i Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

9-12.2.1.c.ii Graph linear and quadratic functions and show intercepts, maxima, and minima.

9-12.2.1.c.iii Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

9-12.2.1.c.iv Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

9-12.2.1.c.v Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

9-12.2.1.c.vi Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

9-12.2.1.c.vi.1 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.

9-12.2.1.c.vi.2 Use the properties of exponents to interpret expressions for exponential functions.

9-12.2.1.c.vi.3 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

9-12.2.1.d Build a function that models a relationship between two quantities

9-12.2.1.d.i Write a function that describes a relationship between two quantities.

9-12.2.1.d.i.1 Determine an explicit expression, a recursive process, or steps for calculation from a context.

9-12.2.1.d.i.2 Combine standard function types using arithmetic operations.

9-12.2.1.d.ii Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

9-12.2.1.e Build new functions from existing functions

9-12.2.1.e.i Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k, and find the value of k given the graphs.

9-12.2.1.e.ii Experiment with cases and illustrate an explanation of the effects on the graph using technology.

9-12.2.1.e.iii Find inverse functions.

9-12.2.1.f Extend the domain of trigonometric functions using the unit circle

9-12.2.1.f.i Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

9-12.2.1.f.ii 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.

9-12.2.2 Quantitative relationships in the real world can be modeled and solved using functions

9-12.2.2.a Construct and compare linear, quadratic, and exponential models and solve problems

9-12.2.2.a.i Distinguish between situations that can be modeled with linear functions and with exponential functions.

9-12.2.2.a.i.1 Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

9-12.2.2.a.i.2 Identify situations in which one quantity changes at a constant rate per unit interval relative to another.

9-12.2.2.a.i.3 Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

9-12.2.2.a.ii Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.

9-12.2.2.a.iii Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

9-12.2.2.a.iv For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

9-12.2.2.b Interpret expressions for functions in terms of the situation they model

9-12.2.2.b.i Interpret the parameters in a linear or exponential function in terms of a context.

9-12.2.2.c Model periodic phenomena with trigonometric functions.

9-12.2.2.c.i Choose the trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

9-12.2.2.d Model personal financial situations

9-12.2.2.d.i Analyze the impact of interest rates on a personal financial plan

9-12.2.2.d.ii Evaluate the costs and benefits of credit

9-12.2.2.d.iii Analyze various lending sources, services, and financial institutions

9-12.2.3 Expressions can be represented in multiple, equivalent forms

9-12.2.3.a Interpret the structure of expressions

9-12.2.3.a.i Interpret expressions that represent a quantity in terms of its context.

9-12.2.3.a.i.1 Interpret parts of an expression, such as terms, factors, and coefficients.

9-12.2.3.a.i.2 Interpret complicated expressions by viewing one or more of their parts as a single entity.

9-12.1.1.c.i Define the complex number i such that i² = -1, and show that every complex number has the form a + bi where a and b are real numbers.

9-12.1.1.c.ii Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

9-12.2.4.c.ii.1 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.