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Skills available for Alaska high school math standards

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Alaska Content Standards: Algebra 1 Alaska Content Standards: Algebra 1
Alaska Content Standards: Algebra 2 Alaska Content Standards: Algebra 2
Alaska Content Standards: Geometry Alaska Content Standards: Geometry
Alaska Content Standards: Mathematics 1 Alaska Content Standards: Mathematics 1
Alaska Content Standards: Mathematics 2 Alaska Content Standards: Mathematics 2
Alaska Content Standards: Mathematics 3 Alaska Content Standards: Mathematics 3
Alaska Content Standards: Precalculus Alaska Content Standards: Precalculus

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9-12.N Number and Quantity

9-12.N-RN The Real Number System
- Extend the properties of exponents to rational exponents.
  - 9-12.N-RN.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)
  - 9-12.N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- Use properties of rational and irrational numbers.
  - 9-12.N-RN.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)
9-12.N-Q Quantities
- Reason quantitatively and use units to solve problems.
  - 9-12.N-Q.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.
  - 9-12.N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.
  - 9-12.N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
- Checkpoint opportunity
  - Checkpoint: Units and quantities (A1-O.6)

9-12.G Geometry

9-12.G-CO Congruence
- Experiment with transformations in the plane.
  - 9-12.G-CO.1 Demonstrates understanding of key geometrical definitions, including angle, circle, perpendicular line, parallel line, line segment, and transformations in Euclidian geometry. Understand undefined notions of point, line, distance along a line, and distance around a circular arc.
    - Angle vocabulary (G-C.1)
  - 9-12.G-CO.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).
  - 9-12.G-CO.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)
  - 9-12.G-CO.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)
  - 9-12.G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
- Checkpoint opportunity
- Understand congruence in terms of rigid motions.
  - 9-12.G-CO.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.
  - 9-12.G-CO.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.
  - 9-12.G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS, and HL) follow from the definition of congruence in terms of rigid motions.
- Checkpoint opportunity
  - Checkpoint: Rigid motion and congruence (G-L.23)
- Prove geometric theorems.
  - 9-12.G-CO.9 Using methods of proof including direct, indirect, and counter examples to prove theorems about lines and angles.
  - 9-12.G-CO.10 Using methods of proof including direct, indirect, and counter examples to prove theorems about triangles.
  - 9-12.G-CO.11 Using methods of proof including direct, indirect, and counter examples to prove theorems about parallelograms.
- Checkpoint opportunity
- Make geometric constructions.
  - 9-12.G-CO.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.
  - 9-12.G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
- Checkpoint opportunity
  - Checkpoint: Geometric constructions (G-U.26)
9-12.G-SRT Similarity, Right Triangles, and Trigonometry
- Understand similarity in terms of similarity transformations.
  - 9-12.G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
    - 9-12.G-SRT.1.a 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)
    - 9-12.G-SRT.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
  - 9-12.G-SRT.2 Given two figures, use the definition of similarity in terms of transformations to explain whether or not they are similar.
  - 9-12.G-SRT.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)
- Prove theorems involving similarity.
  - 9-12.G-SRT.4 Prove theorems about triangles.
  - 9-12.G-SRT.5 Apply congruence and similarity properties and prove relationships involving triangles and other geometric figures.
- Checkpoint opportunity
  - Checkpoint: Triangle similarity and congruence (G-P.18)
- Define trigonometric ratios and solve problems involving right triangles.
  - 9-12.G-SRT.6 Understand 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.G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.
    - Sine and cosine of complementary angles (G-R.8)
  - 9-12.G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
- Checkpoint opportunity
  - Checkpoint: Right triangle trigonometry (G-R.18)
- Apply trigonometry to general triangles.
  - 9-12.G-SRT.9 Derive the formula A = 1/2 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)
  - 9-12.G-SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems.
  - 9-12.G-SRT.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)
9-12.G-C Circles
- Understand and apply theorems about circles.
  - 9-12.G-C.1 Prove that all circles are similar.
    - Similarity of circles (G-P.10)
  - 9-12.G-C.2 Identify and describe relationships among inscribed angles, radii, and chords.
  - 9-12.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
  - 9-12.G-C.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
- Find arc lengths and areas of sectors of circles.
  - 9-12.G-C.5 Use and apply the concepts of arc length and areas of sectors of circles. Determine or 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)
9-12.G-GPE Expressing Geometric Properties with Equations
- Translate between the geometric description and the equation for a conic section.
  - 9-12.G-GPE.1 Determine or 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.
  - 9-12.G-GPE.2 Determine or 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)
- Use coordinates to prove simple geometric theorems algebraically.
  - 9-12.G-GPE.4 Perform simple coordinate proofs. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
    - Classify shapes on the coordinate plane: justify your answer (G-N.13)
    - Determine if a point lies on a circle (G-V.3)
  - 9-12.G-GPE.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).
  - 9-12.G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
  - 9-12.G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
- Checkpoint opportunity
9-12.G-GMD Geometric Measurement and Dimension
- Explain volume formulas and use them to solve problems.
  - 9-12.G-GMD.1 Explain how to find the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
  - 9-12.G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
- Checkpoint opportunity
  - Checkpoint: Volume (G-T.14)
- Visualize relationships between two-dimensional and three-dimensional objects.
  - 9-12.G-GMD.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)
9-12.G-MG Modeling with Geometry
- Apply geometric concepts in modeling situations.
  - 9-12.G-MG.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)
  - 9-12.G-MG.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)
  - 9-12.G-MG.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)

9-12.S Statistics and Probability

9-12.S-CP Conditional Probability and the Rules of Probability
- Understand independence and conditional probability and use them to interpret data.
  - 9-12.S-CP.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)
  - 9-12.S-CP.2 Understand 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.S-CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret 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)
  - 9-12.S-CP.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)
  - 9-12.S-CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
- Use the rules of probability to compute probabilities of compound events in a uniform probability model.
  - 9-12.S-CP.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)
  - 9-12.S-CP.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)
  - 9-12.S-CP.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)
  - 9-12.S-CP.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)
9-12.S-MD Using Probability to Make Decisions
- Use probability to evaluate outcomes of decisions.
  - 9-12.S-MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
  - 9-12.S-MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

9-12.A Algebra

9-12.A-SSE Seeing Structure in Expressions
- Interpret the structure of expressions.
  - 9-12.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
    - 9-12.A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
    - 9-12.A-SSE.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.)
  - 9-12.A-SSE.2 Use the structure of an expression to identify ways to rewrite it.
- Write expressions in equivalent forms to solve problems.
  - 9-12.A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
    - 9-12.A-SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines. For example, x² + 4x +3 = (x +3)(x +1).
    - 9-12.A-SSE.3.b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
    - 9-12.A-SSE.3.c Use the properties of exponents to transform expressions for exponential functions.
      - Evaluate expressions using properties of exponents (A1-W.8)
      - Evaluate an exponential function (A1-Y.1)
  - 9-12.A-SSE.4 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.
- Checkpoint opportunity
  - Checkpoint: Write and interpret equivalent expressions (A1-CC.21)
9-12.A-APR Arithmetic with Polynomials and Rational Expressions
- Perform arithmetic operations on polynomials.
  - 9-12.A-APR.1 Add, subtract, and multiply polynomials. Understand that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
- Checkpoint opportunity
  - Checkpoint: Polynomial operations (A1-AA.13)
9-12.A-CED Creating Equations and Inequalities
- Create equations and inequalities that describe numbers or relationships.
  - 9-12.A-CED.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.
  - 9-12.A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  - 9-12.A-CED.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.
  - 9-12.A-CED.4 Rearrange formulas (literal equations) 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: Problem solving with equations and inequalities (A1-DD.13)
9-12.A-REI Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning.
  - 9-12.A-REI.1 Apply properties of mathematics to justify steps in solving equations in one variable.
- Solve equations and inequalities in one variable.
  - 9-12.A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
  - 9-12.A-REI.4 Solve quadratic equations in one variable.
    - 9-12.A-REI.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)
    - 9-12.A-REI.4.b Solve quadratic equations by inspection (e.g., for x² = 49), 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.
- Checkpoint opportunity
  - Checkpoint: Solve linear equations and inequalities (A1-K.16)
  - Checkpoint: Quadratic equations (A1-CC.20)
- Solve systems of equations.
  - 9-12.A-REI.5 Show 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)
  - 9-12.A-REI.6 Solve systems of linear equations exactly and approximately, e.g., with graphs or algebraically, focusing on pairs of linear equations in two variables.
  - 9-12.A-REI.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)
  - 9-12.A-REI.8 Represent a system of linear equations as a single matrix equation in a vector variable.
    - Solve a system of equations using augmented matrices (A1-V.12)
    - Solve a system of equations using augmented matrices: word problems (A1-V.13)
  - 9-12.A-REI.9 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
- Represent and solve equations and inequalities graphically.
  - 9-12.A-REI.10 Understand 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).
  - 9-12.A-REI.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.
  - 9-12.A-REI.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: Solve equations using graphs and tables (A1-Q.22)
  - Checkpoint: Systems of equations and inequalities (A1-V.16)

9-12.F Functions

9-12.F-IF Interpreting Functions
- Understand the concept of a function and use function notation.
  - 9-12.F-IF.1 Understand that a function from one set (called the domain) to another set (called the range) 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).
  - 9-12.F-IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  - 9-12.F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
- Interpret functions that arise in applications in terms of the context.
  - 9-12.F-IF.4 For a function that models a relationship between two quantities,
    - 9-12.F-IF.4.a interpret key features of graphs and tables in terms of the quantities, and
    - 9-12.F-IF.4.b sketch graphs showing key features given a verbal description of the relationship.
  - 9-12.F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - 9-12.F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
- Checkpoint opportunity
  - Checkpoint: Function concepts (A1-Q.20)
  - Checkpoint: Average rate of change (A1-Q.21)
- Analyze functions using different representations.
  - 9-12.F-IF.7 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.F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.
    - 9-12.F-IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
    - 9-12.F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
  - 9-12.F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - 9-12.F-IF.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.
    - 9-12.F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.
  - 9-12.F-IF.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
  - Checkpoint: Features of functions (A1-DD.14)
  - Checkpoint: Graph and analyze functions (A1-EE.13)
9-12.F-BF Building Functions
- Build a function that models a relationship between two quantities.
  - 9-12.F-BF.1 Write a function that describes a relationship between two quantities.
    - 9-12.F-BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
    - 9-12.F-BF.1.b Combine standard function types using arithmetic operations.
  - 9-12.F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
- Checkpoint opportunity
  - Checkpoint: Sequences (A1-P.13)
- Build new functions from existing functions.
  - 9-12.F-BF.3 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 (both positive and negative); find the value of k given the graphs.
  - 9-12.F-BF.4 Find inverse functions.
    - 9-12.F-BF.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-DD.16)
  - Checkpoint: Function transformations (A1-EE.14)
9-12.F-LE Linear, Quadratic, and Exponential Models
- Construct and compare linear, quadratic, and exponential models and solve problems.
  - 9-12.F-LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
    - 9-12.F-LE.1.a Show that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
    - 9-12.F-LE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
    - 9-12.F-LE.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
  - 9-12.F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or input-output table of values.
  - 9-12.F-LE.3 Observe using graphs and tables 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)
    - Compare linear, exponential, and quadratic growth (A1-DD.12)
- Interpret expressions for functions in terms of the situation they model.
  - 9-12.F-LE.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)

9-12.S Statistics and Probability

9-12.S-ID Interpreting Categorical and Quantitative Data
- Summarize, represent, and interpret data on a single count or measurement variable.
  - 9-12.S-ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
  - 9-12.S-ID.2 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.S-ID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
    - Box plots (A1-N.9)
    - Identify an outlier and describe the effect of removing it (A1-MM.5)
- Checkpoint opportunity
  - Checkpoint: Line plots, histograms, and box plots (A1-N.10)
  - Checkpoint: Compare data sets (A1-MM.17)
- Summarize, represent, and interpret data on two categorical and quantitative variables.
  - 9-12.S-ID.5 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 (A1-LL.3)
    - Find conditional probabilities using two-way frequency tables (A1-LL.4)
  - 9-12.S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
    - 9-12.S-ID.6.a 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.
    - 9-12.S-ID.6.b Informally assess the fit of a function by plotting and analyzing residuals.
      - Interpret a scatter plot (A1-MM.8)
    - 9-12.S-ID.6.c Fit a linear function for a scatter plot that suggests a linear association.
      - Write equations for lines of best fit (A1-MM.12)
- Checkpoint opportunity
  - Checkpoint: Two-way frequency tables (A1-LL.11)
- Interpret linear models.
  - 9-12.S-ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
    - Interpret regression lines (A1-MM.14)
    - Analyze a regression line of a data set (A1-MM.15)
  - 9-12.S-ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.
    - Match correlation coefficients to scatter plots (A1-MM.10)
    - Calculate correlation coefficients (A1-MM.11)
  - 9-12.S-ID.9 Distinguish between correlation and causation.
    - Correlation and causation (A1-MM.16)
- Checkpoint opportunity
  - Checkpoint: Linear modeling (A1-MM.18)

9-12.N Number and Quantity

9-12.N-CN The Complex Number System
- Perform arithmetic operations with complex numbers.
  - 9-12.N-CN.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
    - Introduction to complex numbers (A2-I.1)
  - 9-12.N-CN.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- Use complex numbers in polynomial identities and equations.
  - 9-12.N-CN.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)
  - 9-12.N-CN.8 Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x – 2i).
  - 9-12.N-CN.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
    - Fundamental Theorem of Algebra (A2-L.17)

9-12.A Algebra

9-12.A-SSE Seeing Structure in Expressions
- Interpret the structure of expressions.
  - 9-12.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
    - 9-12.A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
    - 9-12.A-SSE.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.)
  - 9-12.A-SSE.2 Use the structure of an expression to identify ways to rewrite it.
- Write expressions in equivalent forms to solve problems.
  - 9-12.A-SSE.4 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.
    - Find the sum of a finite geometric series (A2-CC.16)
9-12.A-APR Arithmetic with Polynomials and Rational Expressions
- Perform arithmetic operations on polynomials.
  - 9-12.A-APR.1 Add, subtract, and multiply polynomials. Understand that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
    - Add and subtract polynomials (A2-L.2)
    - Multiply polynomials (A2-L.3)
- Understand the relationship between zeros and factors of polynomials.
  - 9-12.A-APR.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).
  - 9-12.A-APR.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.
- Use polynomial identities to solve problems.
  - 9-12.A-APR.4 Prove polynomial identities and use them to describe numerical relationships.
  - 9-12.A-APR.5 Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.
- Rewrite rational expressions.
  - 9-12.A-APR.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)
  - 9-12.A-APR.7 Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system similar to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.
    - Multiply and divide rational expressions (A2-O.5)
    - Add and subtract rational expressions (A2-O.6)
9-12.A-CED Creating Equations and Inequalities
- Create equations and inequalities that describe numbers or relationships.
  - 9-12.A-CED.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.
  - 9-12.A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  - 9-12.A-CED.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.
  - 9-12.A-CED.4 Rearrange formulas (literal equations) to highlight a quantity of interest, using the same reasoning as in solving equations.
    - Solve multi-variable equations (A2-B.6)
9-12.A-REI Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning.
  - 9-12.A-REI.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)
- Represent and solve equations and inequalities graphically.
  - 9-12.A-REI.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.

9-12.F Functions

9-12.F-IF Interpreting Functions
- Interpret functions that arise in applications in terms of the context.
  - 9-12.F-IF.4 For a function that models a relationship between two quantities,
    - 9-12.F-IF.4.a interpret key features of graphs and tables in terms of the quantities, and
    - 9-12.F-IF.4.b sketch graphs showing key features given a verbal description of the relationship.
  - 9-12.F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  - 9-12.F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) 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)
- Analyze functions using different representations.
  - 9-12.F-IF.7 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.F-IF.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
    - 9-12.F-IF.7.c Graph polynomial functions, identifying zeros (using technology) or algebraic methods when suitable factorizations are available, and showing end behavior.
    - 9-12.F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
  - 9-12.F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - 9-12.F-IF.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.
    - 9-12.F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.
  - 9-12.F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
9-12.F-BF Building Functions
- Build a function that models a relationship between two quantities.
  - 9-12.F-BF.1 Write a function that describes a relationship between two quantities.
    - 9-12.F-BF.1.b Combine standard function types using arithmetic operations.
- Build new functions from existing functions.
  - 9-12.F-BF.3 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 (both positive and negative); find the value of k given the graphs.
  - 9-12.F-BF.4 Find inverse functions.
    - 9-12.F-BF.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.
9-12.F-LE Linear, Quadratic, and Exponential Models
- Construct and compare linear, quadratic, and exponential models and solve problems.
  - 9-12.F-LE.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.
9-12.F-TF Trigonometric Functions
- Extend the domain of trigonometric functions using the unit circle.
  - 9-12.F-TF.1 Understand 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)
  - 9-12.F-TF.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.
- Model periodic phenomena with trigonometric functions.
  - 9-12.F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
- Prove and apply trigonometric identities.
  - 9-12.F-TF.8 Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios.
    - Trigonometric identities I (A2-BB.3)
    - Trigonometric identities II (A2-BB.4)

9-12.S Statistics and Probability

9-12.S-ID Interpreting Categorical and Quantitative Data
- Summarize, represent, and interpret data on a single count or measurement variable.
  - 9-12.S-ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
9-12.S-IC Making Inferences and Justifying Conclusions
- Understand and evaluate random processes underlying statistical experiments.
  - 9-12.S-IC.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
    - Identify biased samples (A2-FF.1)
  - 9-12.S-IC.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.
- Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
  - 9-12.S-IC.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
    - Experiment design (A2-FF.12)
  - 9-12.S-IC.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
    - Find confidence intervals for population means (A2-FF.9)
    - Find confidence intervals for population proportions (A2-FF.10)
  - 9-12.S-IC.5 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 (A2-FF.13)
  - 9-12.S-IC.6 Evaluate reports based on data.
    - Interpret confidence intervals for population means (A2-FF.11)
    - Analyze the results of an experiment using simulations (A2-FF.13)
9-12.S-MD Using Probability to Make Decisions
- Use probability to evaluate outcomes of decisions.
  - 9-12.S-MD.6 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
  - 9-12.S-MD.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
    - Choose the better bet (A2-EE.9)

9-12.N Number and Quantity

9-12.N-CN The Complex Number System
- Perform arithmetic operations with complex numbers.
  - 9-12.N-CN.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- Represent complex numbers and their operations on the complex plane.
  - 9-12.N-CN.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
  - 9-12.N-CN.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 – √3i)³ = 8 because (1 – √3i) has modulus 2 and argument 120°.
  - 9-12.N-CN.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
    - Midpoints in the complex plane (PC-S.7)
    - Distance in the complex plane (PC-S.8)
9-12.N-VM Vector and Matrix Quantities
- Represent and model with vector quantities.
  - 9-12.N-VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
  - 9-12.N-VM.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
    - Find the component form of a vector (PC-U.2)
    - Find the component form of a three-dimensional vector (PC-V.2)
  - 9-12.N-VM.3 Solve problems involving velocity and other quantities that can be represented by vectors.
- Perform operations on vectors.
  - 9-12.N-VM.4 Add and subtract vectors.
    - 9-12.N-VM.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
    - 9-12.N-VM.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
      - Find the magnitude and direction of a vector sum (PC-U.9)
    - 9-12.N-VM.4.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
  - 9-12.N-VM.5 Multiply a vector by a scalar.
    - 9-12.N-VM.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).
      - Multiply a vector by a scalar (PC-U.10)
      - Scalar multiples of three-dimensional vectors (PC-V.4)
    - 9-12.N-VM.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|·||v||. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
      - Find the magnitude of a vector scalar multiple (PC-U.11)
      - Determine the direction of a vector scalar multiple (PC-U.12)
- Perform operations on matrices and use matrices in applications.
  - 9-12.N-VM.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
  - 9-12.N-VM.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
    - Multiply a matrix by a scalar (PC-L.4)
  - 9-12.N-VM.8 Add, subtract, and multiply matrices of appropriate dimensions.
  - 9-12.N-VM.9 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
    - Properties of matrices (PC-L.8)
  - 9-12.N-VM.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
  - 9-12.N-VM.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
  - 9-12.N-VM.12 Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

9-12.A Algebra

9-12.A-REI Reasoning with Equations and Inequalities
- Solve systems of equations.
  - 9-12.A-REI.8 Represent a system of linear equations as a single matrix equation in a vector variable.
    - Solve a system of equations using augmented matrices (PC-I.8)
    - Solve a system of equations using augmented matrices: word problems (PC-I.9)
  - 9-12.A-REI.9 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

9-12.F Functions

9-12.F-IF Interpreting Functions
- Analyze functions using different representations.
  - 9-12.F-IF.7 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.F-IF.7.d Graph rational functions, identifying zeros and discontinuities (asymptotes/holes) using technology, and algebraic methods when suitable factorizations are available, and showing end behavior.
      - Rational functions: asymptotes and excluded values (PC-E.1)
9-12.F-BF Building Functions
- Build a function that models a relationship between two quantities.
  - 9-12.F-BF.1 Write a function that describes a relationship between two quantities.
    - 9-12.F-BF.1.c Compose functions.
      - Composition of functions (PC-A.11)
- Build new functions from existing functions.
  - 9-12.F-BF.4 Find inverse functions.
    - 9-12.F-BF.4.b Verify by composition that one function is the inverse of another.
      - Identify inverse functions (PC-A.12)
      - Check whether two rational functions are inverses (PC-E.3)
    - 9-12.F-BF.4.c Read values of an inverse function from a graph or a table, given that the function has an inverse.
      - Find values of inverse functions from tables (PC-A.13)
      - Find values of inverse functions from graphs (PC-A.14)
    - 9-12.F-BF.4.d Produce an invertible function from a non-invertible function by restricting the domain.
  - 9-12.F-BF.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
9-12.F-TF Trigonometric Functions
- Extend the domain of trigonometric functions using the unit circle.
  - 9-12.F-TF.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number.
    - Find trigonometric ratios using right triangles (PC-M.5)
    - Find trigonometric ratios of special angles (PC-M.7)
  - 9-12.F-TF.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
    - Symmetry and periodicity of trigonometric functions (PC-O.2)
- Model periodic phenomena with trigonometric functions.
  - 9-12.F-TF.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
    - Inverses of trigonometric functions (PC-M.9)
    - Inverses of trigonometric functions using a calculator (PC-M.10)
  - 9-12.F-TF.7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
    - Solve trigonometric equations (PC-M.11)
- Prove and apply trigonometric identities.
  - 9-12.F-TF.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

9-12.G Geometry

9-12.G-GPE Expressing Geometric Properties with Equations
- Translate between the geometric description and the equation for a conic section.
  - 9-12.G-GPE.3 Derive the equations of ellipses and hyperbolas given foci and directrices.
    - Write equations of ellipses in standard form (PC-P.9)
    - Write equations of hyperbolas in standard form (PC-P.12)
9-12.G-GMD Geometric Measurement and Dimension
- Explain volume formulas and use them to solve problems.
  - 9-12.G-GMD.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

9-12.S Statistics and Probability

9-12.S-MD Using Probability to Make Decisions
- Calculate expected values and use them to solve problems.
  - 9-12.S-MD.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
    - Write a discrete probability distribution (PC-Y.2)
    - Graph a discrete probability distribution (PC-Y.3)
  - 9-12.S-MD.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
    - Expected values of random variables (PC-Y.4)
  - 9-12.S-MD.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
    - Write the probability distribution for a game of chance (PC-Y.7)
    - Expected values for a game of chance (PC-Y.8)
  - 9-12.S-MD.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
- Use probability to evaluate outcomes of decisions.
  - 9-12.S-MD.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
    - 9-12.S-MD.5.a Find the expected payoff for a game of chance.
      - Expected values for a game of chance (PC-Y.8)
    - 9-12.S-MD.5.b Evaluate and compare strategies on the basis of expected values.
      - Choose the better bet (PC-Y.9)

9-12.N Number and Quantity

9-12.N-Q Quantities
- Reason quantitatively and use units to solve problems.
  - 9-12.N-Q.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.
  - 9-12.N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.
  - 9-12.N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

9-12.A Algebra

9-12.A-SSE Seeing Structure in Expressions
- Interpret the structure of expressions.
  - 9-12.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
    - 9-12.A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
      - Sort factors of variable expressions (A1-I.2)
    - 9-12.A-SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.
9-12.A-CED Creating Equations and Inequalities
- Create equations and inequalities that describe numbers or relationships.
  - 9-12.A-CED.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.
  - 9-12.A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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9-12.N Number and Quantity

9-12.N-RN The Real Number System

Extend the properties of exponents to rational exponents.

9-12.N-RN.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.

9-12.N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Use properties of rational and irrational numbers.

9-12.N-RN.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

9-12.N-Q Quantities

Reason quantitatively and use units to solve problems.

9-12.N-Q.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.

9-12.N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.

9-12.N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Checkpoint opportunity

9-12.G Geometry

9-12.G-CO Congruence

Experiment with transformations in the plane.

9-12.G-CO.1 Demonstrates understanding of key geometrical definitions, including angle, circle, perpendicular line, parallel line, line segment, and transformations in Euclidian geometry. Understand undefined notions of point, line, distance along a line, and distance around a circular arc.

9-12.G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

9-12.G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

9-12.G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Checkpoint opportunity

Understand congruence in terms of rigid motions.

9-12.G-CO.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.

9-12.G-CO.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.

9-12.G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS, and HL) follow from the definition of congruence in terms of rigid motions.

Checkpoint opportunity

Prove geometric theorems.

9-12.G-CO.9 Using methods of proof including direct, indirect, and counter examples to prove theorems about lines and angles.

9-12.G-CO.10 Using methods of proof including direct, indirect, and counter examples to prove theorems about triangles.

9-12.G-CO.11 Using methods of proof including direct, indirect, and counter examples to prove theorems about parallelograms.

Checkpoint opportunity

Make geometric constructions.

9-12.G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Checkpoint opportunity

9-12.G-SRT Similarity, Right Triangles, and Trigonometry

Understand similarity in terms of similarity transformations.

9-12.G-SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:

9-12.G-SRT.1.a 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.G-SRT.1.b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

9-12.G-SRT.2 Given two figures, use the definition of similarity in terms of transformations to explain whether or not they are similar.

9-12.G-SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Checkpoint opportunity

Prove theorems involving similarity.

9-12.G-SRT.4 Prove theorems about triangles.

9-12.G-SRT.5 Apply congruence and similarity properties and prove relationships involving triangles and other geometric figures.

Checkpoint opportunity

Define trigonometric ratios and solve problems involving right triangles.

9-12.G-SRT.6 Understand 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.G-SRT.7 Explain and use the relationship between the sine and cosine of complementary angles.

9-12.G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Checkpoint opportunity

Apply trigonometry to general triangles.

9-12.G-SRT.9 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

9-12.G-SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems.

9-12.G-SRT.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).

Checkpoint opportunity

9-12.G-C Circles

Understand and apply theorems about circles.

9-12.G-C.1 Prove that all circles are similar.

9-12.G-C.2 Identify and describe relationships among inscribed angles, radii, and chords.

9-12.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

9-12.G-C.4 Construct a tangent line from a point outside a given circle to the circle.

Checkpoint opportunity

Find arc lengths and areas of sectors of circles.

Checkpoint opportunity

9-12.G-GPE Expressing Geometric Properties with Equations

Translate between the geometric description and the equation for a conic section.

9-12.G-GPE.1 Determine or 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.

9-12.G-GPE.2 Determine or derive the equation of a parabola given a focus and directrix.

Checkpoint opportunity

Use coordinates to prove simple geometric theorems algebraically.

9-12.G-GPE.4 Perform simple coordinate proofs. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

9-12.G-GPE.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).

9-12.G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

9-12.G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Checkpoint opportunity

9-12.A-SSE.3.a Factor a quadratic expression to reveal the zeros of the function it defines. For example, x² + 4x +3 = (x +3)(x +1).

9-12.A-REI.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.