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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-RN.A The Real Number System: Extend the properties of exponents to rational exponents.
- HS.N-RN.A.1 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.
  - Evaluate integers raised to positive rational exponents (A1-W.11)
- HS.N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
HS.N-RN.B The Real Number System: Use properties of rational and irrational numbers.
- HS.N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
- Checkpoint opportunity
  - Checkpoint: Radicals and rational exponents (A1-FF.9)
HS.N-CN.A The Complex Number System: Perform arithmetic operations with complex numbers.
- HS.N-CN.A.1 Define 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)
- HS.N-CN.A.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HS.N-CN.C The Complex Number System: Use complex numbers in polynomial identities and equations.
- HS.N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.
  - Solve a quadratic equation using square roots (A2-K.6)
  - Using the discriminant (A2-K.12)
- HS.N-CN.C.8 Extend polynomial identities to the complex numbers.
  - Complex conjugate theorem (A2-L.12)
  - Conjugate root theorems (A2-L.13)
- HS.N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

1 Number and Quantity

HS.N-Q.A Quantities: Reason quantitatively and use units to solve problems.
- HS.N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- HS.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
- HS.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
- Checkpoint opportunity
  - Checkpoint: Units and quantities (A1-O.6)

1 Number and Quantity

HS.N-CN.A The Complex Number System: Perform arithmetic operations with complex numbers.
- HS.N-CN.A.1 Define 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)
- HS.N-CN.A.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HS.N-CN.C The Complex Number System: Use complex numbers in polynomial identities and equations.
- HS.N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.
  - Solve a quadratic equation using square roots (A2-K.6)
  - Using the discriminant (A2-K.12)
- HS.N-CN.C.8 Extend polynomial identities to the complex numbers.
- HS.N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  - Fundamental Theorem of Algebra (A2-L.17)

3 Data, Statistics, and Probability

HS.S-CP.A Conditional Probability & the Rules of Probability: Understand independence and conditional probability and use them to interpret data.
- HS.S-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
  - Theoretical and experimental probability (G-X.1)
  - Outcomes of compound events (G-X.2)
- HS.S-CP.A.2 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.
- HS.S-CP.A.3 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.
  - Find conditional probabilities (G-X.11)
  - Independence and conditional probability (G-X.12)
- HS.S-CP.A.4 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.
  - Find probabilities using two-way frequency tables (G-X.9)
  - Find conditional probabilities using two-way frequency tables (G-X.13)
- HS.S-CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
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.
- HS.S-CP.B.6 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.
  - Find conditional probabilities using two-way frequency tables (G-X.13)
  - Geometric probability (G-X.14)
- HS.S-CP.B.7 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 (G-X.15)
- HS.S-CP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A) P(B | A) = P(B) P(A | B), and interpret the answer in terms of the model.
  - Find conditional probabilities (G-X.11)
- HS.S-CP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems.
- Checkpoint opportunity
  - Checkpoint: Understand independence and conditional probability (G-X.16)
  - Checkpoint: Probabilities of compound events (G-X.17)
HS.S-MD.B Using Probability to Make Decisions: Use probability to evaluate outcomes of decisions.
- HS.S-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
- HS.S-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

1 Number and Quantity

HS.N-RN.A The Real Number System: Extend the properties of exponents to rational exponents.
- HS.N-RN.A.1 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.
  - Evaluate integers raised to positive rational exponents (A1-W.11)
  - Evaluate integers raised to rational exponents (A1-W.12)
- HS.N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
HS.N-RN.B The Real Number System: Use properties of rational and irrational numbers.
- HS.N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
- Checkpoint opportunity
  - Checkpoint: Radicals and rational exponents (A1-FF.9)
HS.N-Q.A Quantities: Reason quantitatively and use units to solve problems.
- HS.N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- HS.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
- HS.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
- Checkpoint opportunity
  - Checkpoint: Units and quantities (A1-O.6)

1 Number and Quantity

HS.N-CN.C The Complex Number System: Use complex numbers in polynomial identities and equations.
- HS.N-CN.C.8 Extend polynomial identities to the complex numbers.
- HS.N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  - Fundamental Theorem of Algebra (A2-L.17)

4 Geometry

HS.G-CO.A Congruence: Experiment with transformations in the plane.
- HS.G-CO.A.1 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.
  - Angle vocabulary (G-C.1)
- HS.G-CO.A.2 Represent transformations in the plane using e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
- HS.G-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
  - Transformations that carry a polygon onto itself (G-L.13)
- HS.G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  - Rotate polygons about a point (G-L.7)
- HS.G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools (e.g., graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will carry a given figure onto another.
- Checkpoint opportunity
HS.G-CO.B Congruence: Understand congruence in terms of rigid motions.
- HS.G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
- HS.G-CO.B.7 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.
- HS.G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
- Checkpoint opportunity
  - Checkpoint: Rigid motion and congruence (G-L.23)
HS.G-CO.C Congruence: Prove geometric theorems.
- HS.G-CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
- HS.G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
- HS.G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
- Checkpoint opportunity
HS.G-CO.D Congruence: Make geometric constructions.
- HS.G-CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
- HS.G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
- Checkpoint opportunity
  - Checkpoint: Geometric constructions (G-U.26)
HS.G-SRT.A Similarity, Right Triangles, and Trigonometry: Understand similarity in terms of similarity transformations.
- HS.G-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor.
  - HS.G-SRT.A.1.a 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.
    - Dilations: graph the image (G-L.15)
    - Dilations and parallel lines (G-L.20)
  - HS.G-SRT.A.1.b Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
- HS.G-SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; 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.
- HS.G-SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
  - Similarity rules for triangles (G-P.8)
- Checkpoint opportunity
  - Checkpoint: Dilations (G-L.24)
  - Checkpoint: Similarity transformations (G-P.17)
HS.G-SRT.B Similarity, Right Triangles, and Trigonometry: Prove theorems involving similarity.
- HS.G-SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
- HS.G-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- Checkpoint opportunity
  - Checkpoint: Triangle similarity and congruence (G-P.18)
HS.G-SRT.C Similarity, Right Triangles, and Trigonometry: Define trigonometric ratios and solve problems involving right triangles.
- HS.G-SRT.C.6 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.
- HS.G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
  - Sine and cosine of complementary angles (G-R.8)
- HS.G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
- Checkpoint opportunity
  - Checkpoint: Right triangle trigonometry (G-R.18)
HS.G-SRT.D Similarity, Right Triangles, and Trigonometry: Apply trigonometry to general triangles.
- HS.G-SRT.D.9 Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
  - Area of a triangle: sine formula (G-R.16)
- HS.G-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems.
- HS.G-SRT.D.11 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).
  - Solve a triangle (G-R.15)
- Checkpoint opportunity
  - Checkpoint: Laws of Sines and Cosines (G-R.19)
HS.G-C.A Circles: Understand and apply theorems about circles.
- HS.G-C.A.1 Prove that all circles are similar.
  - Similarity of circles (G-P.10)
- HS.G-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
- HS.G-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
- HS.G-C.A.4 Construct a tangent line from a point outside a given circle to the circle.
  - Construct a tangent line to a circle (G-U.18)
- Checkpoint opportunity
HS.G-C.B Circles: Find arc lengths and areas of sectors of circles.
- HS.G-C.B.5 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; derive the formula for the area of a sector.
- Checkpoint opportunity
  - Checkpoint: Arc length and area of sectors (G-U.25)
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.1 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.
- HS.G-GPE.A.2 Derive the equation of a parabola given a focus and directrix.
- Checkpoint opportunity
  - Checkpoint: Equations of circles (G-V.19)
  - Checkpoint: Equations of parabolas (G-V.20)
HS.G-GPE.B Expressing Geometric Properties with Equations: Use coordinates to prove simple geometric theorems algebraically.
- HS.G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically.
  - Classify shapes on the coordinate plane: justify your answer (G-N.13)
  - Determine if a point lies on a circle (G-V.3)
- HS.G-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
- HS.G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
- HS.G-GPE.B.7 Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.
- Checkpoint opportunity
HS.G-GMD.A Geometric Measurement and Dimension: Explain volume formulas and use them to solve problems.
- HS.G-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
- HS.G-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
- Checkpoint opportunity
  - Checkpoint: Volume (G-T.14)
HS.G-GMD.B Geometric Measurement and Dimension: Visualize relationships between two-dimensional and three-dimensional objects.
- HS.G-GMD.B.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
  - Cross sections of three-dimensional figures (G-H.4)
  - Solids of revolution (G-H.5)
- Checkpoint opportunity
  - Checkpoint: Cross sections and solids of revolution (G-H.6)
HS.G-MG.A Modeling with Geometry: Apply geometric concepts in modeling situations.
- HS.G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
  - Pythagorean theorem (G-Q.1)
- HS.G-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
  - Calculate density, mass, and volume (G-W.9)
- HS.G-MG.A.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).
- Checkpoint opportunity
  - Checkpoint: Geometric modeling and design (G-W.10)
  - Checkpoint: Density (G-W.11)

2 Algebra and Functions

HS.A-SSE.A Seeing Structure in Expressions: Interpret the structure of expressions.
- HS.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
  - HS.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
  - HS.A-SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
    - Interpret parts of quadratic expressions: word problems (A2-K.)
- HS.A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
HS.A-SSE.B Seeing Structure in Expressions: Write expressions in equivalent forms to solve problems.
- HS.A-SSE.B.4 Use the formula for the sum of a finite geometric series (when the common ratio is not 1) to solve problems.
  - Find the sum of a finite geometric series (A2-CC.16)
HS.A-APR.A Arithmetic with Polynomials & Rational Expressions: Perform arithmetic operations on polynomials.
- HS.A-APR.A.1 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.
  - Add and subtract polynomials (A2-L.2)
  - Multiply polynomials (A2-L.3)
HS.A-APR.B Arithmetic with Polynomials & Rational Expressions: Understand the relationship between zeros and factors of polynomials.
- HS.A-APR.B.2 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.)
- HS.A-APR.B.3 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.
HS.A-APR.C Arithmetic with Polynomials & Rational Expressions: Use polynomial identities to solve problems.
- HS.A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.
- HS.A-APR.C.5 Know and apply the Binomial Theorem for the expansion of 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.)
HS.A-APR.D Arithmetic with Polynomials & Rational Expressions: Rewrite rational expressions.
- HS.A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
  - Divide polynomials using long division (A2-L.4)
  - Simplify rational expressions (A2-O.4)
- HS.A-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expressions; add, subtract, multiply, and divide rational expressions.
  - Multiply and divide rational expressions (A2-O.5)
  - Add and subtract rational expressions (A2-O.6)
HS.A-CED.A Creating Equations: Create equations that describe numbers or relationships.
- HS.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
- HS.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales.
- HS.A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
- HS.A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  - Solve multi-variable equations (A2-B.6)
HS.A-REI.A Reasoning with Equations & Inequalities: Understand solving equations as a process of reasoning and explain the reasoning.
- HS.A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  - Solve radical equations (A2-M.14)
  - Solve rational equations (A2-O.7)
HS.A-REI.D Reasoning with Equations & Inequalities: Represent and solve equations and inequalities graphically.
- HS.A-REI.D.11 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 e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
HS.F-IF.B Interpreting Functions: Interpret functions that arise in applications in terms of the context.
- HS.F-IF.B.4 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.
- HS.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
- HS.F-IF.B.6 Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.
  - Find the slope of a linear function (A2-D.8)
  - Average rate of change (A2-D.12)
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.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
  - HS.F-IF.C.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
  - HS.F-IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
- HS.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  - HS.F-IF.C.8.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.
  - HS.F-IF.C.8.b Use the properties of exponents to interpret expressions for exponential functions.
- HS.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
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.b Combine standard function types using arithmetic operations.
HS.F-BF.B Building Functions: Build new functions from existing functions.
- HS.F-BF.B.3 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.
- HS.F-BF.B.4 Find inverse functions.
  - HS.F-BF.B.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
HS.F-LE.A Linear, Quadratic & Exponential Models: Construct and compare linear, quadratic, and exponential models and solve problems.
- HS.F-LE.A.4 For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
HS.F-TF.A Trigonometric Functions: Extend the domain of trigonometric functions using the unit circle.
- HS.F-TF.A.1 Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
  - Convert between radians and degrees (A2-Y.1)
  - Radians and arc length (A2-Y.2)
- HS.F-TF.A.2 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.
HS.F-TF.B Trigonometric Functions: Model periodic phenomena with trigonometric functions.
- HS.F-TF.B.5 Model periodic phenomena with trigonometric functions with specified amplitude, frequency, and midline.
HS.F-TF.C Trigonometric Functions: Prove and apply trigonometric identities.
- HS.F-TF.C.8 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.
  - Trigonometric identities I (A2-BB.3)
  - Trigonometric identities II (A2-BB.4)

2 Algebra and Functions

HS.A-SSE.A Seeing Structure in Expressions: Interpret the structure of expressions.
- HS.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
  - HS.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
    - Sort factors of variable expressions (A1-I.2)
  - HS.A-SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
HS.A-CED.A Creating Equations: Create equations that describe numbers or relationships.
- HS.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
- HS.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales.
- HS.A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
- HS.A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  - Rearrange multi-variable equations (A1-I.11)
- Checkpoint opportunity
  - Checkpoint: Represent constraints (A1-V.17)
  - Checkpoint: Expressions, equations, and inequalities (A1)
HS.A-REI.A Reasoning with Equations & Inequalities: Understand solving equations as a process of reasoning and explain the reasoning.
- HS.A-REI.A.1 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. Construct a viable argument to justify a solution method.
HS.A-REI.B Reasoning with Equations & Inequalities: Solve equations and inequalities in one variable.
- HS.A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
- Checkpoint opportunity
  - Checkpoint: Solve linear equations and inequalities (A1-K.16)
HS.A-REI.C Reasoning with Equations & Inequalities: Solve systems of equations.
- HS.A-REI.C.5 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.
  - Solve a system of equations using elimination (A1-V.10)
  - Solve a system of equations using elimination: word problems (A1-V.11)
- HS.A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
HS.A-REI.D Reasoning with Equations & Inequalities: Represent and solve equations and inequalities graphically.
- HS.A-REI.D.10 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 (which could be a line).
- HS.A-REI.D.11 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 e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
- HS.A-REI.D.12 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.
- Checkpoint opportunity
  - Checkpoint: Systems of equations and inequalities (A1-V.16)
  - Checkpoint: Solve equations using graphs and tables (A1)
HS.F-IF.A Interpreting Functions: Understand the concept of a function and use function notation.
- HS.F-IF.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).
- HS.F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- HS.F-IF.A.3 Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
HS.F-IF.B Interpreting Functions: Interpret functions that arise in applications in terms of the context.
- HS.F-IF.B.4 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.
- HS.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
- HS.F-IF.B.6 Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.
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.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
  - HS.F-IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
    - Graph exponential functions (A1-Y.2)
    - Match exponential functions and graphs II (A1-Y.4)
- HS.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
  - Compare linear functions: graphs and equations (A1-T.15)
  - Compare linear functions: tables, graphs, and equations (A1-T.16)
- Checkpoint opportunity
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.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
  - HS.F-BF.A.1.b Combine standard function types using arithmetic operations.
- HS.F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
HS.F-BF.B Building Functions: Build new functions from existing functions.
- HS.F-BF.B.3 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.
  - Transformations of linear functions (A1-T.29)
  - Function transformation rules (A1-EE.7)
- Checkpoint opportunity
  - Checkpoint: Sequences (A1-P.13)
  - Checkpoint: Build functions (A1)
HS.F-LE.A Linear, Quadratic & Exponential Models: Construct and compare linear, quadratic, and exponential models and solve problems.
- HS.F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
  - HS.F-LE.A.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
  - HS.F-LE.A.1.b Identify situations in which one quantity changes at a constant rate per unit interval relative to another.
  - HS.F-LE.A.1.c Identify situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
- HS.F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
- HS.F-LE.A.3 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.
  - Compare linear and exponential growth (A1-DD.11)
HS.F-LE.B Linear, Quadratic, & Exponential Models: Interpret expressions for functions in terms of the situation they model.
- HS.F-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context.
- Checkpoint opportunity
  - Checkpoint: Compare linear and exponential functions (A1-DD.15)

2 Algebra and Functions

HS.A-SSE.A Seeing Structure in Expressions: Interpret the structure of expressions.
- HS.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
  - HS.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
  - HS.A-SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
    - Interpret parts of quadratic expressions: word problems (A1-CC.)
- HS.A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
HS.A-SSE.B Seeing Structure in Expressions: Write expressions in equivalent forms to solve problems.
- HS.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  - HS.A-SSE.B.3.a Factor a quadratic expression to reveal the zeros of the function it defines.
  - HS.A-SSE.B.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
    - Complete the square (A1-CC.9)
    - Write a quadratic function in vertex form (A1-CC.13)
  - HS.A-SSE.B.3.c Use the properties of exponents to transform expressions for exponential functions.
- Checkpoint opportunity
  - Checkpoint: Write and interpret equivalent expressions (A1)
HS.A-APR.A Arithmetic with Polynomials & Rational Expressions: Perform arithmetic operations on polynomials.
- HS.A-APR.A.1 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.
- Checkpoint opportunity
  - Checkpoint: Polynomial operations (A1-AA.13)
HS.A-CED.A Creating Equations: Create equations that describe numbers or relationships.
- HS.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
- HS.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales.
- HS.A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  - Solve multi-variable equations (A2-B.6)
HS.A-REI.B Reasoning with Equations & Inequalities: Solve equations and inequalities in one variable.
- HS.A-REI.B.4 Solve quadratic equations in one variable.
  - HS.A-REI.B.4.a 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.
    - Complete the square (A1-CC.9)
  - HS.A-REI.B.4.b 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. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
HS.A-REI.C Reasoning with Equations & Inequalities: Solve systems of equations.
- HS.A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
  - Solve a system of linear and quadratic equations by graphing (A1-CC.18)
  - Solve a system of linear and quadratic equations (A1-CC.19)
- Checkpoint opportunity
  - Checkpoint: Quadratic equations (A1)
HS.F-IF.B Interpreting Functions: Interpret functions that arise in applications in terms of the context.
- HS.F-IF.B.4 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.
- HS.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
- HS.F-IF.B.6 Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.
  - Rate of change: graphs (A1-Q.19)
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.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
  - HS.F-IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
- HS.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  - HS.F-IF.C.8.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.
  - HS.F-IF.C.8.b Use the properties of exponents to interpret expressions for exponential functions.
- HS.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
- Checkpoint opportunity
  - Checkpoint: Interpret features of functions (A1)
  - Checkpoint: Graph and analyze functions (A1)
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.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
  - HS.F-BF.A.1.b Combine standard function types using arithmetic operations.
HS.F-BF.B Building Functions: Build new functions from existing functions.
- HS.F-BF.B.3 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.
- HS.F-BF.B.4 Find inverse functions.
  - HS.F-BF.B.4.a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
    - Find the inverse of a function (A1-Q.11)
- Checkpoint opportunity
  - Checkpoint: Build functions (A1)
HS.F-LE.A Linear, Quadratic & Exponential Models: Construct and compare linear, quadratic, and exponential models and solve problems.
- HS.F-LE.A.3 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.
  - Compare linear, exponential, and quadratic growth (A1-DD.12)
HS.F-TF.C Trigonometric Functions: Prove and apply trigonometric identities.
- HS.F-TF.C.8 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.
  - Trigonometric identities I (A2-BB.3)
  - Trigonometric identities II (A2-BB.4)

2 Algebra and Functions

HS.A-SSE.A Seeing Structure in Expressions: Interpret the structure of expressions.
- HS.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.
  - HS.A-SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
    - Interpret parts of quadratic expressions: word problems (A2-K.)
    - Polynomial vocabulary (A2-L.1)
  - HS.A-SSE.A.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
    - Interpret parts of quadratic expressions: word problems (A2-K.)
- HS.A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
HS.A-SSE.B Seeing Structure in Expressions: Write expressions in equivalent forms to solve problems.
- HS.A-SSE.B.4 Use the formula for the sum of a finite geometric series (when the common ratio is not 1) to solve problems.
  - Find the sum of a finite geometric series (A2-CC.16)
HS.A-APR.A Arithmetic with Polynomials & Rational Expressions: Perform arithmetic operations on polynomials.
- HS.A-APR.A.1 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.
  - Add and subtract polynomials (A2-L.2)
  - Multiply polynomials (A2-L.3)
HS.A-APR.B Arithmetic with Polynomials & Rational Expressions: Understand the relationship between zeros and factors of polynomials.
- HS.A-APR.B.2 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.)
- HS.A-APR.B.3 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.
HS.A-APR.C Arithmetic with Polynomials & Rational Expressions: Use polynomial identities to solve problems.
- HS.A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.
- HS.A-APR.C.5 Know and apply the Binomial Theorem for the expansion of 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.)
HS.A-APR.D Arithmetic with Polynomials & Rational Expressions: Rewrite rational expressions.
- HS.A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
- HS.A-APR.D.7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expressions; add, subtract, multiply, and divide rational expressions.
  - Multiply and divide rational expressions (A2-O.5)
  - Add and subtract rational expressions (A2-O.6)
HS.A-CED.A Creating Equations: Create equations that describe numbers or relationships.
- HS.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  - Write exponential functions: word problems (A2-T.)
  - Solve quadratic equations: word problems (PC-C.)
- HS.A-CED.A.2 Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales.
- HS.A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
- HS.A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  - Solve multi-variable equations (A2-B.6)
HS.A-REI.A Reasoning with Equations & Inequalities: Understand solving equations as a process of reasoning and explain the reasoning.
- HS.A-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
  - Solve radical equations (A2-M.14)
  - Solve rational equations (A2-O.7)
HS.A-REI.D Reasoning with Equations & Inequalities: Represent and solve equations and inequalities graphically.
- HS.A-REI.D.11 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 e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
HS.F-IF.B Interpreting Functions: Interpret functions that arise in applications in terms of the context.
- HS.F-IF.B.4 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.
- HS.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - Domain and range of polynomials (A2-L.16)
  - Domain and range of radical functions (A2-M.12)
- HS.F-IF.B.6 Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.
  - Average rate of change (A2-D.12)
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.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
    - Graph an absolute value function (A2-D.13)
    - Graph piecewise-defined functions (A2-D.17)
  - HS.F-IF.C.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
  - HS.F-IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
- HS.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  - HS.F-IF.C.8.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.
  - HS.F-IF.C.8.b Use the properties of exponents to interpret expressions for exponential functions.
- HS.F-IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
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.b Combine standard function types using arithmetic operations.

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1 Number and Quantity

HS.N-RN.A The Real Number System: Extend the properties of exponents to rational exponents.

HS.N-RN.A.1 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.

HS.N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

HS.N-RN.B The Real Number System: Use properties of rational and irrational numbers.

HS.N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Checkpoint opportunity

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

HS.N-CN.A.1 Define 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.

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

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

HS.N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.

HS.N-CN.C.8 Extend polynomial identities to the complex numbers.

HS.N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

1 Number and Quantity

HS.N-Q.A Quantities: Reason quantitatively and use units to solve problems.

HS.N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

HS.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

HS.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Checkpoint opportunity

1 Number and Quantity

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

HS.N-CN.A.1 Define 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.

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

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

HS.N-CN.C.7 Solve quadratic equations with real coefficients that have complex solutions.

HS.N-CN.C.8 Extend polynomial identities to the complex numbers.

HS.N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

3 Data, Statistics, and Probability

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

HS.S-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

HS.S-CP.A.2 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.

HS.S-CP.A.3 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.

HS.S-CP.A.4 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.

HS.S-CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

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.

HS.S-CP.B.6 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.

HS.S-CP.B.7 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.

HS.S-CP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A) P(B | A) = P(B) P(A | B), and interpret the answer in terms of the model.

HS.S-CP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

Checkpoint opportunity

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

HS.S-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

HS.S-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

1 Number and Quantity

HS.N-RN.A The Real Number System: Extend the properties of exponents to rational exponents.

HS.N-RN.A.1 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.

HS.N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

HS.N-RN.B The Real Number System: Use properties of rational and irrational numbers.

HS.N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Checkpoint opportunity

HS.N-Q.A Quantities: Reason quantitatively and use units to solve problems.

HS.N-Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

HS.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

HS.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Checkpoint opportunity

1 Number and Quantity

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

HS.N-CN.C.8 Extend polynomial identities to the complex numbers.

HS.N-CN.C.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

4 Geometry

HS.G-CO.A Congruence: Experiment with transformations in the plane.

HS.G-CO.A.1 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.

HS.G-CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

HS.G-CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

HS.G-CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools (e.g., graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will carry a given figure onto another.

Checkpoint opportunity

HS.G-CO.B Congruence: Understand congruence in terms of rigid motions.

HS.G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

HS.G-CO.B.7 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.

HS.G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Checkpoint opportunity

HS.G-CO.C Congruence: Prove geometric theorems.

HS.G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Checkpoint opportunity

HS.G-CO.D Congruence: Make geometric constructions.

HS.G-CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.