Curriculum Design Template Content Area: Mathematics Course Title: Algebra 2/Trig HN Grade Level: 11 Functions, Equations and Graphs Linear Systems Quadratic Equations and Functions Marking Period 1 Polynomials and Polynomial Functions Radical Functions an d Rational Expressions Exponential and Logarithmic Functions Marking Period 2 Rational Functions Sequence and Series Probability and Statistics Marking Period 3 Periodic Functions and Trigonometry Trigonometric Identities and Equations Matrices Marking Period 4 Date Created: August 2012 Board Approved on: August 27, 2012
Honors Algebra 2/Trig Unit Title Lessons to include Time Tools of Algebra Functions, equations, and graphs Linear Systems Quadratic Equation and Function Polynomials and Polynomial functions Radical functions and Rational Expressions Exponential and logarithmic functions Common Core Standards Math 1.1 Properties of Real Numbers Daily N.RN.3 1.3 Solving Equations Wormups A.CED.1, A.CED.4 1.5 Absolute Value Equations and A.SSE.1.b, A.CED.1 1.6 Probability review 2.2 Linear equations A.CED.2, F.IF.7,F.IF.8,F.IF.9 2.5 Absolute value functions and graphs 2 weeks F.IF.7, F.IF.7.b, F.BF.3 2.7 Two-variable, inequalitites A.CED.2, F.IF.7.b 3.2 Solving by elimination A.CED.2, A.CED.3, A.REI.5, A.REI.6 3.3 Solving by graphing 2 weeks A.CED.2, A.CED.3, A.REI.6, A.REI.11 3.6 Systems with 3 variables Extens A.REI.6 5.1 Modeling data w/quadratic funct. F.IF.4, F.IF.5 5.2Properties of Parabolas A.SSE.2 5.3 Transforming Parabolas Extend A.SSE.2 5.4 Factoring quadratic expressions A.SSE.2 5 weeks 5.5 Quadratic Equations A.SSE.1.a, A.APR.3, A.CED.1 5.6 Complex numbers Extends N.CN.2 5.7Completing the square Reviews A.REI.4.b 5.8 The Quadratic Formula 6.2 Polynomials and Linear factors A.SSE.1, A.APR.3, F.IF.7.c, F.BF.1 6.4 Solving Polynomial Equations A.SSE.2, A.REI.11 3 weeks N.CN.7, N.CN.8, N.CN.9, 6.6 The fundamental thm of algebra A.APR.3 6.7 Permutations and combinations 7.1 roots and radical expressions A.SSE.2 7.2 multiply/divide rational expressions A.SSE.2 7.4 rational exponents N.RN.1, N.RN.2 7.5 Solving square roots 4 weeks A.CED.4, A.REI.2 7.7 Inverses F.BF.4.a, F.BF.4.c 7.8 Graphing Square roots and other radical functions A.SSE.1.b, A.CED.2, F.IF.7, 8.2 Properties of Exponential functions 3 weeks F.IF.7.e, F.IF.8, F.BF.1, 8.4 Properties of Logarithms F.LE.4
Midterm Review & exam weeks Rational Functions Sequence and Series Probability and Statistics Periodic functions and Trigonometry Trigonometric Identities and Equations Matrices Final review and Exam week 8.5 Exponential & logarithmic functions A.REI.11, F.LE.4 8.6 Natural logarithms F.LE.4 Chapters 2,3,5-8 1 weeks all 9.1 Inverse Variation A.CED.2, A.CED.4 9.2 The Reciprocal Function Family A.CED.2, F.BF.1, F.BF.3 9.3 Rational Functions and Graphs A.CED.2, F.IF.7, F.BF.1, F.BF.1.B 9.4 Rational Expressions 4 weeks A.SSE.1, A.SSE.1.A, A.SSE.1B, A.SSE.2, A.APR.7 9.5 Adding/Subtract Rational Express A.APR.7 9.6 Solving Rational Equations A.APR,6, A.APR.7, A.CED.1, A.REI.2, A.REI.11 11.2 Arithmetic Sequences F.IF.3 11.3 Geometric Sequences A.SSE.4 2 weeks 11.4 Arithmetic Series F.IF.3 11.5 Geometric Series A.SSE.4 12.1 Probability Distributions S.CP.2, S.CP.5, S.CP.7 12.2 Conditional Probability S.CP.3, S.CP.4, S.CP.5, S.CP.6, 2 weeks S.CP.8 12.3 Analyzing Data S.IC.6 12.4 Standard Deviation S.ID.4, S.IC.6 13.2 Angles and the unit circle F.TF.2 13.3 Radian measure F.TF.1 13.4 The sine function 4 weeks F.IF.4, F.IF.7.e, F.TF.2, F.TF.5 13.5 The Cosine Function F.IF.4, F.IF.7.e, F.TF.2, F.TF.5 13.6 The Tangent function F.IF.7.e, F.TF.2, F.TF.5 14.1 Trigonometric identities F.TF.8 14.3 Right Triangles & trig ratios G.SRT.6, G.SRT.8 3 weeks 14.4 Area and Law of Sines G.SRT.9, G.SRT.10, G.SRT.11 14.5 Law of Cosines G.SRT.10, G.SRT.11 4.2 Adding and subtracting matrices N.VM.8, N.VM.10 N.VM.6, N.VM.7, N.VM.8, 4.3 Matrix multiplication N.VM.9 3 weeks 4.5 Determinants, and Inverses N.VM.10, N.VM.12 4.7 Inverse matrices, and systems N.VM.8 4.8 Augmented Systems and Matrices Chapters 7,8,13,14 2 weeks All
Course Title: Honors Algebra 2 Grade Level: 10 th & 11 th Overarching Essential Questions What are Functions? What are linear systems and how are they used? What are matrices and how are they used in life? What is a quadratic equation and what is its function? What are radical expressions and functions? What are logarithmic and exponential functions? What does it mean to be periodic? What is trigonometry and how is it used? What are the trig identities? Can we solve trig identities using methods on both sides of the equation to solve? What is a permutation and what is a combination? Overarching Enduring Understanding Students in Honors Algebra 2 will learn about functions, linear systems, and
matrices and how they are used in everyday life. They will also become fluent with quadratics, polynomials, radical expressions, logarithmic and exponential functions, as well as the basics in trigonometry. They will also begin to explore concepts that will propel them to the next sequence which is Honors Pre-Calculus Course Description Honors Algebra 2 is a continuation of the skills learned in Algebra I. This course will cover such topics as linear and quadratic functions, quadratic relations, linear systems, and powers and roots. The course will also explore areas such as trigonometry, exponents, and logarithms. Graphing calculators will be used extensively in this course. This course is designed for the college bound student who intends to attend a 4-year college and/or a STEM career. The students selected for this class will be in the top 10% mathematically in their respective graduating class. Tools of Algebra (Chapter 1) Essential Questions How can you use the properties of real numbers to simplify algebraic expressions? How do you solve an equation or inequality? How do you solve an absolute values equation or inequality? Can you use absolute value inequalities to create a word problem that warrants an absolute value graph? Key Terms Variable, properties, equation, absolute value, inequality Objectives Students will be able to: Graph and order real numbers To identify properties of real numbers To solve equations To solve problems by writing equations To write and solve absolute value equations and inequalities Work with equations that solve profit and loss problems Standards associate with objectives MA.N.RN.3 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. MA.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. MA.A.CED.4 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.
MA.A.SSE.1.b Interpret expressions that represent a quantity in terms of its context. 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. Suggested Lesson Activities Show uses for Venn diagrams (page 5) of properties of real numbers Work with solving equations with an Algebra 1 review Show solving inequalities with a graph (example 4, page 28) Students work in groups of 2 and solve each other s absolute value graph Differentiation /Customizing learning (strategies) Work in groups and have students use a Venn diagram to show the examples of real numbers Allow students to create their own word bank using real numbers Students need to explore further activities of absolute value inequalities that translating to graphing Functions, Equations and Graphs (Chapter 2) Essential Questions What is a function? How is slope used and can you put it into linear form? What is the difference between dependent and independent variables? What is point slope form? What types of graphs are produced by absolute value equations? Can you find the vertex for an absolute value equation When and how would you shade a two variable inequality? Use the stat-plot function to see what type of equation certain domains and ranges fall into Key Terms Function, slope, point-slope, dependent and independent variables, vertex, shading, domain, range Objectives Students will be able to: Graph a function with and without a calculator Find slope of a given line Work with independent and dependent variables Graph an absolute value inequality Shade a 2 variable inequality Explain why certain inequalities have a limited domain and/or range Standards associate with objectives MA.A.CED.2- Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MA.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. MA.F.IF.7.b - Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. MA.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MA.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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 explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Suggested Lesson Activities Work with problems from page 63 (all examples) for linear equations Use the graphing application or Geometer Sketchpad to show and calculate slope Show how we find the vertex of an absolute value equations Show domains and ranges of all graphs Work with shading on an absolute value inequality Use graphs to match up functions with a given equation Differentiation /Customizing learning (strategies) Allow students to work in groups WITHOUT a calculator and graph different absolute value equations. Some students may want to work with negative coefficients, while other may not be ready for this. Have students work on a short essay to explain the differences in absolute value inequalities. Students can work collaboratively working on practice graphs they have made up for one another and match a practice multiple choice to them Use on-line video tutor to show additional ways to translate equations from vertex to vertex Linear System (Chapter 3) Essential Questions Name and show how to use the methods for solving systems of equations What is the best way to check your solutions to a given set of equation? When graphing a set of lines, how close is it feasible to get to a given solution? Using the substitution method, can we also use the row reduction method in conjunction, and how is this done? Where can we use row reduction in real-life and explain your process of solving? Show how to solve a 3 x 3 set of equations and use 2 methods to do so? Compare and contrast the many methods of solving equations and determine a checklist of advantages and disadvantages Key Terms Coefficients, row reduction, elimination, graphing, additive inverses, linear equations, non-
linear equations Objectives Students will be able to: Solve sets of 2 x 2 equations using the row reduction method Solve sets of equations using both the substitution method and row reduction Work with sets of equations using the graphing method WITHOUT a calculator Solve sets of 3 x 3 equations using the elimination method Use the graphing method to find the point of intersection given 3 equations with 3 unknowns Use technology to solve a 3 x 3 equation set Differentiate between multiple methods of solving equations and develop criteria for their usage Standards associate with objectives MA.A.CED.2- Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MA.A.CED.3- 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. MA.A.REI.5- 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 MA.A.REI.6- Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. MA.A.REI.11- 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. Suggested Lesson Activities Work with numerous examples on page 128 to solve sets of equations Put example 17 on the board and use 3 methods to solve for a solution Discuss inconsistent sets of equations Page 138 (57-62) do higher level forms of equations with non whole number coefficients Use technology to solve for solutions to systems of equations graphically Differentiation /Customizing learning (strategies) Allow students of advanced abilities to work with non-whole number coefficients and use the word problems from section 3.4 to work on Use group work as a tool to let students of varying abilities to learn in a cooperative environment
Allow students that are struggling to work on their algebraic skills by giving reinforcement problems for homework Collaboratively have students work on systems of equations with infinitely many solutions and no solutions at all Matrices (Chapter 4) Essential Questions How do matrices coincide to linear equations and explain their uses? Do matrices have a correlation to our modern computers? What is the real-world application of matrices Explain how we use matrix inverses to solve systems of equations Find the determinant of a given matrix and expand it to an inverse Can you think of a word problems where matrices would be useful? Use augmented matrices and Cramer s rule to discuss the differences in equations Key Terms Matrix, inverse, determinant, coefficient, cramer s rule, equal matrices, non-solution matrices, consistent, inconsistent, square matrix, multiplicative identity matrix Objectives Students will be able to: Add and subtract matrices Multiply matrices of varying order Use matrix multiplication to solve systems of equations Find the determinant and inverse of a matrix Use the TI-84 to solve matrices and to find their determinant and inverse Work on varying degrees of difficulty of word problems using matrices to solve Use Augmented Matrices and Cramer s rule for descriptions of systems Standards associate with objectives MA.N.VM.6- Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. MA.N.VM.7- Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. MA.N.VM.8 Add, subtract, and multiply matrices of appropriate dimensions. MA.N.VM.9 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. MA.N.VM.10 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. MA.N.VM.12- Work with 2 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. Suggested Lesson Activities Use numerous examples from section 4.2 to work on the addition and subtraction of matrices Page 178 (10-14) show all parts of differences of matrices Example 6 on section 4.3 on the smartboard to work with variable usage Work with various problems to find whether a product is defined or not Use example 2 on page 200 to find determinants of matrices Problems (7-15) on section 4.7 for further review Problems page 225 (18-23) Differentiation /Customizing learning (strategies) Allow students of higher ability to work on word problems where matrices are the main source of finding solutions Use group work as a tool to allow students to cooperatively work on matrix multiplication Allow students to work collaboratively to solve systems of equations using various methods to check for consistency Quadratic equations and functions (Chapter 5) Essential Questions How do you model data using quadratic functions? Can you prove the quadratic equation and explain the main step involved? Explain the properties of parabolas and give examples? Which ways do parabolas transform? WITHOUT a calculator, explain how to find the vertex of a parabola? Do you know how to complete the square, show your work? When are complex numbers used? How do you solve a quadratic without the middle term? Work with complex numbers and show how the quadratic equation and the discriminant are related Key Terms Quadratic, parabola, vertex, transform, quadratic equation, complex numbers, root, nth term, function, minimum, maximum, discriminant Objectives Students will be able to Model data using a quadratic
Know what a parabola is and find its vertex Find the vertex of a parabola, with and without a calculator Know the properties of a parabola Work with parabolas of higher dimension coefficients Transform a parabola (shrink, stretch) Move a parabola or switch its direction Factor all types of quadratic expressions Use the quadratic equation to solve for solutions Work with complex numbers Solve for extraneous solutions Prove the quadratic equation using the complete the square method Standards associate with objectives MA.N.CN.2 Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. MA.A.SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients. MA.A.SSE.2 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 ). MA.A.APR.3 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. MA.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. MA.A.REI.4.b - Solve quadratic equations by inspection (e.g., for x2 = 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. MA.F.IF.4- 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. MA.F.IF.5 - 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 Suggested Lesson Activities Work with examples from page 241 (only 10-15) Show properties of parabolas on the smartboard with vertex Explain maximum and minimum values for any given parabola
Work on example 5 from book, section 5.4 and show all properties Show how complex numbers work and give quadratic examples Factor quadratics and show examples with and without solutions Page 263, all example problems to be worked on cooperatively Show the difference of two perfect squares Use square roots to solve quadratics if possible Collaboratively students may work together and fill in missing steps for the quadratic equation proof. Use practice coefficients for the quadratic equation proof and solve Differentiation /Customizing learning (strategies) Allow students of varying levels to work collaboratively on the proof of the quadratic equation Show how completing the square will help to make the proof go much more smoothly Students can spiral review and see if they can prove other formulas from passed lessons to see the relations with proofs and mathematical formulas. Polynomials and polynomial functions (Chapter 6) Essential Questions Can you write an equation in factored form? What are the zeroes of an equation? Discuss the relative maximum and minimums of a given function? Name one of the many ways to solve an equation of higher order? Explain how you would factor a perfect square trinomial? Discuss where you would find complex roots? Use the quadratic equation to work with complex numbers? What purposes does the fundamental theorem of algebra have? Use permutations and combination to solve different probability style real world problems Key Terms Zeroes, relative maximum, relative minimum, complex roots, polynomial functions, fundamental theorem of algebra, imaginary roots, permutation, combination Objectives Students will be able to: Find extraneous solutions to given equations Look for irrational roots and be able to decipher why they have irrational solutions Find a 4 th degree polynomial equation with integer coefficients Find all roots for a polynomial equation Explain how and why we use the fundamental theorem of algebra Work with complex roots of given polynomial equations Discuss the role of factorial and use probability problems to model real-world
situations Standards associate with objectives MA.N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. MA.N.CN.8 - (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x 2i). MA.N.CN.9 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. MA.A.SSE.1 - Interpret expressions that represent a quantity in terms of its context. MA.A.SSE.2 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 ). MA.A.APR.3 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. MA.A.REI.11 - 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 MA.F.IF.7.c - Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. MA.F.BF.1 - Write a function that describes a relationship between two quantities. Suggested Lesson Activities Examples on page 317 (16-28) on smart board, graph and find the zeroes Work with relative maximum and minimum of higher order equations Problem #37 on page 317, visual geometry box with Geometer Sketchpad Practice examples 4 and 5 for a classroom discussion Section 6.4, factor differences of cubes, all examples Work with the fundamental theorem of algebra to show complex roots and solutions Page 348 (21-28) and sports related probability questions Differentiation /Customizing learning (strategies) Allow students of differing abilities to work collaboratively to do complex root solutions Review the quadratic equation and do practice problems from the book with respect to complex solutions Use kinesthetic learning by bringing in a cereal box, use variables and see how close the student can come to a decimal approximation Use internet sites to discuss probability using examples when order does and does not matter
Radical functions and Rational Exponents (Chapter 7) Essential Questions Can you explain the absolute value solutions to a square root of power 4? Explain what a conjugate is and how it is used in simplifying? Make a word problem using an expression of a power less than 1. Solve for x in an equation using powers in fractional form? Explain how to expand a binomial by the 3 step process of expansion? Can you explain when you can find extraneous solutions to equations? What type of equations have inverses and why? Do inverses have a tie to extraneous solutions, and if so, why? Give the domain and range of inverse functions and explain their roots? Can you graph functions using square roots without a calculator and explain their given shift Use technology apps to graph functions with varying exponential degrees Key Terms Extraneous solution, inverse, radicand, binomial, square root equation, rational exponent, like radicals, rationalize the denominator, cubic functions Objectives Students will be able to: Discuss the differences between a square root and a principal square root Expand a given binomial in different forms Simplify radical expressions and use absolute values for given solutions Rationalize the denominator of given expressions Use conjugates to simplifying rational expressions Determine when an expression is fully simplifying using conjugates Put given expressions in different forms using a radicand Solve square root equations and other fractional root equations Solve and check for extraneous solutions Graph function with cubed roots and varying Physics applications to the formula involving our Earth s gravity Standards associate with objectives MA.N.RN.1-1. 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. MA.N.RN.2 - Rewrite expressions involving radicals and rational exponents using the properties of exponents. MA.A.SSE.2 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 ). MA.A.CED.4 - 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. MA.REI.2 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. MA.F.BF.4a - 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. MA.F.BF.4c - Read values of an inverse function from a graph or a table, given that the function has an inverse Suggested Lesson Activities Use negative exponent examples and specifically do page 372 (39-54) Expansion of binomial problems, pick examples from section 7.2, example 3 Use smart board to show how absolute value solutions are used when simplifying radical expressions Example 4, section 7.3 using conjugates to simplify expressions of fractional values Show rationalizing of a denominator and explain restrictions, page 382 Work with the 4 main methods of simplifying numbers with rational exponents (4 problems) Show examples from 7.5 on how to solve equations with 2 rational exponents Use example page 416 (example 5) to show the applications of the Earth s gravity application Differentiation /Customizing learning (strategies) Allow more advanced students to simplify IRRATIONAL expressions and rational the denominator Work with solving equations to a more advanced level such as solving solutions with conjugates and then looking for extraneous solutions Use on-line examples from the book reinforcing more basic skills when dealing with rational exponents, both positive and negative Allow the exploring of different inverses of equations and introduce the ideas of calculus and slope tangent Collaboratively work on matching graphs of varying degrees to a different set of equations
Exponential and Logarithmic functions (Chapter 8) Essential Questions What is a logarithm and how are they used to solve equations? What types of bases are used for logarithm and explain the value of e How do we go about using logs in everyday life word problems and show 3 examples of their uses Predict the graph to a logarithmic function by using values Find the asymptote of a logarithmic graph. Simplify logarithms of like bases Use the change of base formula to solve word problems of varying degrees. Solve logarithm problems by graphing and finding the intersection of the graphs Interpolate logarithmic values of equations using logarithms Work on a class project involving a word bank and flashcards for rules of logarithms as exponents Key Terms Logarithm, base e, natural logarithm. Asymptote, logarithmic functions as inverses, compound interest and its uses, graphing using tables Objectives Students will be able to: Label asymptotes of a given logarithmic graph Solve an equation with base e Translate a logarithmic graph with and without a calculator Use logarithmic functions as inverses and label the key parts of the graph Simplify logarithmic with like bases to represent as a single function Explain the properties of logarithms and use them to simplify and solve Expand logarithms without like bases and solve Use the properties of logarithms to evaluate expressions Use a table to show the values of logarithms at given points on a graph Use natural logarithms to solve application style problems Realize the varying levels of exponential functions in practice problem style questions using logarithms Standards associate with objectives MA.A.SSE.1.b Interpret expressions that represent a quantity in terms of its context. 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. MA.A.REI.11 - 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
MA.A.CED.2- Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. MA.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. MA.F.IF.7e - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. MA.F.IF.8 - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MA.F.BF.1 - Write a function that describes a relationship between two quantities. MA.F.LE.4 - 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. Suggested Lesson Activities Use example on page 41 to work with compound interest and discuss Graph logarithmic functions and their parent functions Work with logarithms of base e and solve for x Page 450 (53-61), have students show solutions on smartboard Find the domain and range of given logarithmic functions (451) Find the asymptotes of given logarithmic functions without a calculator Example 2 and 3 page 455, show simplifying logarithms Expansion of logarithms (79-84) page 458 Solve logarithmic functions like example 6, page 463 Use change of base formula to solve logarithmic functions (33-41) page 464 Differentiation /Customizing learning (strategies) Allow students to use the ph balance of given chemicals to calculate how much needs to be mixed into a given solution to find another value Use logarithmic functions of base e to discuss exponential growth and decay. Varying word problems for examples Allow students who are struggling with the material of exponents to review how positive and negative exponents are solved in equation form Use the books and videos on the smart board to reinforce logarithmic properties Periodic Functions and Trigonometry (Chapter 13) Essential Questions How do we use the trigonometric functions to solve special right traingles? Use circle measure (circumference) to find a given angle in degrees Show how the cosine functions oscillates without a calculator. Show how the sine function oscillates and compare it to cosine. Find the period and cycle number for a given trigonometric function.
Sketch one cycle of trigonometric functions using the period and cycle. Identify the range of a given cosine function How do we compare the tangent function to the sine and cosine function? Find the asymptotes of a given tangent function and explain Use a real-world exercise like a ferris wheel to find the indicated angles of a given arc, then find the arc length Key Terms Arc length, sine, cosine, tangent, circumference, cycle, amplitude, period, cycles, asymptote, radians, degrees Objectives Students will be able to: Graph trigonometric functions using period, amplitude, and cycles. Know the unit circles and use reference angles to solve for functions Know the differences between terminal and co-terminal angles. Work between radians and degrees Find the length of a given arc in degrees and radians using s=rα Use different graphs of the sine function to work with the period and amplitude Solve for given values in a cosine function Work with the graph of tangent and know where the asymptotes are Describe why a trigonometric function is undefined Standards associate with objectives MA.F.IF.4- 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. MA.F.IF.7e - Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. MA.F.TF.1 - Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. MA.F.TF.2 - Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. MA.F.TF.5 - Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline Suggested Lesson Activities Using geometer sketchpad 4.0, show examples of graphs of sine functions and show visually their number of cycles Work with examples on page 721 to find exact values of special trigonometric angles Show different examples on smart board calculator like page 725 (1-15) to show
radian measure Use geometer sketchpad to measure arc length, radian measure, and radius of any given circle Page 731 Examples (31-42) to determine the sign of a given trigonometric function Show how to interpolate values of a sine function at given points on a graph Write an equation of a function using a given graph Show using the unit circle where tangent functions are undefined, and find its asymptotes Differentiation /Customizing learning (strategies) Allow students to work with pictures of graphs of non whole coefficients and determine the properties of a trigonometric function Have students review the unit circle and see if they can interpolate values for non special angles Work with real-world word problems to show how trigonometric functions are used and how they will have more than one answer Students who are struggling should review the unit circle and learn to use the TI-84 to aid in their knowledge of trigonometry Reinforce oscillations using geometer sketchpad and show the period and amplitude of basic trigonometric functions. Trigonometric Identities and Equations (Chapter 14) Essential Questions Prove the 3 basic trigonometric identities given the Pythagorean identity Can you simplify trigonometric expressions to its simplest form Explain how you would go about solving a complex trigonometric equation for a given variable Identify the different quadrants tangent function solutions and why Use the law of sines to solve for a given triangle with two sides and an angle Graph the interval from 0 to 2π for a given cosine function Use the law of cosines to interpolate given angle measure Discuss how the area of a triangle can be calculated given 2 sides of an oblique triangle and angle measure in degree Key Terms Theta, law of cosines, law of sines, oblique, co-terminal, terminal, oscillations, Pythagorean trigonometric identity Objectives Students will be able to: Show the oscillation of both the sine and cosine function Use the Pythagorean identity to show variations of its use Discuss the methods used to solve trigonometric identities Use interval oscillations to solve tangent functions
Solve lengths and angles of a triangle using the Law of Sines and Law of Cosines Simplify trigonometric functions to their simplest form Work with equations using trigonometric functions and real numbers Standards associate with objectives MA.F.TF.8 - Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. MA.G.SRT.6 - Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. MA.G.SRT.8 - Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. MA.G.SRT.9 - Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. MA.G.SRT.10 - Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MA.G.SRT.11 - Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Suggested Lesson Activities Use examples 2 and 3, page 780 to analyze trigonometric identities Work with numerous examples on simplifying a given expression Solve 2 sided trigonometric identities and show how the Pythagorean identity is useful Problems on age 787 (16-30) on smartboard, with solutions Finding complete solutions in radians of each equation, page 788, all sample exercises Examples on page 802 using the law of sines, show how they work in degree measure, as well as radians Page 811, examples (24-29) as a review for a formative assessment Differentiation /Customizing learning (strategies) Allow students to make up their own trigonometric identities and have each other try to solve them Use the law of sines in conjunction with intricate word problems dealing with oblique triangles Work with the law of cosines and the law of sines to check all angles and side length Students that are struggling with the material must work on the original trigonometric identities and reinforce their previous skills