Arkansas

RC Radicals and Complex Numbers

• Radical Expressions and Rational Exponents: Students apply properties to radical expressions and rational exponents.

  • A2.RC.1 Explain how extending the properties of integer exponents to rational exponents provides an alternative notation for radicals.

    • Evaluate rational exponents (A2-T.1)
    • Convert between rational exponents and radicals (A2-T.2)

  • A2.RC.2 Apply the properties of exponents to translate between radical and exponential forms of expressions.

    • Multiplication with rational exponents (A2-T.3)
    • Division with rational exponents (A2-T.4)
    • Power rule (A2-T.5)
    • Simplify expressions involving rational exponents I (A2-T.6)
    • Simplify expressions involving rational exponents II (A2-T.7)

  • A2.RC.3 Simplify and perform operations with radical expressions with and without variables; rationalizing denominators should include conjugates.

    • Roots of integers (A2-R.1)
    • Roots of rational numbers (A2-R.2)
    • Find roots using a calculator (A2-R.3)
    • Simplify radical expressions with variables I (A2-R.4)
    • Simplify radical expressions with variables II (A2-R.5)
    • Nth roots (A2-R.6)
    • Multiply radical expressions (A2-R.7)
    • Simplify radical expressions involving fractions (A2-R.8)
    • Add and subtract radical expressions (A2-R.9)
    • Simplify radical expressions using the distributive property (A2-R.10)
    • Simplify radical expressions using conjugates (A2-R.11)

• Complex Numbers: Students apply properties to complex numbers.

  • A2.RC.4 Know there is a complex number i and describe contexts from which complex numbers appear.

    • Introduction to complex numbers (A2-K.1)

  • A2.RC.5 Perform the operations of addition, subtraction, multiplication, and conjugation of complex numbers.

    • Add and subtract complex numbers (A2-K.2)
    • Complex conjugates (A2-K.3)
    • Multiply complex numbers (A2-K.4)
    • Divide complex numbers (A2-K.5)
    • Add, subtract, multiply, and divide complex numbers (A2-K.6)
    • Powers of i (A2-K.8)

  • A2.RC.6 Use polynomial identities with complex numbers.

    • Solve polynomial equations (A2-Q.1)
    • Complex conjugate theorem (A2-Q.5)
    • Conjugate root theorems (A2-Q.6)

FN Functions

• Compositions: Students compose and compare functions.

  • A2.FN.1 Combine functions by addition, subtraction, multiplication, division, and composition to model the relationship between two quantities in mathematical and real-world contexts.

    • Add and subtract polynomials (A2-L.2)
    • Multiply polynomials (A2-L.3)
    • Add and subtract functions (A2-U.1)
    • Multiply functions (A2-U.2)
    • Divide functions (A2-U.3)
    • Composition of linear functions: find a value (A2-U.4)
    • Composition of linear functions: find an equation (A2-U.5)
    • Composition of linear and quadratic functions: find a value (A2-U.6)
    • Composition of linear and quadratic functions: find an equation (A2-U.7)
    • Multiply and divide rational expressions (A2-X.5)
    • Add and subtract rational expressions (A2-X.6)

• Transformations: Students perform transformations in the coordinate plane.

  • A2.FN.2 Decide if a function is even or odd from a graph or an algebraic expression.

    • Even and odd functions (A2-Q.13)

  • A2.FN.3 Explain how restricting the domain of a function allows the creation of its inverse.

    • Find inverse functions and relations (A2-V.4)

  • A2.FN.4 Write and graph the inverse of a given function; understand that the graph of an inverse function is a reflection of the function over the line y = x.

    • Identify inverse functions (A2-V.1)
    • Find values of inverse functions from tables (A2-V.2)
    • Find values of inverse functions from graphs (A2-V.3)
    • Find inverse functions and relations (A2-V.4)

LFE Linear Functions and Equations

• Arithmetic Sequences: Students use arithmetic sequences to model problems.

  • A2.LFE.1 Write and use arithmetic sequences recursively and explicitly to model situations; translate between the two forms when given a graph, a description of the relationship, or two input-output pairs.

    • Find terms of an arithmetic sequence (A2-MM.1)
    • Write a formula for an arithmetic sequence (A2-MM.6)
    • Convert between explicit and recursive formulas: arithmetic sequences (A2)

• Matrices: Students perform operations and matrices.

  • A2.LFE.2 Multiply a matrix by a scalar.

    • Multiply a matrix by a scalar (A2-TT.4)

  • A2.LFE.3 Add and subtract matrices.

    • Add and subtract matrices (A2-TT.3)
    • Add and subtract scalar multiples of matrices (A2-TT.5)

• Systems of Equations: Students solve systems of equations.

  • A2.LFE.4 Solve systems of linear equations in three variables using matrices; use Gaussian elimination or technology.

    • Solve a system of equations using augmented matrices (A2-TT.18)
    • Solve a system of equations using augmented matrices: word problems (A2-TT.19)

QFE Quadratic Functions, Equations and Inequalities

• Create & Solve: Students create and solve problems that model quadratic relationships.

  • A2.QFE.1 Calculate and interpret the average rate of change of a quadratic function represented in a table, graph, or as an equation in the context of mathematical and real-world problems.

  • A2.QFE.2 Solve quadratic equations with complex number solutions.

    • Solve a quadratic equation using square roots (A2-O.1)
    • Solve a quadratic equation by completing the square (A2-O.5)
    • Solve a quadratic equation using the quadratic formula (A2-O.6)
    • Using the discriminant (A2-O.8)

  • A2.QFE.3 Represent and solve real-world problems using quadratic inequalities.

    • Graph solutions to quadratic inequalities (A2-O.9)
    • Solve quadratic inequalities (A2-O.10)

• Graph & Key Features: Students graph and interpret key features of equations that model quadratic relationships.

  • A2.QFE.4 Use the discriminant to determine the number and type of solutions of a quadratic equation.

    • Using the discriminant (A2-O.8)

  • A2.QFE.5 Sketch the graph of a quadratic function given a verbal description and show key features.

    • Characteristics of quadratic functions: graphs (A2-N.1)
    • Graph a quadratic function (A2-N.5)
    • Match quadratic functions and graphs (A2-N.7)
    • Domain and range of quadratic functions: graphs (A2-N.8)

• Systems of Equations: Students solve systems of equations.

  • A2.QFE.6 Solve a system of equations consisting of a linear equation and a nonlinear equation in two variables by choosing substitution or graphically (with or without technology) as appropriate for the system of equations.

    • Solve a system of linear and quadratic equations by graphing: parabolas (A2-P.1)
    • Solve a system of linear and quadratic equations: parabolas (A2-P.2)
    • Solve a system of linear and quadratic equations: circles (A2-P.3)

ELF Exponential & Logarithmic Functions & Equations

• Create & Solve: Students create and solve problems that model exponential and logarithmic relationships.

  • A2.ELF.1 Use the properties of exponents to find equivalent expressions and to solve equations, including those involving rational exponents.

    • Evaluate rational exponents (A2-T.1)
    • Multiplication with rational exponents (A2-T.3)
    • Division with rational exponents (A2-T.4)
    • Power rule (A2-T.5)
    • Simplify expressions involving rational exponents I (A2-T.6)
    • Simplify expressions involving rational exponents II (A2-T.7)
    • Solve equations with rational exponents (A2-T.8)
    • Solve exponential equations by rewriting the base (A2-CC.1)

  • A2.ELF.2 Write and solve equations from real-world problems that can be represented as a logarithmic or exponential function in one variable.

    • Exponential growth and decay: word problems (A2-Z.9)
    • Compound interest: word problems (A2-CC.4)
    • Continuously compounded interest: word problems (A2-CC.5)

  • A2.ELF.3 Write exponential equations that model the relationship between two quantities when given a graph, a written description, or a table of values within a mathematical or real-world context.

    • Write exponential functions: word problems (A2-Z.3)

  • A2.ELF.4 Write and use geometric sequences recursively and explicitly to model situations; translate between the two forms when given a graph, a description of the relationship, or two input-output pairs.

    • Find terms of a geometric sequence (A2-MM.2)
    • Write a formula for a geometric sequence (A2-MM.7)
    • Convert between explicit and recursive formulas for geometric sequences (A2)

• Logarithms: Students define and use logarithms.

  • A2.ELF.5 Translate between logarithmic and exponential forms of an equation.

    • Convert between exponential and logarithmic form: rational bases (A2-AA.1)
    • Convert between natural exponential and logarithmic form (A2-AA.2)
    • Convert between exponential and logarithmic form: all bases (A2-AA.3)

  • A2.ELF.6 Use properties of logarithms to simplify and evaluate logarithmic expressions, with or without technology.

    • Evaluate logarithms (A2-AA.4)
    • Evaluate natural logarithms (A2-AA.5)
    • Change of base formula (A2-AA.6)
    • Evaluate logarithms using a calculator (A2-AA.7)
    • Identify properties of logarithms (A2-AA.8)
    • Product property of logarithms (A2-AA.9)
    • Quotient property of logarithms (A2-AA.10)
    • Power property of logarithms (A2-AA.11)
    • Properties of logarithms: mixed review (A2-AA.12)
    • Evaluate logarithms using properties (A2-AA.13)

  • A2.ELF.7 Use the inverse relationship between exponents and logarithms to solve problems.

    • Convert between exponential and logarithmic form: all bases (A2-AA.3)
    • Solve exponential equations using common logarithms (A2-CC.2)
    • Solve exponential equations using natural logarithms (A2-CC.3)
    • Solve logarithmic equations I (A2-CC.6)
    • Solve logarithmic equations II (A2-CC.7)

• Graph & Key Features: Students graph and interpret key features of exponential and logarithmic models.

  • A2.ELF.8 Determine the domain and range of logarithmic functions in mathematical problems.

    • Domain and range of logarithmic functions (A2-BB.1)

  • A2.ELF.9 Determine reasonable domain and range values of logarithmic functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

  • A2.ELF.10 Sketch the graph of an exponential function given a verbal description and show key features.

    • Graph exponential functions (A2-Z.4)
    • Match exponential functions and graphs (A2-Z.5)

  • A2.ELF.11 Calculate and interpret the average rate of change of an exponential function represented in a table, graph, or as an equation in the context of mathematical and real-world problems.

    • Exponential functions over unit intervals (A2-Z.6)

  • A2.ELF.12 Graph exponential and logarithmic functions with and without context, identifying key features, and determining constraints in a given context.

    • Graph exponential functions (A2-Z.4)
    • Match exponential functions and graphs (A2-Z.5)
    • Graph logarithmic functions (A2-BB.2)

  • A2.ELF.13 Graph and generalize the effect of transformations on exponential and logarithmic functions.

  • A2.ELF.14 Given the graphs of exponential and logarithmic functions, explain the effects of the transformation from the parent function.

PRF Polynomial, Rational, & Other Functions & Equations

• Create & Solve: Students create and solve rational and polynomial equations.

  • A2.PRF.1 Write and solve equations from real-world problems that can be represented as a rational or square root function in one variable.

    • Pythagorean Theorem and its converse (A2-II.1)

  • A2.PRF.2 Solve non-linear formulas for a specified variable.

    • Rearrange multi-variable equations (A2-B.4)

  • A2.PRF.3 Find zeros of polynomial functions with a degree of 3 or higher when suitable factorizations are available in a real-world and mathematical context.

    • Solve polynomial equations (A2-Q.1)
    • Find the roots of factored polynomials (A2-Q.2)

  • A2.PRF.4 Use zeros and an understanding of multiplicity to sketch a graph of a polynomial function with a degree of 3 or higher.

    • Match polynomials and graphs using zeros (A2-Q.10)
    • Match polynomials and graphs (A2-Q.11)

  • A2.PRF.5 Apply the Fundamental Theorem of Algebra to determine the number and potential types of roots of polynomial functions based on the degree of the polynomial.

    • Fundamental Theorem of Algebra (A2-Q.8)

  • A2.PRF.6 Solve rational and radical equations containing one variable specifying extraneous solutions.

    • Solve radical equations (A2-S.4)
    • Solve rational equations (A2-X.8)

• Graph & Key Features: Students graph and identify key features of functions.

  • A2.PRF.7 Determine the domain and range of polynomial and rational functions in mathematical problems.

    • Domain and range of quadratic functions: graphs (A2-N.8)
    • Domain and range of quadratic functions: equations (A2-N.9)
    • Domain and range of polynomials (A2-Q.12)

  • A2.PRF.8 Determine reasonable domain and range values of polynomial and rational functions representing real-world situations, both continuous and discrete; interpret the solution as reasonable or unreasonable in context.

  • A2.PRF.9 Interpret the key features of polynomial functions that model a relationship between two quantities in a given context; translate between different representations of the function, especially graphs, tables, and equations.

    • Characteristics of quadratic functions: graphs (A2-N.1)
    • Characteristics of quadratic functions: equations (A2-N.2)
    • Complete a function table: quadratic functions (A2-N.3)
    • Interpret parts of quadratic expressions: word problems (A2-N.14)
    • Find the roots of factored polynomials (A2-Q.2)
    • Match polynomials and graphs using end behavior (A2-Q.9)
    • Match polynomials and graphs using zeros (A2-Q.10)

  • A2.PRF.10 Sketch the graph of a polynomial function given a verbal description and show key features.

    • Graph a quadratic function (A2-N.5)
    • Match quadratic functions and graphs (A2-N.7)
    • Match polynomials and graphs (A2-Q.11)

  • A2.PRF.11 Calculate and interpret the average rate of change of polynomial functions represented in a table, graph, or as an equation in context of mathematical and real-world problems.

    • Average rate of change: polynomials (A2)

  • A2.PRF.12 Graph functions with and without context, identifying key features and determining constraints in a given context.

    • Graph an absolute value function (A2-H.3)
    • Graph piecewise-defined functions (A2-I.2)
    • Graph a quadratic function (A2-N.5)
    • Match polynomials and graphs using end behavior (A2-Q.9)
    • Match polynomials and graphs using zeros (A2-Q.10)
    • Match polynomials and graphs (A2-Q.11)
    • Graph square root functions (A2-S.2)
    • Rational functions: asymptotes and excluded values (A2-X.1)

  • A2.PRF.13 Graph and generalize the effect of transformations on square root, cubic, and rational functions.

  • A2.PRF.14 Given a graph, explain the effects of the transformation from the parent function.

• Factor Polynomials: Students factor polynomials.

  • A2.PRF.15 Apply the Remainder Theorem to factor and create equivalent forms of polynomial functions.

    • Divide polynomials using long division (A2-L.4)
    • Divide polynomials using synthetic division (A2-L.5)
    • Evaluate polynomials using synthetic division (A2-L.7)
    • Write a quadratic function from its zeros (A2-N.10)
    • Find the roots of factored polynomials (A2-Q.2)
    • Write a polynomial from its roots (A2-Q.3)

  • A2.PRF.16 Verify polynomial identities and use them to describe numerical relationships.

    • Factor sums and differences of cubes (A2-M.8)

• Rational Expressions: Students perform operations on rational expressions.

  • A2.PRF.17 Apply understanding of rational number operations to add, subtract, multiply, and divide by nonzero rational expressions.

    • Multiply and divide rational expressions (A2-X.5)
    • Add and subtract rational expressions (A2-X.6)
    • Simplify complex fractions (A2-X.7)

  • A2.PRF.18 Rewrite simple rational expressions in different forms.

    • Simplify rational expressions (A2-X.4)

  • A2.PRF.19 Divide polynomial expressions using inspection, long division, and synthetic division, with and without a remainder.

    • Divide polynomials using long division (A2-L.4)
    • Divide polynomials using synthetic division (A2-L.5)

• Systems of Equations: Students solve systems of equations.

  • A2.PRF.20 Explain why a solution to the equation f(x) = g(x) is the x-coordinate where the y-coordinate of f(x) and g(x) are the same using graphs, tables, or approximations.

    • Find solutions using a table (A2-A.7)
    • Approximate solutions using a table (A2-A.8)
    • 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)
    • Solve a system of linear and quadratic equations by graphing: parabolas (A2-P.1)

SP Statistics & Probability

• Statistical Experiments & Studies: Students evaluate processes for statistical experiments, make inferences, and justify conclusions from statistical studies.

  • A2.SP.1 Use data from a random sample to make inferences about a population.

    • Identify biased samples (A2-RR.1)
    • Interpret confidence intervals for population means (A2-RR.7)

  • A2.SP.2 Compare theoretical and empirical probabilities using simulations.

    • Theoretical probability (A2)
    • Experimental probability (A2)

  • A2.SP.3 Distinguish between sample surveys, experiments, and observational studies and explain the purpose of randomization in statistical studies.

    • Experiment design (A2-RR.8)

  • A2.SP.4 Read and explain, in context, the validity of data from outside reports by: identifying the variables as quantitative or categorical, describing how the data was collected, indicating any potential biases or flaws, identifying inferences the author of the report made from sample data.

    • Identify biased samples (A2-RR.1)