Pre-Calculus Curriculum

September 8 – June 16

21 Century Career Ready Practice

Cycle I:

Topic:
Units 1-2.
Functions, Polynomial and Rational / Skills:

Determine if a relationship represents a function
Evaluate the value of a function
Determine the domain and range of a function
Evaluate, oven a given interval, the average change of a function
Determine intervals where a function is increasing, decreasing, and constant
Determine the inverse of a function, and whether a function is 1 to 1
Evaluate combinations of and composite functions
Determine if a function is odd, even, or neither, both graphically and algebraically
Graph the 8 basic parent functions
Graph, read, and evaluate piecewise-defined functions
Understand how transformations are represented, both in equations and in graphs
Sketch functions based off their parent functions and corresponding transformations
Identify key characteristics of parent functions, using domain, range, maxima and minima, and intervals of increasing and decreasing
Solve real-world problems using a variety of functions
Examine equations of polynomial functions to determine left and right end behavior
Identify the zeros of a polynomial function and its multiplicity
Identify the domain, range, and degree of polynomial functions
Analyze the graphs of polynomial functions with respect to turning points, zeros, and end behavior
Form polynomials from zeros and graphs
Divide polynomials with long and synthetic division
Compare and contrast the properties of real and imaginary numbers
Perform arithmetic operations on complex numbers.
Find the real and complex zeros of a polynomial
Find the domain, and the vertical and horizontal asymptotes of a rational function

/ Projected # of days:
45 (23 block)

Cycle 2:

Topic:
Units 3-4.
Exponential and Logarithmic Functions, Trigonometry / Skills:

Evaluate exponential functions
Graph exponential and logarithmic functions
Define the number e
Define the domain and range of exponential and logarithmic functions
Change exponential expressions to logarithmic expressions and vice-versa
Expand logarithmic expressions
Condense logarithmic expressions into a single expression
Use properties of logarithms and exponents
Solve problems using any base
Solve real world problems involving interest, growth, and decay
Define the six trigonometric ratios of an angle
Evaluate trigonometric ratios using triangles and/or calculators
Solve triangles (including unknown sides and/or angles) using trigonometric ratios
Convert from radians to degrees and vice-versa
Define trigonometric functions in terms of the unit circle
Prove and work with basic Pythagorean identities
Identify co-terminal and reference angles using degrees and radians
Graph the basic trigonometric functions
Graph the sine, cosine, cosecant, and secant functions
Graph transformations of these basic functions

/ Projected # of days:
45 (23 block)

Cycle 3:

Topic:
Units 5-6
Solving Trigonometric Equations and Trigonometric Applications / Skills:

State the domain and range of trigonometric functions
Define the period of a periodic function, specifically of the trigonometric functions
State the period and amplitude of these basic functions
State the vertical and phase shift of the basic trigonometric functions
Solve trigonometric equations graphically
State the complete solution to a trigonometric equation
Understand that, without a set domain, an infinite amount of angles may satisfy an equation
Define the domain and range of inverse trigonometric functions
Use inverse trigonometric notation
Solve trigonometric equations algebraically
Use the sum and difference formulas
Use the co function identities
Use double angle identities
Use power reducing identities
Use half angle identities
Use product-to-sum identities
Use sum-to-product identities
Solve oblique triangles using the Law of Sines
Solve oblique triangles using the Law of Cosines
Use area formulas (specifically Heron’s formula) to find the area of triangles
Graph a complex number in the complex plane
Find the absolute value of a complex number
Express a complex number in polar form
Perform polar operations
Find the components and magnitude of a vector
Perform scalar multiplication and operations of vectors
Find the dot product of two vectors and the angle between two vectors

/ Projected # of days:
34 (17 block)

Cycle 4:

Topic:
Units 7-8
Analytic Geometry and Systems of Equations / Skills:

Define and write the equation of an ellipse
Identify important characteristics and graph ellipses
Define and write the equation of a hyperbola
Identify important characteristics and graph hyperbolas
Graph and write the equation of a translated conic
Determine the shape of a translated conic without graphing
Define and write the equation of a parabola
Identify important characteristics and graph parabolas
Locate points in a polar coordinate system
Convert between coordinates in rectangular and polar systems
Create graphs of equations in polar coordinates
Define eccentricity of an ellipse, parabola, and a hyperbola
Solve a system of equations using elimination
Solve a system of equations using substitution
Solve systems using matrices
Add, subtract, and multiply matrices by scalars
Multiply two matrices
Define the order of a matrix
Recognize consistent and inconsistent systems
Find the inverse of a matrix
Solve nonlinear systems algebraically
Find the determinant of a matrix

/ Projected # of days:
42 (21 block)

Unit 1—Functions and Their Graphs

Goal(s)(NJCCCS and CCSS):
F.BF.1. Write a function that describes a relationship between two quantities.
F.BF.4. Find inverse functions.
F.BF.4b. Verify by composition that one function is an inverse of another.
F.BF.4c. Read values of an inverse function from a graph or table, given that the function has an inverse.
F.BF.4d. Produce an invertible function from a non-invertible function by restricting a domain.
F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in the simple cases and using technology for more complicated cases.
Essential Questions:
How are functions and their graphs related?
What are some properties and patterns of functions and their related parent functions?
How do patterns and functions help us describe data and real-world physical phenomena?
Skills/Knowledge/Understandings:

Determine if a relationship represents a function
Evaluate the value of a function
Determine the domain and range of a function
Evaluate, oven a given interval, the average change of a function
Determine intervals where a function is increasing, decreasing, and constant
Determine the inverse of a function, and whether a function is 1 to 1
Evaluate combinations of and composite functions
Determine if a function is odd, even, or neither, both graphically and algebraically
Graph the 8 basic parent functions
Graph, read, and evaluate piecewise-defined functions
Understand how transformations are represented, both in equations and in graphs
Sketch functions based off their parent functions and corresponding transformations
Identify key characteristics of parent functions, using domain, range, maxima and minima, and intervals of increasing and decreasing
Solve real-world problems using a variety of functions

Objectives:
1) Using their equations, SWBAT recognize functions and determine its domain, range, and average change over an interval
2) Using their graphs, SWBAT determine parent functions and transform graphs of parents functions
3) Using equations of multiple functions, SWBAT form combinations of functions and composite functions
4) Using its definition, SWBAT define, find, and verify inverse functions
5) Using their equations, SWBAT solve everyday problems that can be modeled using functions
Assessments:
Formative:
Daily exit slips, always including at least one question that requires students to summarize and write, in their own words, what they learned that day. / Summative: Unit 1 exam, involving functions and graphs, operations on functions, inverse functions. / Authentic:
Choose a physical phenomenon that is represented by one of the parent functions. Graph it, denote the domain and range, and find the transformations needed to go from the parent function to this function. Additionally, students need to clearly define what each variable represents in the physical world.
Literacy Connections:
Every exit slip will require students to synthesize their daily learning and write, in their own words, what they learned.
Interdisciplinary Connections:
Physics—Modeling projectile motion
Physics—Using Hooke’s Law of springs
History—Using regression models for prediction and analysis
Social Science—Predicting future US census data
Technology Integration:
TI-84 for plotting the general form of the parent functions
TI-84 for graphing functions and finding the zeros
Smart Board for showing visually how function transformations affect graphs.
Key Vocabulary:
Function
Domain
Range
Implicit Domain
Inverse Function
1-to-1
Parent Function
Composite Function
Piecewise Function
Transformations
Symmetry
Odd/even functions
Increasing/decreasing
Relative extrema
Rate of change
Useful Sites:
Functions and Domain/Range--
Piecewise Functions--
Composition of Functions--
Function Transformations--http://math.kennesaw.edu/~sellerme/sfehtml/classes/math1113/transformation.pdf
Inverse Functions--
Lesson Reviews--
Primary Documents:
Text Crosswalk:
Larson and Hostetler, Brooks/Cole, 7th edition. 2007.
Unit 1 covers pages 1-126.
Page 70 shows graphs of all 8 of the most common parent functions.
The front inside cover of the book shows graphs and properties of the 8 parent functions, as well as of transcendental functions.
A checklist summary of Unit 1 objectives is found on page 116.
An extensive set of review problems is found on pages 117-122.
Page 123 offers a sample Unit Assessment.
Page 124 offers a proof of the midpoint formula.
Pages 125-126 offer challenging problems that require extra critical thinking from students dealing with Unit 1.

*Differentiation:

Unit 2—Polynomial and Rational Functions

Goal(s)(NJCCCS and CCSS):
A.APR.6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x)+r(x)/b(x), where a(x), b(x), q(x) and r(x) are polynomials with the degree of r(x) less than the degree of b(x) using inspection, long division, or, for the more complicated examples, a computer algebra system.
A.APR.7. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication and division by a nonzero rational expression; add, subtract, multiply and divide rational expressions.
F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in the simple cases and using technology for more complicated cases.
F.IF.7d. Graph rational functions, identifying zeros when suitable factorizations are available, and showing end behavior.
N.CN.3. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
N.CN.8. Extend polynomial identities to the complex numbers. For example, rewrite as .
N.CN.9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Essential Questions:
What are common characteristics and properties of polynomials?
How do we go about efficiently graphing polynomials?
How do patterns and polynomials help us describe data and physical phenomena and solve a variety of problems?
What can asymptotes tell us about functions and their behavior?
Skills/Knowledge/Understandings:
1) Examine equations of polynomial functions to determine left and right end behavior
2) Identify the zeros of a polynomial function and its multiplicity
3) Identify the domain, range, and degree of polynomial functions
4) Analyze the graphs of polynomial functions with respect to turning points, zeros, and end behavior
5) Form polynomials from zeros and graphs
6) Divide polynomials with long and synthetic division
7) Compare and contrast the properties of real and imaginary numbers
8) Perform arithmetic operations on complex numbers
9) Find the real and complex zeros of a polynomial
10) Find the domain, and the vertical and horizontal asymptotes of a rational function
Objectives:
1) Using polynomials, SWBAT define rational expressions and divide polynomials to create rational expressions
2) Using its equation, SWBAT determine the zeros (real and complex) of a polynomial
3) Using the equations of rational functions, SWBAT find their intercepts, asymptotes, holes, domain, and range
4) Using their two components, SWBAT perform arithmetic operations on complex numbers
Assessments:
Formative:
Daily exit slips, always including at least one question that requires students to summarize and write, in their own words, what they learned that day. / Summative: Chapter 2 assessment, including end behavior, finding zeros (real and complex), complex operations, the Fundamental Theorem of Algebra, and graphing polynomials. / Authentic:
Create-a-graph. Pick a real-world situation that is modeled by a polynomial or rational function. Then graph it by hand, taking into account the end behavior, zeros, multiplicity, and number of turning points. Label the axis and clearly define what they mean in this specific context.
Literacy Connections:
Students will read an article on population modeling and how their new knowledge can help model and predict future population sizes.
Students will be able to compare and contrast properties of real and imaginary numbers.
Every exit slip will require students to synthesize their daily learning and write, in their own words, what they learned.
Interdisciplinary Connections:
Business-Manufacturing Predictions
History—Population Modeling
Science—Boyle’s Law, photosynthesis
Technology Integration:
TI-84 to graph polynomials and verify solutions
TI-84 to approximate the real zeros of a polynomial
Smart Board for visually showing horizontal and vertical asymptotes.
Key Vocabulary:
Polynomial
Degree
Zeros
Multiplicity
Higher-order polynomial
Leading coefficient
Complex numbers
Complex conjugate
Horizontal asymptote
Vertical asymptote
Real and imaginary components
Fundamental Theorem of Algebra
Rational Function
Useful Sites:
Complex numbers--
Complex numbers video--
End behavior--
Fundamental Theorem of Algebra--
How to Find Asymptotes--
Lesson Reviews--
Primary Documents:
Text Crosswalk:
Larson and Hostetler, Brooks/Cole, 7th edition. 2007.
Unit 2 covers pages 127-216.
A checklist summary of Unit 2 objectives is found on page 207.
A comprehensive set of review problems can be found on pages 208-211.
A sample Unit Assessment can be found on page 212.
Page 213 offers proofs of the Remainder and Factor Theorems.
Page 214 offers proofs of the Linear Factorization Theorem and the Factors of a Polynomial.
Pages 215-216 offer challenging problems that require extra critical thinking from students dealing with Unit 1.

*Differentiation:

Unit 3—Exponential and Logarithmic Functions

Goal(s)(NJCCCS and CCSS):
F.BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
F.IF.7.E. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F.LE.4. For exponential models, express as a logarithm the solution to where a, c, and d are numbers and the base b is 2-10, or e; evaluate the logarithm using technology.
Essential Questions:
How are exponential and logarithmic relationships related?
How are exponential and logarithmic relationships used to model, solve, and understand real world situations?
Skills/Knowledge/Understandings:

Evaluate exponential functions
Graph exponential and logarithmic functions
Define the number e
Define the domain and range of exponential and logarithmic functions
Change exponential expressions to logarithmic expressions and vice-versa
Expand logarithmic expressions
Condense logarithmic expressions into a single expression
Use properties of logarithms and exponents
Solve problems using any base
Solve real world problems involving interest, growth, and decay

Objectives:

Using properties of logarithms and exponential expressions, SWBAT solve exponential and logarithmic equations
Using exponential and logarithmic functions, SWBAT describe and model real-world scenarios mathematically
Using properties of logarithms, SWBAT simplify and evaluate logarithmic expressions
Using their domain and parent functions, SWBAT graph transformations of logarithmic and exponential functions

Assessments:
Formative:
Daily exit slips, always including at least one question that requires students to summarize and write, in their own words, what they learned that day. / Summative:
Teacher-created Chapter 3 assessment (sample can be found on textbook page 275). Multiple choice, short response, and extended response/open ended questions will all be included. Calculators allowed. / Authentic:
How much will college really cost you? Have students research the cost of a College or University that they’re considering attending, and also some basic information about student loans. They will then choose a plan and, using actual numbers and their newfound knowledge, be able to find out just how much their continued studies will cost. This will be presented as a poster.
Literacy Connections:
Authentic Assessment “True College Cost” posters will be orally and visually presented to the class, requiring the students to succinctly summarize their work and practice their public speaking skills.
Every exit slip will require students to synthesize their daily learning and write, in their own words, what they learned.
Interdisciplinary Connections:
Science—Newton’s Law of Cooling
Logistics—Spread of a rumor/virus
Social Science—Population models
Financing—Compound vs. simple interest
Technology Integration:
TI-84 for plotting exponential and logarithmic functions.
TI-84 for evaluating (using tables) and solving logarithmic and exponential functions.
Smart Board for visually showing the 5 main types of models involving exponential and logarithmic functions.
Key Vocabulary:
Logarithm
Exponential
Base
Natural logarithm
Natural base e
Continuous interest
Exponential growth/decay
Useful Sites:
Logarithmic Properties--
Main Exponential and Logarithmic Models-- http://people.richland.edu/james/lecture/m116/logs/models.html
Exponential Growth/Decay Problems-- http://dl.uncw.edu/digilib/mathematics/algebra/mat111hb/eandl/elmodels/elmodels.html
History of the Natural Base ‘e’-- http://www.math.uconn.edu/~glaz/my_articles/theenigmaticnumbere.convergence10.pdf
Lesson Reviews--
Primary Documents:
Text Crosswalk:
Larson and Hostetler, Brooks/Cole, 7th edition. 2007.
Unit 3 covers pages 217-280.
Page 257 introduces the 5 most common types of mathematical models involving exponential and logarithmic functions.
An extensive set of review problems is found on pages 271-274
Page 275 offers a sample Unit Assessment.
Pages 276-277 offers a sample Units 1-3 summative assessment.
Page 278 offers proofs of the properties of logarithms.
Pages 279-280 offer challenging problems that require extra critical thinking from students dealing with Unit 3.

Unit 4—Trigonometry