Arkansas

F Functions

• F.1.MAA The student will graphically, numerically, and algebraically evaluate concepts of different types of functions; include recursively defined functions, series, and sequences; and apply them to programming applications

  • F.1.MAA.1 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. [e.g., the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1]

    • Find terms of an arithmetic sequence (A2-MM.1)
    • Find terms of a geometric sequence (A2-MM.2)
    • Evaluate explicit formulas for sequences (A2-MM.3)
    • Evaluate recursive formulas for sequences (A2-MM.4)
    • Classify formulas and sequences (A2-MM.5)
    • Write a formula for an arithmetic sequence (A2-MM.6)
    • Write a formula for a geometric sequence (A2-MM.7)
    • Write a formula for a recursive sequence (A2-MM.8)
    • Sequences: mixed review (A2-MM.12)
    • Find terms of a recursive sequence (PC-AA.2)

  • F.1.MAA.2 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases: graph exponential and logarithmic functions showing intercepts, end behavior, and trigonometric functions, showing period, midline, and amplitude; graph linear and quadratic functions and show intercepts, maxima, and minima; graph rational functions identifying zeros and asymptotes when suitable factorizations are available and showing end behavior.

    • Graph a linear function (PC)
    • Match exponential functions and graphs (A2-Z.5)
    • Characteristics of quadratic functions (PC-D.2)
    • Graph a quadratic function (PC-D.3)
    • Match quadratic functions and graphs (PC-D.4)
    • Rational functions: asymptotes and excluded values (PC-G.1)
    • Find properties of sine functions (PC-S.1)
    • Graph sine functions (PC-S.4)
    • Find properties of cosine functions (PC-S.6)
    • Graph cosine functions (PC-S.9)
    • Graph sine and cosine functions (PC-S.14)

  • F.1.MAA.3 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

    • Solve a quadratic equation by factoring (PC-D.6)
    • Solve a quadratic equation by completing the square (PC-D.7)
    • Solve exponential equations by rewriting the base (PC-K.1)

  • F.1.MAA.4 Compare properties of two functions each represented in a different way: algebraically, graphically, numerically in tables, or by verbal descriptions (e.g., given a graph of one quadratic function and an algebraic expression for another, determine which has the larger maximum).

    • Compare linear functions: tables, graphs, and equations (A1-N.12)

  • F.1.MAA.5 Write a function that describes a relationship between two quantities: compose functions [e.g., if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time]; combine standard function types using arithmetic operations (e.g., build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential and relate these functions to the model); determine an explicit expression, a recursive process, or steps for calculation from a context.

    • Write a formula for an arithmetic sequence (A2-MM.6)
    • Write a formula for a geometric sequence (A2-MM.7)
    • Write a formula for a recursive sequence (A2-MM.8)
    • Add, subtract, multiply, and divide functions (PC-A.7)
    • Composition of functions (PC-A.8)
    • Find a recursive formula (PC-AA.4)

  • F.1.MAA.6 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

    • Write a formula for an arithmetic sequence (A2-MM.6)
    • Write a formula for a geometric sequence (A2-MM.7)
    • Write a formula for a recursive sequence (A2-MM.8)
    • Find a recursive formula (PC-AA.4)
    • Find recursive and explicit formulas (PC-AA.5)
    • Convert a recursive formula to an explicit formula (PC-AA.6)
    • Convert an explicit formula to a recursive formula (PC-AA.7)
    • Convert between explicit and recursive formulas (PC-AA.8)

  • F.1.MAA.7 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-R.9)

  • F.1.MAA.8 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-R.11)

  • F.1.MAA.9 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-K.1)

  • F.1.MAA.10 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

    • Multiply complex numbers (A2-K.4)
    • Add and subtract complex numbers (PC-V.1)

  • F.1.MAA.11 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

    • Complex conjugates (PC-V.2)
    • Multiply and divide complex numbers (PC-V.5)
    • Add, subtract, multiply, and divide complex numbers (PC-V.6)
    • Absolute values of complex numbers (PC-V.7)
    • Powers of i (PC-V.8)

EF Equations and Formulas

• EF.2.MAA The student will manipulate formulas and equations and apply them to programming applications

  • EF.2.MAA.1 Represent constraints by equations or inequalities and by systems of equations and/or inequalities (two and three variable systems); interpret solutions as viable or nonviable options in a modeling context (e.g., represent inequalities describing nutritional and cost constraints on combinations of different foods).

    • Solve a system of equations by graphing (PC-N.1)
    • Solve a system of equations by graphing: word problems (PC-N.2)
    • Solve a system of equations using substitution (PC-N.4)
    • Solve a system of equations using substitution: word problems (PC-N.5)
    • Solve a system of equations using elimination (PC-N.6)
    • Solve a system of equations using elimination: word problems (PC-N.7)
    • Solve a system of equations in three variables using substitution (PC-N.10)
    • Solve a system of equations in three variables using elimination (PC-N.11)
    • Determine the number of solutions to a system of equations in three variables (PC-N.12)
    • Solve systems of linear inequalities by graphing (PC-O.1)
    • Solve systems of linear and absolute value inequalities by graphing (PC-O.2)
    • Find the vertices of a solution set (PC-O.3)
    • Linear programming (PC-O.4)

  • EF.2.MAA.2 Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations (e.g., rearrange Ohm's law V = IR to highlight resistance R).

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

  • EF.2.MAA.3 Give an informal argument for the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone; use dissection arguments, Cavalieri's principle, and informal limit arguments.

  • EF.2.MAA.4 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

  • EF.2.MAA.5 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

    • Volume of prisms and cylinders (G-U.7)
    • Volume of pyramids and cones (G-U.8)
    • Volume of spheres (G-U.9)

SEM Systems of Equations and Matrices

• SEM.3.MAA The student will create, manipulate, and solve systems of equations and matrices and apply them to programming arrays

  • SEM.3.MAA.1 Represent a system of linear equations as a single matrix equation in a vector variable.

  • SEM.3.MAA.2 Find the inverse of a matrix if it exists and use it to solve systems of linear equations; use technology for matrices of dimension 3 × 3 or greater.

    • Inverse of a 2 x 2 matrix (PC-Q.12)
    • Inverse of a 3 x 3 matrix (PC-Q.13)
    • Solve matrix equations using inverses (PC-Q.15)

  • SEM.3.MAA.3 Use matrices to represent and manipulate data (e.g., to represent payoffs or incidence relationships in a network).

  • SEM.3.MAA.4 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-Q.4)

  • SEM.3.MAA.5 Add, subtract, and multiply matrices of appropriate dimensions.

    • Add and subtract matrices (PC-Q.3)
    • Add and subtract scalar multiples of matrices (PC-Q.5)
    • Multiply two matrices (PC-Q.6)
    • Simplify matrix expressions (PC-Q.7)
    • Solve matrix equations (PC-Q.9)

  • SEM.3.MAA.6 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation but still satisfies the associative and distributive properties.

    • Matrix operation rules (PC-Q.2)
    • Properties of matrices (PC-Q.8)

  • SEM.3.MAA.7 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.

    • Determinant of a matrix (PC-Q.10)
    • Is a matrix invertible? (PC-Q.11)
    • Inverse of a 2 x 2 matrix (PC-Q.12)
    • Inverse of a 3 x 3 matrix (PC-Q.13)
    • Identify inverse matrices (PC-Q.14)
    • Solve matrix equations using inverses (PC-Q.15)

  • SEM.3.MAA.8 Work with 2 × 2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.

    • Identify transformation matrices (PC-Q.16)
    • Transformation matrices: write the vertex matrix (PC-Q.17)
    • Transformation matrices: graph the image (PC-Q.18)

PS Problem Solving

• PS.4.MAA The student will develop and apply logical reasoning skills to solve real-world problems through the development of mathematical models

  • PS.4.MAA.1 Analyze and interpret graphs, charts, and tables in the design and implementation of a computer program.

  • PS.4.MAA.2 Write an algorithm to solve mathematical problems using formulas, equations, and functions.

  • PS.4.MAA.3 Analyze and interpret truth tables from basic statements using Boolean operators (AND, OR, XOR, and NOT).

  • PS.4.MAA.4 Write an algorithm from a mathematical model.

PD Program Design

• PD.5.MAA The student will design a step-by-step plan to solve a given problem

  • PD.5.MAA.1 Translate a mathematical expression into a computer statement which involves writing assignment statements and using the order of operations.

  • PD.5.MAA.2 Implement conditional statements that include if/then, if/then/else, case statements, and Boolean logic.

  • PD.5.MAA.3 Define and differentiate Decision (selection) and Sequence (process).

  • PD.5.MAA.4 Represent an algorithm representation as a flowchart and in pseudocode.

  • PD.5.MAA.5 Use flowchart terminology such as terminals (starts and stops), subroutines, and connectors.

  • PD.5.MAA.6 Develop recursive relationships from mathematical models (e.g., arithmetic and geometric sequences).

    • Find terms of a recursive sequence (PC-AA.2)
    • Find a recursive formula (PC-AA.4)
    • Find recursive and explicit formulas (PC-AA.5)
    • Convert a recursive formula to an explicit formula (PC-AA.6)
    • Convert an explicit formula to a recursive formula (PC-AA.7)
    • Convert between explicit and recursive formulas (PC-AA.8)

  • PD.5.MAA.7 Define and use variable data types (e.g., integers, real, character).

PI Program Implementation

• PI.6.MAA The student will create, edit, and execute programs using a programmable calculator and/or computer spreadsheet application program

  • PI.6.MAA.1 Create, edit, and execute a program utilizing an array in a programmable calculator and spreadsheet application.

  • PI.6.MAA.2 Create, edit, and execute programs using recursions and loops in a programmable calculator and spreadsheet application.

  • PI.6.MAA.3 Create, edit, and execute programs to calculate mathematical formulas (e.g., quadratic formula, volume of a simple solid).

  • PI.6.MAA.4 Develop functional programs from algorithms developed from the mathematical models.

  • PI.6.MAA.5 Create programs using various display modes including tables and graphs.

  • PI.6.MAA.6 Locate, categorize, and implement programming commands.

  • PI.6.MAA.7 Use subroutines to reduce keystrokes and memory use.

  • PI.6.MAA.8 Use a spreadsheet application to sort data using various methods (e.g., bubble, quick, shell).

DMT Data Manipulation and Testing

• DMT.7.MAA The student will manipulate data to adjust and test programs designed using a programmable calculator and/or computer spreadsheet application

  • DMT.7.MAA.1 Compare results from mathematical formulas to their program equivalent.

  • DMT.7.MAA.2 Identify and eliminate error messages using troubleshooting techniques (debug).

  • DMT.7.MAA.3 Understand and differentiate the different error types (e.g., syntax, runtime, logic).

  • DMT.7.MAA.4 Design and investigate best-case or worst-case scenarios of a program.

  • DMT.7.MAA.5 Name a range, one cell or a group of cells; use the name to select cells.

  • DMT.7.MAA.6 Estimate best-case or worst-case scenarios using a spreadsheet application scenario tool.