Correlation of the ALEKS courses Algebra 1, High School Geometry, Algebra 2, and PreCalculus to the Common Core State Standards for High School

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1 Correlation of the ALEKS courses Algebra 1, High School Geometry, Algebra 2, and PreCalculus to the Common Core State Standards for High School Number and Quantity N-RN.1: N-RN.2: N-RN.3: N-Q.1: = ALEKS course topic that addresses the standard = Teacher Directed N-RN: The Real Number System N-Q: Quantities* 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. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. Converting between radical form and exponent form Rewrite expressions involving radicals and rational exponents using the properties of exponents. Square root of a perfect square monomial Simplifying a radical expression: Problem type 1 Simplifying a radical expression: Problem type 2 Square root addition Simplifying a sum of radical expressions Square root multiplication Simplifying a product of radical expressions Simplifying a product of radical expressions using the distributive property Special products with square roots: Conjugates and squaring Simplifying a product of radical expressions: Advanced Rationalizing the denominator of a radical expression Rationalizing the denominator of a radical expression using conjugates Simplifying a higher radical: Problem type 1 Simplifying a higher radical: Problem type 2 Rationalizing the denominator of a higher index radical with variables Simplifying products or quotients of higher index radicals with different indices Rational exponents: Basic Rational exponents: Negative exponents and fractional bases Rational exponents: Products and quotients Rational exponents: Powers of powers 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. Classifying sums and products as rational or irrational 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. Finding unit rates Solving a simple word problem using the formula d = rt ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 1/25

2 N-Q.2: N-Q.3: Solving a word problem involving rates and time conversion Converting between temperatures in Fahrenheit and Celsius Converting between compound units: Basic Converting between compound units: Advanced Solving a distance, rate, time problem using a linear equation Interpreting direct variation from a graph Area of a triangle Distinguishing between area and perimeter Area of a parallelogram Area of a trapezoid Area involving rectangles and triangles Circumference and area of a circle Perimeter involving rectangles and circles Arc length and area of a sector of a circle Area between two concentric circles Area involving rectangles and circles Area involving inscribed figures Area involving rectangles and circles: Advanced problem Volume of a triangular prism Volume of a pyramid Volume of a cylinder Volume of a cone Rate of filling of a solid Surface area of a cube or a rectangular prism Surface area of a solid made of unit cubes Surface area of a triangular prism Surface area of a cylinder Define appropriate quantities for the purpose of descriptive modeling. Translating a sentence into a one-step equation Writing a multi-step equation for a real-world situation Translating a sentence into a simple inequality Writing a simple inequality for a real-world situation Writing a compound inequality Writing a multi-step inequality for a real-world situation Writing an equation and drawing its graph to model a real-world situation Translating a sentence into a one-step expression Translating a sentence into a multi-step equation Translating a sentence into a one-step inequality Translating a sentence into a multi-step inequality Writing and evaluating a function that models a real-world situation Writing an equation that models exponential growth or decay Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Finding the absolute error and percent error of a measurement N-CN: The Complex Number System N-CN.1: N-CN.2: Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. Using i to rewrite square roots of negative numbers Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Simplifying a product or quotient involving roots of negative numbers Adding and subtracting complex numbers ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 2/25

3 N-CN.3: N-CN.4: N-CN.5: N-CN.6: N-CN.7: N-CN.8: N-CN.9: Multiplying complex numbers (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Dividing complex numbers (+) 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. Writing a complex number in trigonometric form (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, ( i) 3 = 8 because ( i) has modulus 2 and argument 120. De Moivre's theorem Finding the nth roots of a number: Problem type 1 Finding the nth roots of a number: Problem type 2 (+) 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. Solve quadratic equations with real coefficients that have complex solutions. Solving a quadratic equation with complex roots (+) Extend polynomial identities to the complex numbers. For example, rewrite x as (x + 2i)(x 2i). Multiplying expressions involving complex conjugates (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Linear factors theorem and conjugate zeros theorem N-VM: Vector & Matrix Quantities N-VM.1: N-VM.2: N-VM.3: N-VM.4: N-VM.4.a: N-VM.4.b: (+) 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). Magnitude of a vector Calculating the magnitude and direction of a vector (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. Translation of a vector (+) Solve problems involving velocity and other quantities that can be represented by vectors. Solving a force problem with vectors (+) Add and subtract vectors. 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. Vector addition and scalar multiplication Vector addition: Geometric approach Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 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. Vector addition and scalar multiplication ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 3/25

4 N-VM.5: N-VM.5.a: N-VM.5.b: N-VM.6: (+) Multiply a vector by a scalar. 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 ). Scalar multiplication of a vector: Geometric Approach Multiplication of a vector by a scalar: Geometric approach 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). Calculating the magnitude and direction of a vector (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. N-VM.7: N-VM.8: N-VM.9: N-VM.10: N-VM.11: N-VM.12: (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. Scalar multiplication of a matrix (+) Add, subtract, and multiply matrices of appropriate dimensions. Scalar multiplication of a matrix Addition and subtraction of matrices Linear combinations of matrices Multiplication of matrices: Basic (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. (+) 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. (+) 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. (+) Work with 2 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. Algebra A-SSE.1: A-SSE.1.a: = ALEKS course topic that addresses the standard = Teacher Directed A-SSE: Seeing Structure in Expressions Interpret expressions that represent a quantity in terms of its context.* Interpret parts of an expression, such as terms, factors, and coefficients. Writing a one-step variable expression for a real-world situation Translating a sentence into a one-step equation Translating a sentence into a two-step expression Writing a simple inequality for a real-world situation Writing a multi-step inequality for a real-world situation Degree and leading coefficient of a polynomial in one variable ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 4/25

5 A-SSE.1.b: Translating a sentence into a multi-step equation Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. Finding the slope and y-intercept of a line given its equation in the form y = mx + b Finding the slope and y-intercept of a line given its equation in the form Ax + By = C Interpreting the parameters of a linear function that models a real-world situation Finding the initial amount and rate of change given an exponential function A-SSE.2: A-SSE.3: A-SSE.3.a: A-SSE.3.b: A-SSE.3.c: Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Factoring a quadratic with leading coefficient 1 Factoring a quadratic polynomial in two variables with leading coefficient greater than 1 Factoring a product of a quadratic trinomial and a monomial Factoring a difference of squares in one variable: Basic Factoring with repeated use of the difference of squares formula Factoring a sum or difference of two cubes Factoring out a monomial from a polynomial: Univariate Factoring out a monomial from a polynomial: Multivariate Factoring out a binomial from a polynomial Factoring a polynomial in one variable by grouping: Problem type 1 Factoring a polynomial in one variable by grouping: Problem type 2 Factoring a multivariate polynomial by grouping: Problem type 1 Factoring a multivariate polynomial by grouping: Problem type 2 Factoring a quadratic polynomial in two variables with leading coefficient 1 Factoring out a constant before factoring a quadratic Factoring a quadratic with leading coefficient greater than 1: Problem type 1 Factoring a quadratic with leading coefficient greater than 1: Problem type 2 Factoring a quadratic with leading coefficient greater than 1: Problem type 3 Factoring a quadratic with a negative leading coefficient Factoring a perfect square trinomial with leading coefficient 1 Factoring a perfect square trinomial with leading coefficient greater than 1 Factoring a perfect square trinomial in two variables Factoring a difference of squares in one variable: Advanced Factoring a polynomial involving a GCF and a difference of squares: Univariate Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* Factor a quadratic expression to reveal the zeros of the function it defines. Finding the roots of a quadratic equation with leading coefficient 1 Finding the roots of a quadratic equation with leading coefficient greater than 1 Finding the x-intercept(s) and the vertex of a parabola Finding the roots of a quadratic equation of the form ax 2 + bx = 0 Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Completing the square Rewriting a quadratic function to find the vertex of its graph Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Understanding the product rule of exponents Evaluating expressions with exponents of zero ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 5/25

6 A-SSE.4: Powers of 10: Negative exponent Writing a positive number without a negative exponent Writing a negative number without a negative exponent Writing a simple algebraic expression without negative exponents Introduction to the product rule of exponents Product rule with positive exponents Product rule with negative exponents Introduction to the quotient rule of exponents Quotients of expressions involving exponents Quotient rule with negative exponents: Problem type 1 Understanding the power rule of exponents Introduction to the power rule of exponents Power rule with positive exponents Power rule with negative exponents: Problem type 1 Power rule with negative exponents: Problem type 2 Using the power and product rules to simplify expressions with positive exponents Using the power, product, and quotient rules to simplify expressions with negative exponents Introduction to the product rule with negative exponents Quotient rule with negative exponents: Problem type 2 Using the power and quotient rules to simplify expressions with positive exponents Using the power and quotient rules to simplify expressions with negative exponents: Problem type 1 Using the power and quotient rules to simplify expressions with negative exponents: Problem type 2 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. For example, calculate mortgage payments.* Sum of the first n terms of a geometric sequence A-APR: Arithmetic with Polynomials & Rational Expressions A-APR.1: A-APR.2: A-APR.3: 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. Simplifying a sum or difference of two univariate polynomials Simplifying a sum or difference of three univariate polynomials Multiplying a monomial and a polynomial: Univariate with positive leading coefficients Multiplying a monomial and a polynomial: Multivariate Multiplying binomials with leading coefficients of 1 Multiplying conjugate binomials: Univariate Multiplying binomials in two variables Squaring a binomial: Univariate Multiplication involving binomials and trinomials in two variables Simplifying a sum or difference of multivariate polynomials Multiplying binomials with leading coefficients greater than 1 Multiplying conjugate binomials: Multivariate Multiplication involving binomials and trinomials in one variable Closure properties of integers and polynomials 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). Using the remainder theorem to evaluate a polynomial Remainder theorem: Advanced 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. Roots of a product of polynomials Finding zeros of a polynomial function written in factored form Finding x- and y-intercepts given a polynomial function ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 6/25

7 A-APR.4: A-APR.5: A-APR.6: A-APR.7: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. (+) 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.1 Binomial formula 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. Dividing a polynomial by a monomial: Univariate Dividing a polynomial by a monomial: Multivariate Polynomial long division: Problem type 1 Polynomial long division: Problem type 2 Polynomial long division: Problem type 3 Simplifying a ratio of polynomials: Problem type 1 Simplifying a ratio of polynomials: Problem type 2 Ratio of multivariate polynomials (+) 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. Multiplying rational expressions: Problem type 1 Multiplying rational expressions: Problem type 2 Dividing rational expressions: Problem type 1 Dividing rational expressions: Problem type 2 Complex fraction: Problem type 1 Complex fraction: Problem type 3 Complex fraction: Problem type 4 Adding rational expressions with common denominators Adding rational expressions with different denominators: ax, bx Adding rational expressions with different denominators: Multivariate Adding rational expressions with different denominators: x+a, x+b Adding rational expressions with different denominators: Quadratic A-CED: Creating Equations* 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. Solving a fraction word problem using a linear equation of the form Ax = B Translating a sentence into a one-step equation Solving a word problem with two unknowns using a linear equation Solving a decimal word problem using a linear equation of the form Ax + B = C Solving a decimal word problem using a linear equation with the variable on both sides Solving a fraction word problem using a linear equation with the variable on both sides Solving a word problem with three unknowns using a linear equation Solving a value mixture problem using a linear equation Solving a percent mixture problem using a linear equation Solving a distance, rate, time problem using a linear equation Translating a sentence into a simple inequality Word problem with linear inequalities Word problem with linear inequalities: Problem type 1 Word problem with linear inequalities: Problem type 2 Solving a word problem using a quadratic equation with rational roots ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 7/25

8 A-CED.2: A-CED.3: A-CED.4: Solving a word problem using a rational equation Evaluating an exponential function that models a real-world situation Solving a word problem using an exponential equation: Problem type 1 Solving a word problem using an exponential equation: Problem type 2 Solving a word problem using an exponential equation: Problem type 3 Solving a word problem using an exponential equation: Problem type 4 Finding the perimeter or area of a rectangle given one of these values Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Writing an equation and drawing its graph to model a real-world situation Graphing a linear equation of the form y = mx Graphing a line given its equation in slope-intercept form: Integer slope Graphing a line given its equation in slope-intercept form: Fractional slope Writing a direct variation equation Writing an equation that models exponential growth or decay 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Solving a word problem using a system of linear inequalities Solving a word problem involving a sum and another simple relationship using a system of linear equations Solving a value mixture problem using a system of linear equations Solving a distance, rate, time problem using a system of linear equations Solving a percent mixture problem using a system of linear equations Solving a tax rate or interest rate problem using a system of linear equations Solving a word problem using a 3 by 3 system of linear equations Solving a word problem using a system of linear equations of the form Ax + By = C Solving a word problem using a system of linear equations of the form y = mx + b Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. Introduction to algebraic symbol manipulation Algebraic symbol manipulation: Problem type 1 Algebraic symbol manipulation: Problem type 2 A-REI: Reasoning with Equations & Inequalities A-REI.1: A-REI.2: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Identifying properties used to solve a linear equation Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solving a proportion of the form x/a = b/c Solving a proportion of the form (x+a)/b = x/c Solving a rational equation that simplifies to a linear equation: Problem type 1 Solving a rational equation that simplifies to a linear equation: Problem type 2 Solving a rational equation that simplifies to a linear equation: Problem type 3 Solving a rational equation that simplifies to a linear equation: Problem type 4 Solving a rational equation that simplifies to a quadratic equation: Problem type 1 Solving a rational equation that simplifies to a quadratic equation: Problem type 2 Solving a rational equation that simplifies to a quadratic equation: Problem type 3 Solving a radical equation that simplifies to a linear equation: One radical, basic Solving a radical equation that simplifies to a linear equation: Two radicals ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 8/25

9 A-REI.3: A-REI.4: A-REI.4.a: Solving a radical equation that simplifies to a quadratic equation: One radical Solving a radical equation that simplifies to a quadratic equation: Two radicals Solving an equation with a root index greater than 2 Solving a proportion of the form (x+a)/b = c/d Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Additive property of equality with whole numbers Additive property of equality with decimals Additive property of equality with integers Additive property of equality with a negative coefficient Multiplicative property of equality with whole numbers Multiplicative property of equality with decimals Multiplicative property of equality with signed fractions Multiplicative property of equality with integers Multiplicative property of equality with fractions Using two steps to solve an equation with whole numbers Using two steps to solve an equation with signed decimals Solving a two-step equation with integers Solving a two-step equation with signed fractions Solving a linear equation with several occurrences of the variable: Variables on the same side and distribution Solving a linear equation with several occurrences of the variable: Variables on both sides and distribution Solving a linear equation with several occurrences of the variable: Variables on both sides and two distributions Solving a linear equation with several occurrences of the variable: Variables on both sides and fractional coefficients Solving a linear equation with several occurrences of the variable: Fractional forms with binomial numerators Solving equations with zero, one, or infinitely many solutions Solving a compound linear inequality: Problem type 1 Introduction to algebraic symbol manipulation Algebraic symbol manipulation: Problem type 1 Additive property of equality with signed fractions Solving a multi-step equation given in fractional form Introduction to solving a linear equation with several occurrences of the variable Solving a linear equation with several occurrences of the variable: Variables on the same side Solving a linear equation with several occurrences of the variable: Variables on both sides Additive property of inequality with whole numbers Additive property of inequality with integers Additive property of inequality with signed fractions Additive property of inequality with signed decimals Multiplicative property of inequality with integers Solving a two-step linear inequality: Problem type 1 Solving a two-step linear inequality: Problem type 2 Solving a two-step linear inequality with a fractional coefficient Solving a linear inequality with multiple occurrences of the variable: Problem type 2 Solving a linear inequality with multiple occurrences of the variable: Problem type 3 Solving inequalities with no solution or all real numbers as solutions Solving a compound linear inequality: Problem type 2 Solve quadratic equations in one variable. 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. ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 9/25

10 A-REI.4.b: A-REI.5: A-REI.6: A-REI.7: A-REI.8: A-REI.9: A-REI.10: Solving a quadratic equation by completing the square Solve quadratic equations by inspection (e.g., for x 2 = 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. Solving equations written in factored form Finding the roots of a quadratic equation with leading coefficient 1 Finding the roots of a quadratic equation with leading coefficient greater than 1 Solving a quadratic equation needing simplification Solving a quadratic equation using the square root property: Problem type 1 Solving a quadratic equation using the square root property: Problem type 2 Applying the quadratic formula: Exact answers Solving a quadratic equation with complex roots Finding the roots of a quadratic equation of the form ax 2 + bx = 0 Solving a quadratic equation using the square root property: Problem type 1 Solving a quadratic equation using the square root property: Problem type 2 Solving a quadratic equation by completing the square Applying the quadratic formula: Decimal answers Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Identifying the operations used to create equivalent systems of equations Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Classifying systems of linear equations from graphs Graphically solving a system of linear equations Solving a simple system using substitution Solving a system of linear equations using elimination with multiplication and addition Solving a system that is inconsistent or consistent dependent Solving a system of 3 equations in 3 unknowns Solving a system of linear equations using elimination with addition Solving a system of linear equations with fractional coefficients Solving a system of linear equations with decimal coefficients Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x 2 + y 2 = 3. Graphically solving a system of linear and quadratic equations Solving a system of linear and quadratic equations (+) Represent a system of linear equations as a single matrix equation in a vector variable. Using the inverse of a matrix to solve a system of linear equations (+) 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). Finding the inverse of a 2x2 matrix Finding the inverse of a 3x3 matrix Using the inverse of a matrix to solve a system of linear equations 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). Graphing a line given its equation in slope-intercept form Graphing an equation involving absolute value in the plane: Advanced Graphing a simple cubic function Graphing a function involving a square root ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 10/25

11 A-REI.11: A-REI.12: Identifying solutions to a linear equation in two variables Graphing a linear equation of the form y = mx Graphing a line given its equation in slope-intercept form: Integer slope Graphing a line given its equation in standard form Graphing an equation involving absolute value in the plane: Basic Graphing an integer function and finding its range for a given domain Graphing a parabola of the form y = ax 2 Graphing a parabola of the form y = ax 2 + c Graphing an exponential function: Problem type 1 Graphing an exponential function: Problem type 2 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.* Using a graphing calculator to solve a system of equations Solving a linear equation by graphing Solving a quadratic equation by graphing 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. Graphing a linear inequality in the plane: Slope-intercept form Graphing a linear inequality in the plane: Standard form Graphing a linear inequality in the plane: Vertical or horizontal lines Graphing a system of linear inequalities Graphing a system of two linear inequalities: Advanced Graphing a system of three linear inequalities Functions F-IF.1: F-IF.2: = ALEKS course topic that addresses the standard = Teacher Directed F-IF: Interpreting Functions 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). Identifying functions from relations Determining whether an equation defines a function Vertical line test Domain and range from ordered pairs Introduction to functions: Notation and graphs Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Function tables Evaluating functions: Problem type 1 Evaluating functions: Problem type 2 Introduction to functions: Notation and graphs Finding inputs and outputs of a function from its graph Graphing a piecewise-defined function Word problem using the maximum or minimum of a quadratic function ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 11/25

12 F-IF.3: F-IF.4: F-IF.5: F-IF.6: F-IF.7: F-IF.7.a: Evaluating an exponential function that models a real-world situation Graphing an integer function and finding its range for a given domain Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. Finding the first terms of a sequence Finding the next terms of an arithmetic sequence with integers Finding the first terms of a sequence given a recursive rule Finding the next terms of a geometric sequence with signed numbers 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Writing an equation and drawing its graph to model a real-world situation Interpreting direct variation from a graph Finding intercepts and zeros of a function given the graph Finding where a function is increasing, decreasing, or constant given the graph Finding local maxima and minima of a function given the graph Even and odd functions Determining the end behavior of the graph of a polynomial function Inferring properties of a polynomial function from its graph Solving a word problem using a quadratic equation with irrational roots Word problem using the maximum or minimum of a quadratic function Evaluating an exponential function that models a real-world situation Choosing a graph to fit a narrative Comparing properties of linear functions given in different forms Finding where a function is increasing, decreasing, or constant given the graph Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* Domain and range from ordered pairs Domain and range from the graph of a continuous function Domain and range from the graph of a piecewise function Domain and range of a linear function that models a real-world situation Graphing an integer function and finding its range for a given domain Domain and range from the graph of a parabola 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.* Finding the average rate of change of a function Finding the average rate of change given the graph of a function Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* Graph linear and quadratic functions and show intercepts, maxima, and minima. Graphing a line given the x- and y-intercepts Graphing a line through a given point with a given slope Graphing a vertical or horizontal line Y-intercept of a line Finding x- and y-intercepts of a line given the equation: Advanced Finding the x-intercept(s) and the vertex of a parabola Graphing a parabola of the form y = (x-a) 2 + c Graphing a parabola of the form y = ax 2 + bx + c: Integer coefficients ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 12/25

13 F-IF.7.b: F-IF.7.c: F-IF.7.d: F-IF.7.e: Graphing a parabola of the form y = ax 2 + bx + c: Rational coefficients Graphing a linear equation of the form y = mx Graphing a line given its equation in slope-intercept form: Integer slope Graphing a line given its equation in slope-intercept form: Fractional slope Graphing a line given its equation in standard form Graphing a line by first finding its x- and y-intercepts Graphing a line by first finding its slope and y-intercept Graphing a line given its equation in point-slope form Finding x- and y-intercepts given the graph of a line on a grid Translating the graph of a parabola: One step Graphing a parabola of the form y = ax 2 Graphing a parabola of the form y = ax 2 + c Solving a quadratic equation by graphing Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graphing an equation involving absolute value in the plane: Advanced Graphing a function involving a square root Graphing a piecewise-defined function Translating the graph of an absolute value function: One step Translating the graph of an absolute value function: Two steps Graphing an equation involving absolute value in the plane: Basic Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graphing a simple cubic function Roots of a product of polynomials Finding zeros of a polynomial function written in factored form Finding x- and y-intercepts given a polynomial function Determining the end behavior of the graph of a polynomial function Matching graphs with polynomial functions (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Finding the asymptotes of a rational function: Problem type 1 Finding the asymptotes of a rational function: Problem type 2 Sketching the graph of a rational function: Problem type 1 Sketching the graph of a rational function: Problem type 2 Graphing rational functions with holes Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. The graph, domain, and range of an exponential function Sketching the graph of an exponential function: Advanced Sketching the graph of a logarithmic function: Basic The graph, domain, and range of a logarithmic function Sketching the graph of a logarithmic function Translating the graph of a logarithmic or exponential function Amplitude and period of sine and cosine functions Sketching the graph of a sine or cosine function: Problem type 1 Sketching the graph of a sine or cosine function: Problem type 2 Sketching the graph of a sine or cosine function: Problem type 3 Sketching the graph of a secant or cosecant function: Problem type 1 Sketching the graph of a tangent or cotangent function: Problem type 1 Sketching the graph of a tangent or cotangent function: Problem type 2 ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 13/25

14 F-IF.8: F-IF.8.a: F-IF.8.b: F-IF.9: F-BF.1: F-BF.1.a: F-BF.1.b: F-BF.1.c: F-BF.2: F-BF.3: Graphing an exponential function: Problem type 1 Graphing an exponential function: Problem type 2 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 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. Finding the maximum or minimum of a quadratic function Word problem using the maximum or minimum of a quadratic function Finding the x-intercept(s) and the vertex of a parabola Rewriting a quadratic function to find the vertex of its graph Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay. Finding the initial amount and rate of change given an exponential function Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F-BF: Building Functions Comparing properties of linear functions given in different forms Comparing properties of quadratic functions given in different forms Write a function that describes a relationship between two quantities.* Determine an explicit expression, a recursive process, or steps for calculation from a context. Writing an equation and drawing its graph to model a real-world situation Writing a function rule given a table of ordered pairs: One-step rules Writing a function rule given a table of ordered pairs: Two-step rules Writing and evaluating a function that models a real-world situation Writing an equation that models exponential growth or decay Combine standard function types using arithmetic operations. For example, 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. Sum, difference, and product of two functions Quotient of two functions Combining functions: Advanced Combining functions to write a new function that models a real-world situation (+) Compose functions. For example, 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. Composition of two functions: Domain and range Composition of two functions: Basic Composition of two functions: Advanced Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.* Arithmetic and geometric sequences: Identifying and writing in standard form Writing an explicit rule for an arithmetic sequence Writing a recursive rule for an arithmetic sequence Writing recursive rules for arithmetic and geometric sequences Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 14/25

15 F-BF.4: F-BF.4.a: F-BF.4.b: F-BF.4.c: F-BF.4.d: explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Even and odd functions Writing an equation for a function after a vertical translation Writing an equation for a function after a vertical and horizontal translation Translating the graph of a function: One step Translating the graph of a function: Two steps Transforming the graph of a function by reflecting over an axis Transforming the graph of a function by shrinking or stretching Transforming the graph of a function using more than one transformation How the leading coefficient affects the shape of a parabola Translating the graph of an absolute value function: One step Translating the graph of an absolute value function: Two steps How the leading coefficient affects the graph of an absolute value function Translating the graph of a parabola: One step Graphing an exponential function: Problem type 1 Graphing an exponential function: Problem type 2 Find inverse functions. 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. For example, f(x) = 2x 3 or f(x) = (x+1)/(x 1) for x 1. Inverse functions: Problem type 1 Inverse functions: Problem type 2 Inverse functions: Problem type 3 (+) Verify by composition that one function is the inverse of another. Determining whether two functions are inverses of each other (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. Inverse functions: Problem type 1 (+) Produce an invertible function from a non-invertible function by restricting the domain. F-BF.5: (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Converting between logarithmic and exponential equations Converting between natural logarithmic and exponential equations Evaluating a logarithmic expression Solving a logarithmic equation: Problem type 1 Solving a logarithmic equation: Problem type 2 Solving a logarithmic equation: Problem type 3 Solving a logarithmic equation: Problem type 4 Solving a logarithmic equation: Problem type 5 Solving an exponential equation: Problem type 1 Solving an exponential equation: Problem type 3 F-LE: Linear, Quadratic, & Exponential Models* F-LE.1: F-LE.1.a: Distinguish between situations that can be modeled with linear functions and with exponential functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Identifying linear functions given ordered pairs Identifying linear, quadratic, and exponential functions given ordered pairs Writing an exponential function rule given a table of ordered pairs ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 15/25

16 F-LE.1.b: F-LE.1.c: F-LE.2: F-LE.3: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Application problem with a linear function: Problem type 1 Application problem with a linear function: Problem type 2 Finding a specified term of an arithmetic sequence given the common difference and first term Identifying linear functions given ordered pairs Writing and evaluating a function that models a real-world situation Interpreting the parameters of a linear function that models a real-world situation Comparing properties of linear functions given in different forms Predictions from the line of best fit Finding the next terms of an arithmetic sequence with integers Identifying arithmetic sequences and finding the common difference Finding a specified term of an arithmetic sequence given the first terms Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Solving a word problem using an exponential equation: Problem type 1 Finding a specified term of a geometric sequence given the common ratio and first term Identifying linear, quadratic, and exponential functions given ordered pairs Finding the initial amount and rate of change given an exponential function Writing an equation that models exponential growth or decay Writing an exponential function rule given a table of ordered pairs Finding the next terms of a geometric sequence with signed numbers Identifying arithmetic and geometric sequences Identifying geometric sequences and finding the common ratio Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Writing an equation and drawing its graph to model a real-world situation Writing an equation of a line given the y-intercept and another point Writing the equation of the line through two given points Slopes of parallel and perpendicular lines: Problem type 2 Writing a function rule given a table of ordered pairs: One-step rules Writing a function rule given a table of ordered pairs: Two-step rules Arithmetic and geometric sequences: Identifying and writing in standard form Writing an equation and graphing a line given its slope and y-intercept Writing an equation in slope-intercept form given the slope and a point Writing an equation in point-slope form given the slope and a point Writing and evaluating a function that models a real-world situation Writing an equation that models exponential growth or decay Writing an exponential function rule given a table of ordered pairs Writing an explicit rule for an arithmetic sequence Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Comparing linear, polynomial, and exponential functions F-LE.4: F-LE.5: 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. Solving an exponential equation: Problem type 3 Solving a word problem using an exponential equation: Problem type 3 Solving a word problem using an exponential equation: Problem type 4 Interpret the parameters in a linear or exponential function in terms of a context. Writing an equation and drawing its graph to model a real-world situation Application problem with a linear function: Problem type 1 ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 16/25

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