Grade 11 Supplemental Math, Curriculum Outline

Grade 11 Supplemental Math, Curriculum Outline

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]

Grade 11 Supplemental Math, page 2–1

Precalculus

Unit 1: Fundamentals

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
1.a Perform basic algebraic operations on exponential expressions, radical expressions, and polynomial expressions.
(4.1-B2, B4,
4.3-A3, D1, D3) / 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”? / 12-4 and 12-3:
[Stewart] Section 1.2 (Exponents and Radicals),
[Stewart] 1.3 (Algebraic Expression)
1.b Recognize rational expressions and perform basic algebraic operations on rational expressions.
(4.1-B1
4.3-A3, D1, D2, D3) / 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?

/ 12-4 and 12-3:
[Stewart] Section 1.4 (Rational Expressions)
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? / 12-4 and 12-3:
[Stewart] Section 1.5 (Equations)

Unit 2: Functions

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
2.a Recognize and evaluate functions.
(4.3-A3, B1, B2, D3) / What is a function? What is the domain and range of a function? When is a variable independent? When is it dependent?
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: the same as. 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). / 12-4:
[Stewart] Section 2.1 (What is a function?)
12-3:
[FST] 2-1 (The Language of Functions)
2.b Graph functions using a graphing calculator.
(4.3-A3, B1, B2, D3) / 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?
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? / 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)
2.c Apply transformations to a given function.
(4.3-B3) / What is a transformation of a function? What different types of transformations are there? What is an odd function? What is an even function?
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 shift the graph of to the right and not to the left as one might expect? / 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

2.d Combine functions to create new functions and identify their resulting domains.
(4.3-B4) / What are the algebraic properties of functions? What is a composite function?
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. / 12-4:
[Stewart] Section 2.7 (Combining Functions)
12-3:
[FST] 3-7 (Composition of Functions)
2.e Identify one-to-one functions and determine their corresponding inverses.
(4.1-B4) / 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. / 12-4:
[Stewart] Section 2.8 (One-to-One Functions and their Inverses)
12-3:
[FST] 3-8 (Inverse Functions)

Unit 3: Polynomial and Rational Functions

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
3.a Identify the key attributes (e.g. degree, zeros, extrema, etc.) of a polynomial function and its graph
(4.3-A3, B5) / 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. / 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)
3.b Divide polynomials using both long and synthetic division.
(4.3-A3, D1) / 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? / 12-4:
[Stewart] Section 3.2 (Dividing Polynomials)
12-3:
[FST] 9.4 (Division and the Remainder Theorem),
[FST] 9.5 (The Factor Theorem)
3.c Find the zeros of polynomials using the Rational Zero Theorem and Descartes Rule of Signs.
(4.3-A3, D2) / 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? / 12-4 and 12-3:
[Stewart] Section 3.3 (Real Zeros of Polynomials)
Unit 3: Polynomial andRational Functions - continued
3.d Perform basic algebraic operations on complex numbers and find complex solutions to quadratic equations.
(4.3-A3) / What is a complex number?
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:
What is the value of? What is the value of?
How do you determine whether a quadratic equation has complex solutions? / 12-4:
[Stewart] Section 3.4 (Complex Numbers)
12-3:
[FST] 9.6 (Complex Numbers)
3.e Apply the Fundamental Theorem of Algebra to finding the zeros of a polynomial.
(4.3-D2) / 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?
The Zeros Theorem asserts that has two zeros, and yet the graph of only intersects the x-axis once. How do you reconcile this apparent conflict? / 12-4:
[Stewart] 3.5 (Complex Zeros and the Fundamental Theorem of Algebra)
12-3:
[FST] 9.7 (The Fundamental Theorem of Algebra)
3.f Identify the key attributes (e.g. asymptotes, intercepts, end behavior, etc.) of a rational function and its graph.
(12-4 only)
(4.3-B5) / 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] 3.6 (Rational Functions)

Unit 4: Exponential and Log Functions

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
4.a Recognize, evaluate, graph and apply transformations to exponential functions.
(4.3-B5) / 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?) / 12-4:
[Stewart] Section 4.1 (Exponential Functions)
12-3:
[FST] 4.3 (Exponential Functions, note: natural exponentials covered in FST 4.6)
4.b Recognize, evaluate, graph and apply transformations to logarithmic functions and convert logarithmic functions to exponential functions (and vice versa).
(4.3-B5) / 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? / 12-4:
[Stewart] Section 4.2 (Logarithmic Functions)
12-3:
[FST] 4.5 (Logarithmic Functions)
[FST] 4.6 (e and Natural Logarithms)
4.c Manipulate (i.e. expand or combine) and evaluate logarithmic expressions using the laws of logarithms.
(4.3-D1) / 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? / 12-4:
[Stewart] Section 4.3 (Laws of Logarithms)
12-3:
[FST] 4.7 (Properties of Logarithms)
Unit 4: Exponential and Log Functions - continued
4.d Solve exponential and logarithmic equations.
(4.3-A3, B1, B4, D3) / 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. / 12-4:
[Stewart] Section 4.4 (Exponential and Logarithmic Equations)
12-3:
[FST] 4.8 (Solving Exponential Equations)
4.e Apply exponential and logarithmic functions to real-life situations.
(4.3-A3, B1, B2, B4, D3) / 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? / 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)

Unit 5: Trigonometric Ratios and Functions

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
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) / 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. / 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)
5.b Find the trigonometric ratio of an acute angle inside a right triangle.
(4.3-B4, D1, D2) / 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. / 12-4:
[Stewart] Section 6.2 (Trigonometry of Right Triangles)
12-3:
[FST] 5.3 (Trigonometric Ratios of Acute Angles)
5.c Find the value of the trigonometric function of an angle (of any size).
(4.3-B4, D1, D2) / 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. / 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)
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) / 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? / 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)
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) / 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? / 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)
5.f Use the Law of Sines and the Law of Cosines to solve triangles and triangle applications.
(4.3-B4, D1, D2) / 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? / 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)

Unit 6: Analytic Trigonometry