# Algebra 2 with Trigonometry Correlation of the ALEKS course Algebra 2 with Trigonometry to the Tennessee Algebra II Standards

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1 Algebra 2 with Trigonometry Correlation of the ALEKS course Algebra 2 with Trigonometry to the Tennessee Algebra II Standards Standard 2 : Number & Operations CLE : CLE : CLE : CLE : : : : = ALEKS course topic that addresses the standard = Teacher Directed Course Level Expectations: 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 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 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 Understand the capabilities and limitations of technology when performing operations, graphing, and solving equations involving complex numbers. Checks for Understanding (Formative/Summative Assessment): 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 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. P. 1/18

2 : : : : : : : : : : 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. 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. 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. Select and use appropriate methods to make estimations without technology when solving contextual problems. 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 : 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. P. 2/18

7 : 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): 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 ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 7/18

10 : : : : : : : 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 ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 10/18

13 SPI : SPI : SPI : SPI : SPI : 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 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 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 Interpret graphs that depict real-world phenomena. Scatter plots and correlation Choosing a graph to fit a narrative Interpreting the graphs of two functions 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 ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 13/18

14 SPI : 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 Solve problems involving the binomial theorem and its connection to Pascal's Triangle, combinatorics, and probability. Binomial formula Standard 4 : Geometry & Measurement CLE : CLE : CLE : CLE : = ALEKS course topic that addresses the standard = Teacher Directed Course Level Expectations: 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 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 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 Know and use the Law of Sines to find missing sides and angles of a triangle, including the ambiguous ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 14/18

16 SPI : SPI : SPI : 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: 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 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. Standard 5 : Data Analysis, Statistics, & Probability = ALEKS course topic that addresses the standard = Teacher Directed ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 16/18

17 Course Level Expectations: CLE : CLE : 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 Evaluate and critique various ways of collecting data and using information based on data published in the media. CLE : CLE : Use data and statistical thinking to draw inferences, make predictions, justify conclusions and identify and explain misleading uses of data. Scatter plots and correlation 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): : Collect, represent and describe both linear and non-linear data developed from contextual situations : : : : : : : : : : 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. 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. Recognize when the correlation coefficient measures goodness of fit and does not imply causation. 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. Design simple experiments to collect data to answer questions of interest. Evaluate published data by considering the source, the design of the study and the analysis and ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 17/18

18 representation (or misrepresentation) of the data : : Investigate bias and the phrasing of questions during data acquisition to formulate reasonable conclusions. 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 : SPI : 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 Compare data sets using graphs and summary statistics. SPI : SPI : SPI : SPI : SPI : SPI : 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 Determine differences between randomized experiments and observational studies. 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. Determine/recognize when the correlation coefficient measures goodness of fit. 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 ALEKS - Copyright 2013 UC Regents and ALEKS Corporation. ALEKS is a registered trademark of ALEKS Corporation. P. 18/18

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