# Algebra 2 with Trigonometry Correlation of the ALEKS course

**Algebra 2 with Trigonometry**

**Correlation of the ALEKS course Algebra 2 with Trigonometry to the **

**Tennessee Algebra II Standards **

**Standard 2 : Number & Operations**

= ALEKS course topic that addresses the standard

TD = Teacher Directed

**Course Level Expectations:**

CLE 3103.2.1: Understand the hierarchy of the complex number system and relationships between the elements, properties and operations.

Using *i* to rewrite square roots of negative numbers

Simplifying a product or quotient involving roots of negative numbers

Adding and subtracting complex numbers

Multiplying complex numbers

Dividing complex numbers

Simplifying a power of *i*

CLE 3103.2.2: Connect numeric, analytic, graphical and verbal representations of both real and complex numbers.

Plotting integers on a number line

Fractional position on a number line

Plotting rational numbers on a number line

Integers and rational numbers

Rational and irrational numbers

Using *i* to rewrite square roots of negative numbers

CLE 3103.2.3: Use appropriate technology (including graphing calculators and computer spreadsheets) to solve problems, recognize patterns and collect and analyze data.

Using a graphing calculator to find zeros of a polynomial function

Using a graphing calculator to find local extrema of a polynomial function

CLE 3103.2.4: Understand the capabilities and limitations of technology when performing operations, graphing, and solving equations involving complex numbers.

TD

**Checks for Understanding (Formative/Summative Assessment):**

3103.2.1: Understand that to solve certain problems and equations, the real number system needs to be extended from real numbers to complex numbers.

Solving a quadratic equation with complex roots

3103.2.2:

3103.2.3:

Define and give examples of each of the types of numbers in the complex number system.

Integers and rational numbers

Rational and irrational numbers

Using *i* to rewrite square roots of negative numbers

Identify and apply properties of complex numbers (including simplification and standard form).

Using *i* to rewrite square roots of negative 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.

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Multiplying complex numbers

Dividing complex numbers

Simplifying a power of *i*

Add and subtract complex numbers.

Adding and subtracting complex numbers

Multiply complex numbers.

Multiplying complex numbers

Define and utilize the complex conjugates to write the quotient of two complex numbers in standard form.

Dividing complex numbers

Graph complex numbers in the complex plane and recognize differences and similarities with the graphical representations of real numbers graphed on the number line.

TD

Solve quadratic equations over the complex number system.

Solving a quadratic equation with complex roots

Find and describe geometrically the absolute value of a complex number.

TD

Draw conclusions based on number concepts, algebraic properties, and/or relationships between expressions and numbers over complex numbers.

Properties of addition

Properties of real numbers

Solving equations with zero, one, or infinitely many solutions

Introduction to algebraic symbol manipulation

Understand the capabilities and limitations of technology. Make estimations without a calculator to detect potential errors.

TD

Select and use appropriate methods to make estimations without technology when solving contextual problems.

TD

Analyze and evaluate contextual situations involving any type of number from the complex number system.

Finding unit rates

Solving a word problem involving rates and time conversion

Finding the sale price given the original price and percent discount

Finding the original price given the sale price and percent discount

Finding the percentage increase or decrease

Computing a percentage from a table of values

Simple interest

Finding the value for a new score that will yield a given mean

Word problem on proportions: Problem type 1

Word problem on proportions: Problem type 2

Word problem involving multiple rates

**State Performance Indicators:**

SPI 3103.2.1: Describe any number in the complex number system.

Integers and rational numbers

Rational and irrational numbers

Using *i* to rewrite square roots of negative numbers

ALEKS - Copyright © 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation.

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SPI 3103.2.2:

SPI 3103.2.3:

Compute with all real and complex numbers.

Integer addition: Problem type 2

Integer subtraction

Signed fraction addition and subtraction: Basic

Signed fraction addition: Advanced

Signed decimal addition with three numbers

Integer multiplication and division

Signed fraction multiplication: Advanced

Exponents and integers: Problem type 1

Exponents and integers: Problem type 2

Multiplying and dividing numbers written in scientific notation

Simplifying a product or quotient involving roots of negative numbers

Adding and subtracting complex numbers

Multiplying complex numbers

Dividing complex numbers

Simplifying a power of *i*

Use the number system, from real to complex, to solve equations and contextual problems.

Solving a quadratic equation with complex roots

**Standard 3 : Algebra**

= ALEKS course topic that addresses the standard

TD = Teacher Directed

**Course Level Expectations:**

CLE 3103.3.1: Understand and apply properties of rational exponents and perform basic operations to simplify algebraic expressions.

Combining like terms: Integer coefficients

Combining like terms: Advanced

Combining like terms in a quadratic expression

Writing a simple algebraic expression without negative exponents

Understanding the product rule of exponents

Introduction to the product rule of exponents

Product rule with positive exponents

Product rule with negative 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

Simplifying a sum or difference of two univariate polynomials

Simplifying a sum or difference of three univariate polynomials

Square root of a perfect square monomial

Simplifying a radical expression: Problem type 1

Simplifying a radical expression: Problem type 2

Simplifying a higher radical: Problem type 2

Simplifying a sum of radical expressions

Simplifying a product of radical expressions

ALEKS - Copyright © 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation.

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Simplifying a product of radical expressions: Advanced

Rational exponents: Basic

Rational exponents: Negative exponents and fractional bases

Rational exponents: Products and quotients

Rational exponents: Powers of powers

Simplifying a ratio of polynomials: Problem type 1

Simplifying a ratio of polynomials: Problem type 2

Ratio of multivariate polynomials

Multiplying rational expressions: Problem type 1

Multiplying rational expressions: Problem type 2

Dividing rational expressions: Problem type 1

Dividing rational expressions: Problem type 2

Adding rational expressions with common denominators

Adding rational expressions with different denominators: ax, bx

Adding rational expressions with different denominators: x+a, x+b

Adding rational expressions with different denominators: Quadratic

Complex fraction: Problem type 1

Complex fraction: Problem type 3

Complex fraction: Problem type 4

CLE 3103.3.2: Understand, analyze, transform and generalize mathematical patterns, relations and functions using properties and various representations.

Evaluating functions: Problem type 1

Domain and range from ordered pairs

Identifying functions from relations

Vertical line test

Determining whether an equation defines a function

Finding inputs and outputs of a function from its graph

Domain and range from the graph of a continuous function

Domain and range from the graph of a piecewise function

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

Finding the first terms of a sequence

Arithmetic and geometric sequences: Identifying and writing in standard form

Arithmetic sequences

Geometric sequences

CLE 3103.3.3: Analyze and apply various methods to solve equations, absolute values, inequalities, and systems of equations over complex numbers.

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 integers

Multiplicative property of equality with signed fractions

Solving a two-step equation with integers

Solving a two-step equation with signed fractions

Using two steps to solve an equation with signed decimals

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 fractional coefficients

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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: Fractional forms with binomial numerators

Solving equations with zero, one, or infinitely many solutions

Solving a linear inequality: Problem type 1

Solving a linear inequality: Problem type 2

Solving a linear inequality: Problem type 3

Solving a linear inequality: Problem type 4

Solving a linear inequality: Problem type 5

Simple absolute value equation

Solving an equation involving absolute value: Basic

Solving an equation involving absolute value: Advanced

Solving an inequality involving absolute value: Basic

Solving an inequality involving absolute value

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

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

Solving a quadratic equation by completing the square

Applying the quadratic formula: Exact answers

Solving a quadratic inequality written in factored form

Solving a quadratic inequality

Solving equations written in factored form

Roots of a product of polynomials

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

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 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 2

Solving an exponential equation: Problem type 3

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 rational inequality: Problem type 1

Solving a rational inequality: Problem type 2

P. 5/18

CLE 3103.3.4: Graph and compare equations and inequalities in two variables. Identify and understand the relationships between the algebraic and geometric properties of the graph.

Graphing a line given its equation in slope-intercept form

Graphing a line given its equation in standard form

Graphing a vertical or horizontal line

Y-intercept of a line

Finding x- and y-intercepts of a line given the equation: Advanced

Finding slope given the graph of a line on a grid

Finding the slope of a line given its equation

Writing an equation of a line given the y-intercept and another point

Writing the equation of the line through two given points

Writing the equations of vertical and horizontal lines through a given point

Graphing a linear inequality in the plane: Standard form

Graphing a linear inequality in the plane: Vertical or horizontal lines

Graphing a linear inequality in the plane: Slope-intercept form

Graphing a parabola of the form y = ax

2

Graphing a simple cubic function

Graphing an equation involving absolute value in the plane: Advanced

Finding the x-intercept(s) and the vertex of a parabola

Rewriting a quadratic function to find the vertex of its graph

Finding the maximum or minimum of a quadratic function

Graphing a parabola of the form y = (x-a)

2

+ c

Graphing a parabola of the form y = ax

2

+ bx + c: Integer coefficients

Classifying the graph of a function

How the leading coefficient affects the shape of a parabola

Writing the equation of a quadratic function given its graph

Graphing a quadratic inequality: Problem type 1

Graphing a quadratic inequality: Problem type 2

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

Inferring properties of a polynomial function from its graph

Matching graphs with polynomial functions

Graphing a function involving a square root

Sketching the graph of an exponential function: Basic

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

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

Graphing rational functions with holes

Matching graphs with rational functions: Two vertical asymptotes

CLE 3103.3.5: Use mathematical models involving equations and systems of equations to represent, interpret and analyze quantitative relationships, change in various contexts, and other real-world phenomena.

Translating a sentence into a one-step equation

Translating a sentence into a two-step expression

Writing a multi-step equation for a real-world situation

Solving a fraction word problem using a linear equation of the form Ax = B

Solving a word problem with two unknowns using a linear equation

Solving a decimal word problem using a linear equation with the variable on both sides

Solving a decimal word problem using a linear equation of the form Ax + B = C

Solving a fraction word problem using a linear equation with the variable on both sides

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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

Writing an equation and drawing its graph to model a real-world situation

Application problem with a linear function: Problem type 1

Application problem with a linear function: Problem type 2

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 quadratic equation with rational roots

Solving a word problem using a quadratic equation with irrational roots

Using a graphing calculator to solve a word problem involving a polynomial of degree 3

Using a graphing calculator to solve a word problem involving a local extremum of a polynomial function

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

Compound interest

Word problem on proportions: Problem type 1

Word problem on proportions: Problem type 2

Word problem involving multiple rates

Solving a word problem using a rational equation

Writing an equation that models variation

Word problem on direct variation

Word problem on inverse variation

Word problem on combined variation

**Checks for Understanding (Formative/Summative Assessment):**

3103.3.1: Perform operations on algebraic expressions and justify the procedures.

Distributive property: Whole number coefficients

Distributive property: Integer coefficients

Understanding the product rule of exponents

Introduction to the product rule of exponents

Product rule with positive exponents

Product rule with negative 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

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 binomials with leading coefficients of 1

Multiplying conjugate binomials: Univariate

P. 7/18

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Multiplying binomials in two variables

Squaring a binomial: Univariate

Multiplication involving binomials and trinomials in two variables

Dividing a polynomial by a monomial: Univariate

Polynomial long division: Problem type 1

Polynomial long division: Problem type 2

Polynomial long division: Problem type 3

Simplifying a sum of radical expressions

Simplifying a product of radical expressions

Simplifying a product of radical expressions: Advanced

Special products with square roots: Conjugates and squaring

Rational exponents: Products and quotients

Rational exponents: Powers of powers

Multiplying rational expressions: Problem type 1

Multiplying rational expressions: Problem type 2

Dividing rational expressions: Problem type 1

Dividing rational expressions: Problem type 2

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

Determine the domain of a function represented in either symbolic or graphical form.

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 of a square root function

Domain of a rational function

Determine and graph the inverse of a function with and without technology.

Inverse functions: Problem type 1

Inverse functions: Problem type 2

Inverse functions: Problem type 3

Analyze the effect of changing various parameters on functions and their graphs.

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

Graph piece-wise and step functions.

Graphing a piecewise-defined function

Simplify expressions and solve equations containing radicals.

Square root of a perfect square

Square root of a rational perfect square

Cube root of an integer

Square root simplification

Square root of a perfect square monomial

Simplifying a radical expression: Problem type 1

Simplifying a radical expression: Problem type 2

Simplifying a higher radical: Problem type 1

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Simplifying a higher radical: 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: Advanced

Simplifying a product of radical expressions using the distributive property

Special products with square roots: Conjugates and squaring

Rationalizing the denominator of a radical expression

Rationalizing the denominator of a radical expression using conjugates

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

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

Solve quadratic equations by factoring, graphing, completing the square, extracting square roots and using the quadratic formula.

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

Solving a quadratic equation by completing the square

Applying the quadratic formula: Exact answers

Solving a quadratic equation with complex roots

Solve a three by three system of linear equations algebraically and by using inverse matrices and determinants with and without technology.

Solving a system of 3 equations in 3 unknowns

Cramer's rule: Problem type 2

Using the inverse of a matrix to solve a system of linear equations

Find an equation for a parabola when given its graph or when given its roots.

Writing a quadratic equation given the roots and the leading coefficient

Writing the equation of a quadratic function given its graph

Given a quadratic equation use the discriminant to determine the nature of the roots.

Discriminant of a quadratic equation

Describe and articulate the characteristics and parameters of a parent function.

TD

Understand the relationship between real zeros of a function and the x-intercepts of its graph.

Finding the x-intercept(s) and the vertex of a parabola

Finding x- and y-intercepts given a polynomial function

Using a graphing calculator to find zeros of a polynomial function

Solve problems using exponential functions requiring the use of logarithms for their solutions.

Solving an exponential equation: Problem type 1

Solving an exponential equation: Problem type 3

Solving a word problem using an exponential equation: Problem type 2

Solving a word problem using an exponential equation: Problem type 4

Define and use arithmetic and geometric sequences and series including using sigma and pi notation.

Arithmetic and geometric sequences: Identifying and writing in standard form

Arithmetic sequences

Geometric sequences

P. 9/18

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Sum of the first n terms of an arithmetic sequence

Sum of the first n terms of a geometric sequence

Find the sum of an geometric series whose common ratio, r, is in the interval (-1,1).

Sum of a geometric series

Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.

Basic properties of logarithms

Writing expressions as a single logarithm

Expanding a logarithmic expression: Problem type 1

Expanding a logarithmic expression: Problem type 2

Change of base for logarithms: Problem type 1

Solving a logarithmic equation: Problem type 2

Solving a logarithmic equation: Problem type 5

Know that the logarithm and exponential functions are inverses and use this information to solve real-world problems.

Converting between logarithmic and exponential equations

Converting between natural logarithmic and exponential equations

Solving a word problem using an exponential equation: Problem type 2

Solving a word problem using an exponential equation: Problem type 4

Solve compound inequalities involving disjunction and conjunction and linear inequalities containing absolute values.

Solving a compound linear inequality: Problem type 1

Solving a compound linear inequality: Problem type 2

Solving an inequality involving absolute value: Basic

Solving an inequality involving absolute value

Solve linear programming problems.

Linear programming

Solving a word problem using linear programming

Understand the relationships between the equations of conic sections and their graphs.

Graphing a parabola with a horizontal or a vertical axis

Writing an equation of a parabola given the vertex and the focus

Graphing a circle given its equation in standard form

Graphing a circle given its equation in general form

Writing an equation of a circle given its center and a point on the circle

Writing an equation of a circle given the endpoints of a diameter

Graphing an ellipse centered at the origin

Graphing an ellipse given its equation in standard form

Graphing an ellipse given its equation in general form

Graphing a hyperbola centered at the origin

Graphing a hyperbola given its equation in standard form

Graphing a hyperbola given its equation in general form

Classifying conics given their equations

Factor polynomials using a variety of methods including the factor theorem, synthetic division, long division, sums and differences of cubes, and grouping.

Factoring out a monomial from a polynomial: Univariate

Factoring out a monomial from a polynomial: Multivariate

Factoring a quadratic with leading coefficient 1

Factoring a perfect square trinomial

Factoring a quadratic with leading coefficient greater than 1

Factoring a quadratic polynomial in two variables with leading coefficient greater than 1

P. 10/18

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Factoring a product of a quadratic trinomial and a monomial

Factoring a difference of squares

Factoring a sum or difference of two cubes

Factoring a polynomial by grouping: Problem type 1

Factoring a polynomial by grouping: Problem type 2

Polynomial long division: Problem type 1

Polynomial long division: Problem type 2

Polynomial long division: Problem type 3

Synthetic division

Determine the number and possible types of zeros for a polynomial function and find the rational roots.

Using the rational zeros theorem to find all zeros of a polynomial: Rational zeros

Descartes' Rule of Signs

Linear factors theorem and conjugate zeros theorem

Understand the connection between the roots, zeros, x-intercepts, factors of polynomials, and solutions of polynomial equations.

Writing a quadratic equation given the roots and the leading coefficient

Finding the x-intercept(s) and the vertex of a parabola

Solving equations written in factored form

Roots of a product of polynomials

Finding zeros of a polynomial function written in factored form

Finding a polynomial of a given degree with given zeros: Real zeros

Finding x- and y-intercepts given a polynomial function

Matching graphs with polynomial functions

Using a graphing calculator to find zeros of a polynomial function

**State Performance Indicators:**

SPI 3103.3.1: Add, subtract and multiply polynomials; divide a polynomial by a lower degree polynomial.

Product rule with positive exponents

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 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

Dividing a polynomial by a monomial: Univariate

Polynomial long division: Problem type 1

Polynomial long division: Problem type 2

Polynomial long division: Problem type 3

SPI 3103.3.2: Solve quadratic equations and systems, and determine roots of a higher order polynomial.

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 equations written in factored form

Roots of a product of polynomials

Finding zeros of a polynomial function written in factored form

Solving a quadratic equation with complex roots

Finding x- and y-intercepts given a polynomial function

Odd root property

P. 11/18

SPI 3103.3.3:

Solving an equation with exponent using the odd-root property

Solving a system of nonlinear equations

Add, subtract, multiply, divide and simplify rational expressions including those with rational and negative exponents.

Quotients of expressions involving exponents

Quotient rule with negative exponents: Problem type 1

Rational exponents: Products and quotients

Simplifying a ratio of polynomials: Problem type 1

Simplifying a ratio of polynomials: Problem type 2

Ratio of multivariate polynomials

Multiplying rational expressions: Problem type 1

Multiplying rational expressions: Problem type 2

Dividing rational expressions: Problem type 1

Dividing rational expressions: Problem type 2

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

Complex fraction: Problem type 1

Complex fraction: Problem type 3

Complex fraction: Problem type 4

SPI 3103.3.4: Use the formulas for the general term and summation of finite arithmetic and both finite and infinite geometric series.

Arithmetic and geometric sequences: Identifying and writing in standard form

Arithmetic sequences

Geometric sequences

Sum of the first n terms of an arithmetic sequence

Sum of the first n terms of a geometric sequence

Sum of a geometric series

SPI 3103.3.5:

SPI 3103.3.6:

SPI 3103.3.7:

SPI 3103.3.8:

Describe the domain and range of functions and articulate restrictions imposed either by the operations or by the contextual situations which the functions represent.

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

Range of a quadratic function

Domain of a square root function

Domain of a rational function

Combine functions (such as polynomial, rational, radical and absolute value expressions) by addition, subtraction, multiplication, division, or by composition and evaluate at specified values of their variables.

Sum, difference, and product of two functions

Quotient of two functions

Composition of two functions: Basic

Composition of two functions: Advanced

Identify whether a function has an inverse, whether two functions are inverses of each other, and/or explain why their graphs are reflections over the line y = x.

Horizontal line test

Determining whether two functions are inverses of each other

Inverse functions: Problem type 1

Inverse functions: Problem type 3

Solve systems of three linear equations in three variables.

P. 12/18

SPI 3103.3.9:

Solving a system of 3 equations in 3 unknowns

Graph the solution set of two or three linear or quadratic inequalities.

Graphing a system of linear inequalities

Graphing a system of nonlinear inequalities: Problem type 1

SPI 3103.3.10: Identify and/or graph a variety of functions and their translations.

Graphing a line given its equation in slope-intercept form

Graphing a line given its equation in standard form

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

Graphing a parabola of the form y = ax

2

Graphing a simple cubic function

Graphing an equation involving absolute value in the plane: Advanced

Graphing a parabola of the form y = (x-a)

2

+ c

Graphing a parabola of the form y = ax

2

+ bx + c: Integer coefficients

Graphing a function involving a square root

Sketching the graph of an exponential function: Basic

The graph, domain, and range of an exponential function

Sketching the graph of a logarithmic function: Basic

The graph, domain, and range of a logarithmic function

Translating the graph of a logarithmic or exponential function

Sketching the graph of a rational function: Problem type 1

SPI 3103.3.11: Graph conic sections (circles, parabolas, ellipses and hyperbolas) and understand the relationship between the standard form and the key characteristics of the graph.

Graphing a parabola with a horizontal or a vertical axis

Graphing a circle given its equation in standard form

Graphing a circle given its equation in general form

Graphing an ellipse centered at the origin

Graphing an ellipse given its equation in standard form

Graphing an ellipse given its equation in general form

Graphing a hyperbola centered at the origin

Graphing a hyperbola given its equation in standard form

Graphing a hyperbola given its equation in general form

Classifying conics given their equations

SPI 3103.3.12: Interpret graphs that depict real-world phenomena.

Scatter plots and correlation

Choosing a graph to fit a narrative

Interpreting the graphs of two functions

SPI 3103.3.13: Solve contextual problems using quadratic, rational, radical and exponential equations, finite geometric series or systems of equations.

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

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Solving a word problem using a quadratic equation with rational roots

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

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

Compound interest

Word problem on proportions: Problem type 1

Word problem on proportions: Problem type 2

Solving a word problem using a rational equation

Word problem on inverse variation

SPI 3103.3.14: Solve problems involving the binomial theorem and its connection to Pascal's Triangle, combinatorics, and probability.

Binomial formula

**Standard 4 : Geometry & Measurement**

= ALEKS course topic that addresses the standard

TD = Teacher Directed

**Course Level Expectations:**

CLE 3103.4.1: Understand the trigonometric functions and their relationship to the unit circle.

Sine, cosine, and tangent ratios

Finding trigonometric ratios given a right triangle

Trigonometric functions and special angles: Problem type 1

Trigonometric functions and special angles: Problem type 2

Trigonometric functions and special angles: Problem type 3

CLE 3103.4.2: Know and use the basic identities of sine, cosine, and tangent as well as their reciprocals.

Sine, cosine, and tangent ratios

Finding trigonometric ratios given a right triangle

Using a trigonometric ratio to find a side length in a right triangle

Using trigonometry to find distances

Using a trigonometric ratio to find an angle measure in a right triangle

Using trigonometry to find angles of elevation or depression

Solving a right triangle

Trigonometric functions and special angles: Problem type 1

Trigonometric functions and special angles: Problem type 2

Trigonometric functions and special angles: Problem type 3

Finding values of trigonometric functions given information about an angle: Problem type 1

CLE 3103.4.3: Graph all six trigonometric functions and identify their key characteristics.

Amplitude and period of sine and cosine functions

Amplitude, period, and phase shift 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

CLE 3103.4.4: Know and use the Law of Sines to find missing sides and angles of a triangle, including the ambiguous

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case.

Solving a triangle with the law of sines: Problem type 1

Solving a triangle with the law of sines: Problem type 2

Solving a word problem using the law of sines

CLE 3103.4.5: Use trigonometric concepts, properties and graphs to solve problems.

Using a trigonometric ratio to find a side length in a right triangle

Using trigonometry to find distances

Using a trigonometric ratio to find an angle measure in a right triangle

Using trigonometry to find angles of elevation or depression

Solving a right triangle

Finding values of trigonometric functions given information about an angle: Problem type 1

Finding values of trigonometric functions given information about an angle: Problem type 2

Finding values of trigonometric functions given information about an angle: Problem type 3

Solving a triangle with the law of sines: Problem type 1

Solving a triangle with the law of sines: Problem type 2

Solving a word problem using the law of sines

Solving a triangle with the law of cosines

Solving a word problem using the law of cosines

Writing the equation of a sine or cosine function given its graph: Problem type 1

**Checks for Understanding (Formative/Summative Assessment):**

3103.4.1: Convert between radians and degrees and vice versa.

Converting between degree and radian measure: Problem type 1

Converting between degree and radian measure: Problem type 2

3103.4.2:

3103.4.3:

3103.4.4:

3103.4.5:

Determine the period and the amplitude of a periodic function.

Amplitude and period of sine and cosine functions

Amplitude, period, and phase shift of sine and cosine functions

Extend the trigonometric functions to periodic functions on the real line by defining them as functions on the unit circle.

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

Understand the relationship between the radius, the central angle, and radian measure.

Arc length and central angle measure

Determine the domain and range of the six trigonometric functions given a graph.

TD

3103.4.6: Know and be able to use the fundamental trigonometric identities, including the Pythagorean identities, reciprocal identities, sum of sine and cosine, and odd and even identities.

Finding trigonometric ratios given a right triangle

Using a trigonometric ratio to find a side length in a right triangle

Using trigonometry to find distances

Using a trigonometric ratio to find an angle measure in a right triangle

Using trigonometry to find angles of elevation or depression

Solving a right triangle

Finding values of trigonometric functions given information about an angle: Problem type 1

Finding values of trigonometric functions given information about an angle: Problem type 2

Finding values of trigonometric functions given information about an angle: Problem type 3

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Simplifying trigonometric expressions

Using cofunction identities

Sum and difference identities: Problem type 1

Sum and difference identities: Problem type 2

Sum and difference identities: Problem type 3

Verifying a trigonometric identity

Proving trigonometric identities: Problem type 1

**State Performance Indicators:**

SPI 3103.4.1: Exhibit knowledge of unit circle trigonometry.

Trigonometric functions and special angles: Problem type 1

Trigonometric functions and special angles: Problem type 2

Trigonometric functions and special angles: Problem type 3

SPI 3103.4.2:

SPI 3103.4.3:

Match graphs of basic trigonometric functions with their equations.

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

Describe and articulate the characteristics and parameters of parent trigonometric functions to solve contextual problems.

TD

**Standard 5 : Data Analysis, Statistics, & Probability**

= ALEKS course topic that addresses the standard

TD = Teacher Directed

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**Course Level Expectations:**

CLE 3103.5.1: Describe, interpret, and apply quantitative data.

Mode of a data set

Finding the mode and range of a data set

Mean and median of a data set

Weighted mean

Percentiles

Population standard deviation

CLE 3103.5.2: Evaluate and critique various ways of collecting data and using information based on data published in the media.

TD

CLE 3103.5.3: Use data and statistical thinking to draw inferences, make predictions, justify conclusions and identify and explain misleading uses of data.

Scatter plots and correlation

CLE 3103.5.4: Develop an understanding of probability concepts in order to make informed decisions.

Probability of an event

Outcomes and event probability

Die rolling

Experimental and theoretical probability

**Checks for Understanding (Formative/Summative Assessment):**

3103.5.1: Collect, represent and describe both linear and non-linear data developed from contextual situations.

TD

3103.5.2:

3103.5.3:

3103.5.4:

3103.5.5:

3103.5.6:

3103.5.7:

3103.5.8:

3103.5.9:

Organize and display data using appropriate methods (including spreadsheets and technology tools) to detect patterns and departures from patterns.

Box-and-whisker plots

Read and interpret data from a two-way table.

Computing a percentage from a table of values

Understand the impact of various sampling methods and use them to draw valid conclusions.

TD

Calculate measures of central tendency and spread (variance and standard deviation).

Mode of a data set

Finding the mode and range of a data set

Mean and median of a data set

Population standard deviation

Use technology to find the appropriate regression equation for both linear and non-linear data.

TD

Recognize when the correlation coefficient measures goodness of fit and does not imply causation.

TD

Know the Empirical Rule for one, two and three standard deviations for a normal distribution.

Word problem involving calculations from a normal distribution

Use data to detect patterns.

TD

3103.5.10:

3103.5.11:

Design simple experiments to collect data to answer questions of interest.

TD

Evaluate published data by considering the source, the design of the study and the analysis and

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3103.5.12:

3103.5.13: representation (or misrepresentation) of the data.

TD

Investigate bias and the phrasing of questions during data acquisition to formulate reasonable conclusions.

TD

Apply both theoretical and experimental probability to analyze the likelihood of an event.

Probability of an event

Outcomes and event probability

Die rolling

Experimental and theoretical probability

**State Performance Indicators:**

SPI 3103.5.1: Compute, compare and explain summary statistics for distributions of data including measures of center and spread.

Mode of a data set

Finding the mode and range of a data set

Mean and median of a data set

Weighted mean

Population standard deviation

SPI 3103.5.2: Compare data sets using graphs and summary statistics.

TD

SPI 3103.5.3:

SPI 3103.5.4:

Analyze patterns in a scatter-plot and describe relationships in both linear and non-linear data.

Scatter plots and correlation

Sketching the line of best fit

Apply the characteristics of the normal distribution.

Word problem involving calculations from a normal distribution

SPI 3103.5.5: Determine differences between randomized experiments and observational studies.

TD

SPI 3103.5.6: Find the regression curve that best fits both linear and non-linear data (using technology such as a graphing calculator) and use it to make predictions.

TD

SPI 3103.5.7: Determine/recognize when the correlation coefficient measures goodness of fit.

TD

SPI 3103.5.8: Apply probability concepts such as conditional probability and independent events to calculate simple probability.

Probability of an event

Outcomes and event probability

Die rolling

Experimental and theoretical probability

Probability of independent events

Probability of dependent events

Probabilities of draws with replacement

Independent events: Basic

Conditional probability: Basic

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