Idaho

N Number and Quantity

• N-CN The Complex Number System

  • Use complex numbers in polynomial identities and equations.

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

      • Solve polynomial equations (A2-L.8)
      • Complex conjugate theorem (PC-D.7)
      • Conjugate root theorems (PC-D.8)

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

      • Fundamental Theorem of Algebra (A2-L.17)

A-SSE Algebra

• A-SSE Seeing Structure in Expressions

  • Interpret the structure of expressions

    • A-SSE.1 Interpret expressions that represent a quantity in terms of its context.

      • A-SSE.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)

      • 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.)

    • A-SSE.2 Use the structure of an expression to identify ways to rewrite it.

      • Factor out a monomial (A2-J.1)
      • Factor using a quadratic pattern (A2-J.4)
      • Pascal's triangle and the Binomial Theorem (A2-L.19)
      • Binomial Theorem I (A2-L.20)
      • Binomial Theorem II (A2-L.21)
      • Simplify radical expressions with variables I (A2-M.4)
      • Simplify radical expressions with variables II (A2-M.5)
      • Simplify radical expressions using conjugates (A2-M.11)
      • Simplify rational expressions (A2-O.4)

  • Write expressions in equivalent forms to solve problems

    • 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)

• A-APR Arithmetic with Polynomials and Rational Expressions

  • Perform arithmetic operations on polynomials

    • A-APR.1 Understand 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)

  • Understand the relationship between zeros and factors of polynomials

    • 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).

      • Write a quadratic function from its zeros (A2-K.16)
      • Evaluate polynomials using synthetic division (A2-L.7)
      • Find the roots of factored polynomials (A2-L.9)
      • Write a polynomial from its roots (A2-L.10)

    • 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.

      • Solve a quadratic equation using the zero product property (A2-K.7)
      • Match quadratic functions and graphs (A2-K.14)
      • Find the roots of factored polynomials (A2-L.9)
      • Match polynomials and graphs (A2-L.15)

  • Use polynomial identities to solve problems

    • A-APR.4 Prove polynomial identities and use them to describe numerical relationships.

      • Factor sums and differences of cubes (A2-J.6)
      • Add and subtract polynomials (A2-L.2)
      • Multiply polynomials (A2-L.3)
      • Solve equations with sums and differences of cubes (PC-D.14)

    • 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.

      • Pascal's triangle (A2-L.18)
      • Pascal's triangle and the Binomial Theorem (A2-L.19)
      • Binomial Theorem I (A2-L.20)
      • Binomial Theorem II (A2-L.21)

  • Rewrite rational expressions

    • 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 by monomials (A1-HH.4)
      • Divide polynomials using long division (A2-L.4)
      • Simplify rational expressions (A2-O.4)

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

      • Multiply and divide rational expressions (A2-O.5)
      • Add and subtract rational expressions (A2-O.6)

• A-CED Creating Equations

  • Create equations that describe numbers or relationships

    • 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.

      • Write exponential functions: word problems (A2-T.)
      • Solve quadratic equations: word problems (PC-C.)

    • A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

      • Graph quadratic functions in standard form (A1-CC.14)
      • Write the equation of a linear function (A2-D.10)
      • Graph a quadratic function (A2-K.5)
      • Write equations of parabolas in vertex form using properties (A2-U.6)
      • Graph sine functions (A2-AA.4)
      • Write equations of cosine functions using properties (A2-AA.8)
      • Graph cosine functions (A2-AA.9)
      • Graph sine and cosine functions (A2-AA.14)

    • A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

      • Write a linear inequality: word problems (A2-C.3)
      • Solve a system of equations by graphing: word problems (A2-E.3)
      • Solve a system of equations using substitution: word problems (A2-E.7)
      • Solve a system of equations using elimination: word problems (A2-E.9)
      • Solve a system of equations using any method: word problems (A2-E.11)

    • A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

      • Solve multi-variable equations (A2-B.6)

• A-REI Reasoning with Equations and Inequalities

  • Understand solving equations as a process of reasoning and explain the reasoning

    • 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

    • 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.

      • Find solutions using a table (A2-D.6)
      • Approximate solutions using a table (A2-D.7)
      • Solve a system of equations by graphing (A2-E.2)
      • Solve a system of equations by graphing: word problems (A2-E.3)
      • Find the number of solutions to a system of equations (A2-E.4)

F-IF Functions

• F-IF Interpreting Functions

  • Interpret functions that arise in applications in terms of the context

    • F-IF.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 given a verbal description of the relationship.

      • Identify graphs: word problems (A2-D.)
      • Find the slope of a linear function (A2-D.8)
      • Graph a linear function (A2-D.9)
      • Characteristics of quadratic functions: graphs (A2-K.1)
      • Characteristics of quadratic functions: equations (A2-K.2)
      • Graph a quadratic function (A2-K.5)
      • Match quadratic functions and graphs (A2-K.14)
      • Match polynomials and graphs (A2-L.15)
      • Rational functions: asymptotes and excluded values (A2-O.1)
      • Find the constant of variation (A2-R.5)
      • Match exponential functions and graphs (A2-T.4)

    • F-IF.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)

    • 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.

      • Average rate of change (A2-D.12)

  • Analyze functions using different representations

    • 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.

      • F-IF.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)

      • F-IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

        • Graph quadratic functions in standard form (A1-CC.14)
        • Graph a quadratic function (A2-K.5)
        • Match polynomials and graphs (A2-L.15)

      • F-IF.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

        • Graph exponential functions (A2-T.3)
        • Match exponential functions and graphs (A2-T.4)
        • Graph logarithmic functions (A2-T.)
        • Find properties of sine functions (A2-AA.1)
        • Graph sine functions (A2-AA.4)
        • Graph translations of sine functions (A2-AA.5)
        • Find properties of cosine functions (A2-AA.6)
        • Graph cosine functions (A2-AA.9)
        • Graph translations of cosine functions (A2-AA.10)
        • Graph sine and cosine functions (A2-AA.14)
        • Graph translations of sine and cosine functions (A2-AA.15)

    • F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

      • 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.

        • Characteristics of quadratic functions: graphs (A2-K.1)
        • Solve a quadratic equation by factoring (A2-K.8)
        • Solve a quadratic equation by completing the square (A2-K.10)

      • F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.

        • Evaluate exponential functions (A2-T.2)
        • Match exponential functions and graphs (A2-T.4)
        • Solve exponential equations by rewriting the base (A2-T.5)

    • F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

• F-BF Building Functions

  • Build a function that models a relationship between two quantities

    • F-BF.1 Write a function that describes a relationship between two quantities.

      • F-BF.1.b Combine standard function types using arithmetic operations.

        • Add and subtract polynomials (A2-L.2)
        • Multiply polynomials (A2-L.3)
        • Multiply and divide rational expressions (A2-O.5)
        • Add and subtract rational expressions (A2-O.6)
        • Add, subtract, multiply, and divide functions (PC-A.10)

  • Build new functions from existing functions

    • 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. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

      • Function transformation rules (A2-Q.1)
      • Translations of functions (A2-Q.2)
      • Reflections of functions (A2-Q.3)
      • Dilations of functions (A2-Q.4)
      • Transformations of functions (A2-Q.5)
      • Describe function transformations (A2-Q.6)

    • F-BF.4 Find inverse functions.

      • 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 inverse functions and relations (A2-P.11)
        • Solve exponential equations using common logarithms (A2-T.6)
        • Solve exponential equations using natural logarithms (A2-T.7)
        • Solve logarithmic equations I (A2-T.8)
        • Solve logarithmic equations II (A2-T.9)

• F-LE Linear, Quadratic, and Exponential Models

  • Construct and compare linear, quadratic, and exponential models and solve problems

    • F-LE.4 For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

      • Convert between exponential and logarithmic form: rational bases (A2-S.1)
      • Convert between natural exponential and logarithmic form (A2-S.2)
      • Convert between exponential and logarithmic form: all bases (A2-S.3)
      • Evaluate logarithms (A2-S.4)
      • Solve exponential equations using common logarithms (A2-T.6)
      • Solve exponential equations using natural logarithms (A2-T.7)

• F-TF Trigonometric Functions

  • Extend the domain of trigonometric functions using the unit circle

    • 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)

    • 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.

      • Coterminal angles (A2-Y.5)
      • Reference angles (A2-Y.6)
      • Find trigonometric ratios using the unit circle (A2-Z.6)

  • Model periodic phenomena with trigonometric functions

    • F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

      • Find properties of sine functions (A2-AA.1)
      • Write equations of sine functions from graphs (A2-AA.2)
      • Write equations of sine functions using properties (A2-AA.3)
      • Find properties of cosine functions (A2-AA.6)
      • Write equations of cosine functions from graphs (A2-AA.7)
      • Write equations of cosine functions using properties (A2-AA.8)

G Geometry

• G-SRT Similarity, Right Triangles, and Trigonometry

  • Apply trigonometry to general triangles

    • G-SRT.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 (A2-Z.22)

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

      • Law of Sines (A2-Z.19)
      • Law of Cosines (A2-Z.20)
      • Area of a triangle: Law of Sines (A2-Z.23)

    • 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 (A2-Z.21)

• G-GMD Geometric Measurement and Dimension

  • Visualize relationships between two-dimensional and three-dimensional objects

    • 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)

• G-MG Modeling with Geometry

  • Apply geometric concepts in modeling situations

    • 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)

    • 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)

    • 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).

S Statistics and Probability

• S-ID Interpreting Categorical and Quantitative Data

  • Summarize, represent, and interpret data on a single count or measurement variable

    • 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.

      • Find probabilities using the normal distribution I (A2-EE.11)
      • Find z-values (A2-EE.13)
      • Distributions of sample means (A2-EE.15)

• S-IC Making Inferences and Justifying Conclusions

  • Understand and evaluate random processes underlying statistical experiments

    • 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)

    • 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

    • 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)

    • 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)

    • 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)

    • 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)

• S-MD Using Probability to Make Decisions

  • Use probability to evaluate outcomes of decisions

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

    • 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)