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Course Title: Precalculus, Level 4 and Level 3 Grade: 11 or 12 Length of Course: One Year (5 credits) Prerequisites: Algebra, Geometry, Algebra 2 Description: This Precalculus course aims at preparing students for success in college-level Calculus. To succeed in Calculus, students must first and foremost acquire a thorough understanding of functions – particularly the properties, behavior and manipulation of important functions such as polynomial, exponential, logarithmic and trigonometric functions. Beyond functions, students must also then have a firm understanding of analytic trigonometry, of sequences and series and of limits. As such, this course focuses solely on the in-depth treatment of these topics and these topics only, as their mastery is considered critical for success in Calculus. For the level 3 track, a unit on Probability and Counting (in lieu of the more rigorous treatment of limits prescribed in the curriculum) is included to prepare students for college-level Statistics. This course strives to give students a proper balance between the comprehension of key concepts and the mastery of skills. With that in mind, this curriculum guide clearly defines the learning objectives for each unit in terms of the key skills and key concepts that must be mastered within each unit. Evaluation: Student performance will be measured using a variety of instructor-specific quizzes and chapter tests as well as a common departmental Midterm and Final Exam. Assessments will equally emphasize measurement of the degree to which required skills have been mastered as well as how well key concepts have been understood. Scope and Sequence: Unit sequencing is designed to allow a Level 4 student to seamlessly transfer to Level 3 without gaps in coverage. A pacing guide for both Level 4 and Level 3 is attached. Text: Level 4: Precalculus Mathematics for Calculus 5th Edition, James Stewart, Lothar Redlin and Saleem Watson, Thomson Brooks/Cole 2006 [Stewart] Level 3: Functions, Statistics and Trigonometry, Rubenstein, Schultz, etal. Scott, Foresman and Company 1992 [FST] Reference Texts: Advanced Mathematics, Precalculus with Discrete Mathematics and Data Analysis, Richard G. Brown, Houghton Mifflin Company 1992 [Brown] Precalculus with Trigonometry, Concepts and Applications, Paul A. Foerster, Key Curriculum Press 2003 [Foerster] Precalculus with Limits: A Graphical Approach 4th Edition, Larson, McDougal Littell [Larson] Precalculus Unit 1: Fundamentals Learning Objectives The student will … 1.a Perform basic algebraic operations on exponential expressions, radical expressions, and polynomial expressions. (4.1-B2, B4, 4.3-A3, D1, D3) 1.b Recognize rational expressions and perform basic algebraic operations on rational expressions. (4.1-B1 4.3-A3, D1, D2, D3) Content Outline Key Definitions, Skills and Concepts What is an exponent? What is a radical? What is a rational exponent? What is a polynomial? Skills check, ability to: Simplify expressions with positive, negative and fractional exponents Simplify expressions with roots of degree 2 or higher Switch back and forth from radical notation to fractional exponent notation Express numbers in scientific notation Add, subtract and multiply polynomials Factor polynomials using a variety of techniques, such as factoring formulas, trial and error or factoring a common monomial Concept check: How can our understanding of exponent notation be used to prove each of the “exponent rules”? What is a rational expression? Skills check, ability to: Simplify rational expressions by canceling common factors from both numerator and denominator Multiply, divide, add and subtract rational expressions Simplify compound fractions Rationalize numerator or denominator using conjugate radical Concept check: How is simple fractional arithmetic similar to manipulating rational expressions? Are the following statements correct? If not, why not? Instructional Materials 12-4 and 12-3: [Stewart] Section 1.2 (Exponents and Radicals), [Stewart] 1.3 (Algebraic Expression) 12-4 and 12-3: [Stewart] Section 1.4 (Rational Expressions) a2 b2 a b a 1 b 1 (a b) 1 1.c Solve linear and quadratic equations. (4.1-A3, B1 4.3-A3, D2, D3) What is a linear equation? What is a quadratic equation? Skills check, ability to: Solve a variety of linear equations Solve multivariate equations for a given variable Solve quadratic equations by factoring, completing the square and the quadratic formula Solve equations with rational expressions and radicals Concept check: How do you know when something is a solution to an equation? Derive the quadratic formula. Why is the discriminant important? Precalculus, page 2 12-4 and 12-3: [Stewart] Section 1.5 (Equations) Precalculus Unit 2: Functions Learning Objectives The student will … Content Outline Key Definitions, Skills and Concepts 2.a Recognize and evaluate functions. What is a function? What is the domain and range of a function? When is a variable independent? When is it dependent? (4.3-A3, B1, B2, D3) Skills check, ability to: Evaluate functions (including piecewise defined functions) Concept check: What does it mean to say that f is a function of x? True or false: f (x ) the same as f x . Explain why. Give examples of functions in real life. Explain why your examples are functions. Represent functions using machine diagrams and arrow diagrams. Represent a given function verbally, algebraically, graphically (visually) and numerically (i.e. using a table of values). 2.b Graph functions using a graphing calculator. What is an ordered pair? What is a graph of a function? When is a function increasing? When is a function decreasing? What is the average rate of change of a function? (4.3-A3, B1, B2, D3) Skills check, ability to: Graph a function using a graphing calculator (student should be able to enter functions in a graphing calculator, change the window size and display the corresponding table of values) Find the values of a function from a graph Find the domain and range of a function from a graph Test whether a given equation or a graph is a function (vertical line test) Graph piecewise defined functions Use a graph to find intervals where a function increases or decreases. Calculate the average rate of change of a function Concept check: Why does the vertical line test work in telling us whether an equation or a graph is or is not a function? What is the relationship between the average rate of change of a function to its slope if that function happens to be linear? What is the relationship if that function happens to be non-linear? Precalculus, page 3 Instructional Materials 12-4: [Stewart] Section 2.1 (What is a function?) 12-3: [FST] 2-1 (The Language of Functions) 12-4: [Stewart] Section 2.2 (Graphs of Functions), [Stewart] Section 2.3 (Increasing and Decreasing Functions) 12-3: [FST] 3-1 (Using an Automatic Grapher) Precalculus Learning Objectives The student will … Content Outline Key Definitions, Skills and Concepts 2.c Apply transformations to a given function. What is a transformation of a function? What different types of transformations are there? What is an odd function? What is an even function? (4.3-B3) Skills check, ability to: Recognize and graph horizontal translation or shift: i.e. graphing f(x-c) and f(x+c) from f(x) Recognize and graph vertical translation or shift: i.e. graphing f(x) – c and f(x) + c from f(x) Recognize and graph reflections: i.e. graphing f(-x) and –f(x) Recognize and graph vertical and horizontal stretching and shrinking: i.e. graphing cf(x) and f(cx) Recognize when functions are symmetric and the type of symmetry that they exhibit (even or odd) Concept check: If c > 0, why does f ( x c) shift the graph of f (x ) to the right and not to the left as one might expect? Instructional Materials 12-4: [Stewart] Section 2.4 (Transformations of Functions) 12-3: [FST] 3-2 (The Graph Translation Theorem) [FST] 3-4 (Symmetries of Graphs) [FST] 3-5 (The Graph Scale Change Theorem) Unit 2: Functions – continued What are the algebraic properties of functions? What is a composite function? 2.d Combine functions to create new functions and identify their resulting domains. (4.3-B4) 2.e Identify one-to-one functions and determine their corresponding inverses. (4.1-B4) Skills check, ability to: Perform addition, subtraction, division and multiplication of functions (algebraically and graphically) Determine the domain of the resulting combined functions Find composite functions and their corresponding domains Concept check: Why is the domain of the combined function the intersection of each respective function’s domain when functions are added, subtracted or multiplied? Explain how one finds the domain of a composite function. Why is the domain of f(g(x) not necessarily the same as the domain of g(f(x))? Give an example. What is a one-to-one function? What is the definition of an inverse of a function? What is the inverse function property? Skills check, ability to: Test for whether a given function is one-to-one and the existence of an inverse function (horizontal line test) Verify whether two functions are inverses Find the inverse of a one-to-one function (algebraic, graphical, and numerical method) and its corresponding domain and range Concept check: Why does a function that is not one-to-one not have an inverse? Explain how one finds an inverse. Precalculus, page 4 12-4: [Stewart] Section 2.7 (Combining Functions) 12-3: [FST] 3-7 (Composition of Functions) 12-4: [Stewart] Section 2.8 (One-to-One Functions and their Inverses) 12-3: [FST] 3-8 (Inverse Functions) Precalculus Unit 3: Polynomial and Rational Functions Learning Objectives The student will … 3.a Identify the key attributes (e.g. degree, zeros, extrema, etc.) of a polynomial function and its graph (4.3-A3, B5) 3.b Divide polynomials using both long and synthetic division. (4.3-A3, D1) 3.c Find the zeros of polynomials using the Rational Zero Theorem and Descartes Rule of Signs. (4.3-A3, D2) Content Outline Key Definitions, Skills and Concepts What is a polynomial function? What is a zero (in the context of a polynomial function)? What are extrema? What is a local minimum? What is a local maximum? What is meant by end behavior? Skills check, ability to: State the degree of a polynomial Find the end behaviors of polynomials Find the zeros of a polynomial Sketch polynomials using the zeros and a table of values Use a graphing tool to find critical points such as local extrema and zeros Concept check: Describe how you go about finding the end behavior of a polynomial function. What effect does a “multiple zero” have on the graph of a polynomial? Consider a polynomial with 3 real zeros, how many local extrema do you expect to find. Explain why. What is polynomial long division? What is synthetic division? Skills check, ability to: Divide polynomials using long and synthetic division Find the value of a polynomial using the remainder theorem Create a polynomial with a given set of zeros Concept check: What do the remainder and factor theorems say and how are they useful in helping us find zeros? What are the Rational Zeros Theorem and Descartes Rule of Signs? Skills check, ability to: Use the Rational Zero Theorem and Descartes Rule of Signs to find the zeros of a polynomial Concept check: Explain how the Rational Zero Theorem works. Why is the Rational Zero Theorem primarily concerned with finding the factors of the leading coefficient and the constant term? What are all the tools that we can use to find real zeros of polynomials? Precalculus, page 5 Instructional Materials 12-4: [Stewart] Section 3.1 (Polynomial Functions and Their Graphs) 12-3: [FST] 9.3 (Graphs of Polynomial Functions) [FST] 9.5 (The Factor Theorem) 12-4: [Stewart] Section 3.2 (Dividing Polynomials) 12-3: [FST] 9.4 (Division and the Remainder Theorem), [FST] 9.5 (The Factor Theorem) 12-4 and 12-3: [Stewart] Section 3.3 (Real Zeros of Polynomials) Precalculus Learning Objectives The student will … Unit 3: Polynomial and Rational Functions - continued 3.d Perform basic algebraic operations on complex numbers and find complex solutions to quadratic equations. (4.3-A3) 3.e Apply the Fundamental Theorem of Algebra to finding the zeros of a polynomial. (4.3-D2) Content Outline Key Definitions, Skills and Concepts What is a complex number? 12-3: [FST] 9.6 (Complex Numbers) 2 What is the value of i ? What is the value of i ? How do you determine whether a quadratic equation has complex solutions? What is the Fundamental Theorem of Algebra? Skills check, ability to: Find all (including complex) zeros of a polynomial Use the Conjugate Zeros Theorem to find some roots of a polynomial Concept Check: What does the Fundamental Theorem of Algebra say? What does the Zeros Theorem say? In your own words, explain why the Zeros Theorem is true if you assume that the Fundamental Theorem of Algebra is true? 2 (4.3-B5) 12-4: [Stewart] Section 3.4 (Complex Numbers) Skills check, ability to: Recognize complex numbers and their parts Add, subtract, multiply and divide complex numbers Simplify expressions with square roots of negative numbers Find complex solutions to quadratic equations Concept Check: The Zeros Theorem asserts that y x has two zeros, and yet the graph of once. How do you reconcile this apparent conflict? 3.f Identify the key attributes (e.g. asymptotes, intercepts, end behavior, etc.) of a rational function and its graph. (12-4 only) Instructional Materials Precalculus, page 6 12-3: [FST] 9.7 (The Fundamental Theorem of Algebra) y x 2 only intersects the x-axis What is a rational function? What is an asymptote? What is a vertical asymptote? What is a horizontal asymptote? What is a slant asymptote? Skills check, ability to: Use transformation rules to sketch graphs of simple rational functions Sketch rational functions by finding intercepts and asymptotes Find slant asymptotes by dividing the polynomials Concept Check: What accounts for the vertical asymptotes in a rational function? How does dividing a rational function give an equation for its end behavior? 12-4: [Stewart] 3.5 (Complex Zeros and the Fundamental Theorem of Algebra) 12-4: [Stewart] 3.6 (Rational Functions) Precalculus Unit 4: Exponential and Log Functions Learning Objectives The student will … 4.a Recognize, evaluate, graph and apply transformations to exponential functions. (4.3-B5) 4.b Recognize, evaluate, graph and apply transformations to logarithmic functions and convert logarithmic functions to exponential functions (and vice versa). (4.3-B5) 4.c Manipulate (i.e. expand or combine) and evaluate logarithmic expressions using the laws of logarithms. (4.3-D1) Content Outline Key Definitions, Skills and Concepts What is an exponential function? What is a natural exponential function? Skills check, ability to: Express an exponential function in standard form (y=a^x used in Stewart or y=a*b^x used in other texts) Evaluate exponential functions (including natural exponential functions) Graph exponential functions (including natural exponential functions) Identify and distinguish graphs of exponential functions. Apply transformations of exponential functions Concept check: What distinguishes an exponential function from a linear function? Give a verbal representation of an exponential function. What is the number e and when is it used? (Or, what is so natural about the number e?) What is a logarithmic function? What is a common logarithm? What is a natural logarithm? Skills check, ability to: Switch back and forth from logarithmic to exponential expressions. Evaluate logarithms using basic properties of logarithms Graph logarithmic functions Apply transformations of logarithmic functions (reflections, vertical translation and horizontal translation) Evaluate common logarithms Evaluate natural logarithms Find the domain of a logarithmic function Concept check: How are logarithmic functions related to exponential functions? Why is the domain of a logarithmic function restricted? What are the laws of logarithms? Skills check, ability to: Use the laws of logarithms to evaluate logarithmic expressions Expand and combine logarithmic expressions Evaluate logarithms using the change of base formula Use the change of base formula to graph a logarithmic function. Concept check: How do the laws of exponents give rise to the laws of logarithms? Precalculus, page 7 Instructional Materials 12-4: [Stewart] Section 4.1 (Exponential Functions) 12-3: [FST] 4.3 (Exponential Functions, note: natural exponentials covered in FST 4.6) 12-4: [Stewart] Section 4.2 (Logarithmic Functions) 12-3: [FST] 4.5 (Logarithmic Functions) [FST] 4.6 (e and Natural Logarithms) 12-4: [Stewart] Section 4.3 (Laws of Logarithms) 12-3: [FST] 4.7 (Properties of Logarithms) Precalculus Learning Objectives The student will … Unit 4: Exponential and Log Functions - continued 4.d Solve exponential and logarithmic equations. (4.3-A3, B1, B4, D3) 4.e Apply exponential and logarithmic functions to reallife situations. (4.3-A3, B1, B2, B4, D3) Content Outline Key Definitions, Skills and Concepts What is an exponential equation? What is a logarithmic equation? Skills check, ability to: Solve equations that involve variables in the exponent (algebraically and graphically) Solve equations that involve logarithms of a variable (algebraically and graphically). Solve more complicated compound interest problems (e.g. finding the term for an investment to double) Concept check: Why are logarithms useful in solving exponential equations? Describe the steps involved in solving a typical logarithmic equation. What is an exponential growth model? What is an exponential decay model? What are logarithmic scales? Skills check, ability to: Apply exponential growth models to real life situations: e.g. predicting the future (and past) size of a population growing exponentially Apply exponential decay models to real life situations: e.g. calculating the amount of mass remaining of a radioactively decaying substance after t units of time. Convert relative magnitudes measured in logarithmic scales to relative magnitudes measured in linear scales Concept check: What does it mean for something to grow or decay exponentially? How do we know we can use an exponential growth or decay function to model physical phenomena? Can a half-life decay model be alternatively expressed using a different decay factor? Give an example. Why are logarithmic scales useful? Precalculus, page 8 Instructional Materials 12-4: [Stewart] Section 4.4 (Exponential and Logarithmic Equations) 12-3: [FST] 4.8 (Solving Exponential Equations) 12-4: [Stewart] Section 4.5 (Modeling with Exponential and Logarithmic Functions) 12-3: [FST] 4.4 (Finding Exponential Models) [FST] 4.9 (Exponential and Logarithmic Modeling) Precalculus Unit 5: Trigonometric Ratios and Functions Learning Objectives The student will … 5.a Use radian measure to measure the size of an angle (or amount of rotation) and convert between degree and radian measure. (4.3-B4, D1, D2) 5.b Find the trigonometric ratio of an acute angle inside a right triangle. (4.3-B4, D1, D2) 5.c Find the value of the trigonometric function of an angle (of any size). (4.3-B4, D1, D2) Content Outline Key Definitions, Skills and Concepts What are radian and degree measures for angles? What are coterminal angles? What are arc length and sector areas? Skills check, ability to: Convert between degree and radian measure Find coterminal angles Find arc length and sector areas Concept check: Using a piece of string, demonstrate how to create an angle of measure 1, 2 and 3 radians on a circle. Hint: how is the radius of circle related to its circumference? In your own words, explain the concept of radian measure. Hint: Think about a circle of radius 1. Derive the formulas for the arc length and sector area when using an angle in radian measure. What are the six right triangle trigonometric ratios? What are special triangles? Skills check, ability to: Find exact values of the trigonometric ratios when given two lengths of a right triangle Use the trigonometric ratios to solve right triangles Find the trigonometric values of special right triangles (45-45-90 and 30-60-90) without the use of a calculator Use the inverse function on the calculator to solve for angles in applications problems Concept check: Justify why trigonometric ratios within a right triangle makes sense using the geometric theorem of similarity. What are the six trigonometric ratios as defined when the angle is placed in standard position? What is a reference angle? What are the basic trig identities? Skills check, ability to: Find reference angle for any angle in standard position Find the exact value of any special angle, including nonacute special angles Determine in what quadrant an angle must lie given the signs of the trigonometric functions Find the exact values of the trig functions when given one of the values Find the area of a triangle using the SAS formula Concept check: To determine sin 150, sin 210, sin 330 and sin 570, I only need to know the value of sin 30. Is this true or false and why? Prove the Pythagorean theorem. Precalculus, page 9 Instructional Materials 12-4: [Stewart] Section 6.1 (Angle Measure) 12-3: [FST] 5.1 (Measures of Angles and Rotations), [FST] 5.2 (Lengths of Arcs and Areas of Sectors) 12-4: [Stewart] Section 6.2 (Trigonometry of Right Triangles) 12-3: [FST] 5.3 (Trigonometric Ratios of Acute Angles) 12-4: [Stewart] Section 6.3 (Trigonometric Functions of Angles) 12-3: [FST] 5.4 (The Sine, Cosine and Tangent Functions), [FST] 5.5 (Exact Values of Trigonometric Functions) Precalculus Learning Objectives The student will … Unit 5: Trigonometric Ratios and Functions - continued 5.d Use the unit circle to find the trigonometric ratios of a given angle (of any size) and find the terminal point of a given rotation around the unit circle. (4.3-B4, D1, D2) 5.e Recognize and sketch graphs of trigonometric functions and identify their key attributes (e.g. amplitude, period, horizontal translation, etc.). (4.3-B4, D1, D2) 5.f Use the Law of Sines and the Law of Cosines to solve triangles and triangle applications. (4.3-B4, D1, D2) Content Outline Key Definitions, Skills and Concepts What is the unit circle? What are the even/odd properties? Skills check, ability to: Use the unit circle to find the values of the six trig functions for special angles Find a terminal point on the unit circle when given a variety of information Determine whether a trigonometric function is even or odd Concept Check: How are the trig functions defined for the unit circle and how is this consistent with the definitions we have seen so far? If you are told to compare cos 77 and cos 82 and say which one is bigger without using a calculator. How would you do it? What is a periodic graph? What is meant by amplitude, period and phase shift? Skills check, ability to: Recognize the graphs of the six trigonometric functions Graph transformations of the six trigonometric functions State the amplitude, period and phase shift of a given trigonometric function Write the trigonometric function for a given graph Concept Check: Why is the graph of any trignometric function periodic? Which graphs have asymptotes and why do they exist? How do the period, amplitude and phase shift relate to our earlier studies of transforming functions? What is the Law of Sines? What is the Law of Cosines? What is the ambiguous case? Skills check, ability to: Use the Law of Sines and Law of Cosines to solve for all possible triangles when given a set of conditions Use Law of Sines and Cosines to solve problems involving bearing and direction Concept Check: How do we prove the Law of Sines and Law of Cosines? How is the Law of Cosines related to the Pythagorean theorem? Why is it possible for more than one triangle to exist when given two sides lengths and a non-included angle measure? How do you check for other possibilities? Precalculus, page 10 Instructional Materials 12-4: [Stewart] Section 5.1 (The Unit Circle), [Stewart] 5.2 (The Trigonometric Functions of Real Numbers) 12-3: [FST] 5.5 (Exact Values of Trigonometric Functions), [FST] 5.7 (Properties of Sines, Cosines and Tangents) 12-4: [Stewart] 5.3 (Trigonometric Graphs), [Stewart] 5.4 (More Trigonometric Graphs) 12-3: [FST] 5.6 (Graphs of the Sine, Cosine and Tangent Functions), [Stewart] 5.3 (Trigonometric Graphs) 12-4: [Stewart] 6.4 (The Law of Sines), [Stewart] 6.5 (The Law of Cosines) 12-3: [FST] 5.8 (The Law of Cosines), [FST] 5.9 (The Law of Sines) Precalculus Unit 6: Analytic Trigonometry Learning Objectives The student will … 6.a Recognize and prove trigonometric identities and simplify trigonometric expressions using trigonometric identities. (4.3-A3, B4, D1, D2, D3) Content Outline Key Definitions, Skills and Concepts What is a trigonometric identity? What are the following identities: reciprocal identities, Pythagorean identities, and even and odd identities? 12-4 only: What are the cofunction identitites? Skills check, ability to: Simplify trigonometric expressions using factoring, common denominators and trigonometric identities Prove trigonometric identities Concept check: When is a trigonometric equation not an identity? Give an example. True or false: 6.b Evaluate and graph inverse trigonometric functions. sin 2 x cos 2 y 1 . Give your reasoning. What is the definition of the inverse sine function? The inverse cosine function? And the inverse tangent function? 12-4 only: What is the definition of the inverse secant function? The inverse cosecant function? The inverse cotangent function? Instructional Materials 12-4: [Stewart] Section 7.1 (Trigonometric Identities) 12-3: [FST] 5-7 (Properties of Sines, Cosines, and Tangents) [FST] 6-7 (The Secant, Cosecant and Cotangent Functions) 12-4: [Stewart] Section 7.4 (Inverse Trigonometric Functions) (4.3-B4, D1, D2, D3) Skills check, ability to: Evaluate inverse sine, inverse cosine and inverse tangent expressions Graph inverse sine, inverse cosine and inverse tangent functions Evaluate trigonometric functions composed of trigonometric and inverse trigonometric functions Concept check: Why are the domains and ranges of inverse functions restricted? 6.c Solve trigonometric equations. (4.3-B4, D1, D2, D3) Skills check, ability to: Solve trigonometric equations by isolating the trigonometric function to find a single solution and using the known periodicity of the function to find other solutions Solve trigonometric equations by graphing the trigonometric function to find a single solution and using the known periodicity of the function to find other solutions Solve trigonometric equations by factoring the quadratic equivalents of the trigonometric equation Solve trigonometric equations by using trigonometric identities Concept check: Describe the steps involved in solving a trigonometric equation. Why does the calculator only give you one solution when you use an inverse trigonometric function? Why is this sometimes not the right solution? Precalculus, page 11 12-3: [FST] 6-5 (Inverse Circular Functions) 12-4: [Stewart] Section 7.5 (Trigonometric Equations) 12-3: [FST] 6-6 (Solving Trigonometric Equations) Precalculus Learning Objectives The student will … Unit 6: Analytic Trigonometry continued 6.d Use the angle addition and angle subtraction formulas for sine, cosine and tangent to evaluate trigonometric equations. (12-4 only topic) Content Outline Key Definitions, Skills and Concepts What are the angle addition and angle subtraction formulas for sine, cosine and tangent? Skills check, ability to: Use angle addition and angle subtraction formulas to evaluate trigonometric expressions Convert sums of sines and cosines into an angle sum of sine or cosine Concept check: How are the angle addition formulas derived? (optional) Instructional Materials 12-4: [Stewart] Section 7.2 (Addition and Subtraction Formulas) 12-3: Not applicable (4.3-A3, D1, D2, D3) 6.e Use the double angle and half-angle formulas for sine, cosine and tangent to evaluate trigonometric equations. (12-4 only topic) (4.3-A3, D1, D2, D3) What are the double angle and half-angle formulas for sine, cosine and tangent? Skills check, ability to: Use half-angle formulas to lower the power of a trigonometric expression Concept check: How are the angle addition, double angle and half-angle formulas related? Derive double angle formulas for sine, cosine and tangent from the corresponding angle sum formula Derive the half-angle formula for sine, cosine and tangent from the corresponding double angle formula Precalculus, page 12 12-4: [Stewart] Section 7.3 (DoubleAngle, Half-Angle, and SumProduct Formulas) 12-3: Not applicable Precalculus Unit 7: Sequences and Series Learning Objectives The student will … 7.a Recognize sequences and summation notation and express sequences in both explicit and recursive forms. (4.3-A3) 7.b Distinguish between arithmetic and geometric sequences and find their formulas and partial/infinite sums (4.3-A3) 7.c Expand binomial expressions using the Binomial Theorem Note: 12-3 students do not need to cover this topic as part of this unit, this topic is included in the unit on Probability and Counting Content Outline Key Definitions, Skills and Concepts What is a sequence? What are the terms of a sequence? What are the partial sums? What are explicit and recursive formulas? What is sigma notation? 12-3 only: What is the limit of a sequence? Skills check, ability to: Use a sequence's explicit rule to find any term of a sequence Find a sequence's explicit rule when given the terms of the sequence Use a recursive rule to find the first five terms of a sequence Write the recursive rule for a given sequence Find the limit of a sequence Find the partial sums of a sequence when given either its rule or the first few terms of the sequence Evaluate an expression using sigma notation Write a series in sigma notation Concept check: When a sequence is expressed with an explicit formula, what is the domain of that sequence? Is it possible to define a sequence both recursively and explicitly? What are arithmetic and geometric sequences? What are the formulas for the partial sums of arithmetic and geometric series? What is the formula for the sum of an infinite geometric series? Skills check, ability to: Find explicit and recursive formulas for given arithmetic and geometric sequences Find the partial sums of arithmetic and geometric series Recognize when an infinite geometric series has a sum and find it Concept check: Derive the formulas for the partial sums of arithmetic and geometric series How does the formula for the partial sum of a geometric series give rise to the formula for the sum of an infinite geometric series? What is the binomial theorem? What is Pascal's Triangle? What is factorial? What is nCr? Skills check, ability to: Expand a binomial expression using the binomial theorem and either Pascal's triangle or nCr Use the binomial theorem to find just one specified term of the expansion of a given binomial expression Factor an expression using the binomial theorem Concept check: Why is the binomial theorem an easier way to expand binomial expressions? How does one construct Pascal's Triangle and what are some of the other interesting properties of the triangle? (4.3-A3) Precalculus, page 13 Instructional Materials 12-4: [Stewart] Section 11.1 (Sequences and Summation Notation) 12-3: [FST] 8.1 (Formulas for Sequences), [FST] 8.2 (Limits of Sequences) 12-4: [Stewart] Section 11.2 (Arithmetic Sequences), [Stewart] 11.3 (Geometric Sequences) 12-3: [FST] 8.3 (Arithmetic Series), [FST} 8.4 Geometric Series), [FST} 8.5 Infinite Series 12-4: [Stewart] Section 11.6 (The Binomial Theorem) Precalculus Unit 8: Limits: A Preview of Calculus (Recommended for 12-4 only) Learning Objectives The student will … 8.a Understand and explain the concept of a limit and estimate the value of a given limit numerically and graphically. (4.3-A2) 8.b Find the value of a given limit algebraically using limit laws. (4.3-A2) 8.c Understand and explain the concept of a derivative and find the derivative of a given function using the definition of a derivative. Content Outline Key Definitions, Skills and Concepts What is the definition of a limit? What is the definition of a one-sided limit? Skills check, ability to: Estimate limits using a calculator (by way of table of values or a graph) Identify situations when the limit fails to exist Concept check: Is the calculator always a reliable way to find limits? When does a limit fail to exist and why? What are the limit laws? Skills check, ability to: Evaluate limits using limit laws Evaluate limits by algebraic simplification, by rationalization of terms and by direct substitution Evaluate limits of piecewise defined functions Concept check: Is it possible for a function to have a limit at a given point and yet not be defined at that point? Give examples. Under what conditions can you use direct substitution to evaluate a limit? What is the definition of a tangent line? What is the definition of a derivative? Skills check, ability to: Find the tangent line of a curve at a given point Find the derivative of a function at a given point Find the derivative of a function Calculate instantaneous velocity or instantaneous rate of change Concept check: In your own words, explain how the definition of a derivative describes a way to measure instantaneous slope? What is the difference between the average slope of a curve and the instantaneous slope of a curve? How are the tangent line of a curve at a given point and the derivative of a curve at a given point related to each other? What is the difference between the derivative of a function at a given point and the derivative of a function? Why can’t you use direct substitution when using the definition of a derivative to evaluate the derivative of a function? Precalculus, page 14 Instructional Materials 12-4: [Stewart] Section 12.1 (Finding Limits Numerically and Graphically) 12-4: [Stewart] Section 12.2 (Finding Limits Algebraically) 12-4: [Stewart] Section 12.3 (Tangent Lines and Derivatives) Precalculus Unit 9: Probability and Counting (Recommended for 12-3 only – to be covered if time permits) Learning Objectives The student will … 9.a Compute simple probabilities by counting the number of favorable events and the size of the sample space. (4.4-B1, B3, B4, B5, B6) 9.b Compute probabilities using addition and/or multiplication counting principles. (4.4-B1, B3, B4, B5, B6) 9.c Compute probabilities of compound events. (4.4- B4) 9.d Distinguish between permutations and combinations and compute permutations and combinations. (4.4-C1, C2, C3, C4) Content Outline Key Definitions, Skills and Concepts What is the definition of sample space? What is the definition of a favorable event? What is meant by a fair event? Instructional Materials 12-3: [FST] 7-1 (Probability of a Simple, Discrete Event) Skills check, ability to: List all the outcomes of sample space for a given probabilistic experiment List all the favorable events for a given probabilistic experiment Compute simple probabilities from knowledge of the size of favorable events and size of the sample space Concept check: Why are values of probabilities always inclusively between zero and one? Explain what is wrong, if anything, with the following statement: I flip a coin three times and it comes up head two times, the probability of getting heads must be 2/3? What is the addition counting principle? What is the multiplication counting principle? What is a mutually exclusive event? What is a complementary event? Skills check, ability to: Compute probabilities using addition counting principles Determine whether events are mutually exclusive Identify complementary events Concept check: Why does the addition counting principle require the subtraction of the intersection of the added events? What is an independent event? What is a dependent event? What is a conditional probability? 12-3: [FST] 7-2 (Addition Counting Principles) [FST] 7-3 (Multiplication Counting Principles) 12-3: [FST] 7-4 (Compound Events) Skills check, ability to: Determine whether events are independent or dependent Use Baye’s theorem to calculate conditional probabilities. Concept check: When is P(A)*P(B/A) equivalent to finding P(A)*P(B)? What is a permutation? What is a combination? Skills check, ability to: Find the number of ways to arrange or select objects when order matters Find the number of ways to arrange or select objects when order does not matter Concept check: What is the difference between a permutation and a combination? Why does the formula for combination involve an extra division by n factorial? Precalculus, page 15 12-3: [FST] 7-5 (Advanced Counting: Permutations) [FST] 8-6 (Advanced Counting: Combinations) Precalculus Learning Objectives The student will … Unit 5: Probability and Counting - continued Content Outline Key Definitions, Skills and Concepts What is Pascal’s Triangle? 12-3: [FST] 8-7 (Pascal’s Triangle) 9.e Construct Pascal’s Triangle and recognize its properties. (4.4-C4) Skills check, ability to: Locate the numerical properties represented by the pattern in Pascal’s triangle. 9.f Expand binomial expressions using the binomial theorem and determine probabilities in binomial experiments. What is the binomial theorem? What is a binomial probability? (4.4-D2) Instructional Materials Skills check, ability to: Expand binomials using “n choose r” Use the binomial theorem to solve counting problems Determine probabilities in situations involving binomial experiments Precalculus, page 16 12-3: [FST] 8-8 (Binomial Theorem) [FST] 8-9 (Binomial Probabilities) Precalculus Unit Sequencing and Pacing Timeframe Quarter 1 Quarter 2 12-4 Unit Sequencing and Pacing 12-3 Unit Sequencing and Pacing Unit 1: Fundamentals 1.a Algebraic Notation and Manipulation (w/ focus on factoring) 1.b Rational Expressions 1.c Linear and Quadratic Equations Unit 1: Fundamentals 1.a Algebraic Notation and Manipulation (w/ focus on factoring) 1.b Rational Expressions 1.c Linear and Quadratic Equations Unit 2: Functions 2.a Recognizing and evaluating functions 2.b Graphs of Functions 2.c Transformations of Functions 2.d Combinations and compositions of functions 2.e One-to-One Functions and their Inverses Unit 2: Functions 2.a Recognizing and evaluating functions 2.b Graphs of Functions 2.c Transformations of Functions 2.d Combinations and compositions of functions 2.e One-to-One Functions and their Inverses Unit 3: Polynomial Functions 3.a Attributes of Polynomial Functions and their Graphs 3.b Long and Synthetic Polynomial Division 3.c Finding Zeros of Polynomials: Rational Zero Theorem and Descartes Rule of Signs 3.d Complex Number Operations and Solutions 3.e Fundamental Theorem of Algebra 3.f Attributes of Rational Functions and their Graphs Unit 3: Polynomial Functions 3.a Attributes of Polynomial Functions and their Graphs 3.b Long and Synthetic Polynomial Division Unit 4: Exponential and Log Functions 4.a Exponential Functions 4.b Logarithmic Functions 4.c Laws of Logarithms 4.d Solving Exponential and Logarithmic Equations 4.e Modeling with Exponential and Logarithmic Functions Unit 3: Polynomial Functions 3.c Finding Zeros of Polynomials: Rational Zero Theorem and Descartes Rule of Signs 3.d Complex Number Operations and Solutions 3.e Fundamental Theorem of Algebra Unit 5: Trigonometric Ratios and Functions 5.a Angle Measure: Radians and Degrees 5.b Right Triangle Trigonometry 5.c Trigonometric Functions of Angles (of Any Size) 5.d The Unit Circle Unit 4: Exponential and Log Functions 4.a Exponential Functions 4.b Logarithmic Functions 4.c Laws of Logarithms 4.d Solving Exponential and Logarithmic Equations 4.e Modeling with Exponential and Logarithmic Functions Unit 5: Trigonometric Ratios and Functions 5.a Angle Measure: Radians and Degrees 5.b Right Triangle Trigonometry Midterm Midterm Precalculus, page 17 Precalculus Timeframe Quarter 3 12-4 Unit Sequencing and Pacing 12-3 Unit Sequencing and Pacing Unit 5: Trigonometric Ratios and Functions 5.e Graphing Trigonometric Functions 5.f Law of Sines and Law of Cosines Unit 5: Trigonometric Ratios and Functions 5.c Trigonometric Functions of Angles (of Any Size) 5.d The Unit Circle 5.e Graphing Trigonometric Functions 5.f Law of Sines and Law of Cosines Unit 6: Analytic Trigonometry 6.a Trigonometric Identities 6.b Inverse Trigonometric Functions 6.c Solving Trigonometric Equations 6.d Addition and Subtraction Formulas 6.e Double Angle and Half-Angle Formulas Quarter 4 Unit 6: Analytic Trigonometry 6.a Trigonometric Identities 6.b Inverse Trigonometric Functions 6.c Solving Trigonometric Equations Unit 7: Sequences and Series 7.a Sequences and Summation Notation 7.b Arithmetic and Geometric Sequences and Series 7.c Binomial Theorem Unit 7: Sequences and Series 7.a Sequence Formulas and Limits of a Sequence 7.b Arithmetic, Geometric and Infinite Series Unit 9: Probability and Counting (time permitting) 9.a Probability of a Simple, Discrete Event 9.b Addition and Multiplication Counting Principles 9.c Compound Events 9.d Advanced Counting: Permutations and Combinations 9.e Pascal’s Triangle 9.f Binomial Theorem and Binomial Probabilities Unit 8: Limits: A Preview of Calculus 8.a Finding Limits Numerically and Graphically 8.b Finding Limits Algebraically 8.c Instantaneous Slope and Tangent Lines Final Final Precalculus, page 18