Hazlet Township Public Schools
COURSE OF STUDY
FOR
Pre-Calculus Honors
June 2010
Michael Bernstein
Anthony Petruzzi
Dara Van Pelt
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 1: The Library of Functions
BRIEF SUMMARY OF UNIT: This unit introduces the student to the library of functions. The students will be able to manipulate functions with both rigid
and non-rigid transformations. They will investigate the composition of more than one function, the inverse function, and use functions in modeling real-world
data.
LINK TO CONTENT STANDARDS:
High School: Functions
Understand the concept of a function and use function notation.
1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly
one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph
of f is the graph of the equation y = f(x).
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the
Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Interpret functions that arise in applications in terms of the context.
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.★
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.
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate
of change from a graph.★
Analyze functions using different representations.
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more
complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
1
COURSE TITLE: Pre-Calculus Honors
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
•
What are the important defining
characteristics and representations of a
function?
How are these functions used to model
real-world problems?
GUIDING QUESTIONS:
•
How do you use the slope of a line and use
it to write the equation of a line?
•
How is a graph of a function used to
determine the key elements of that
function?
•
How do you write equations and draw
graphs for the simple transformations of
functions?
•
How do you combine two functions to
form a new function?
GRADE(S): 11-12
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
•
•
•
•
•
•
•
the definition of slope
that there are different forms of the
equation of a line including slopeintercept form, standard form, and pointslope form.
the definitions of a function
the domain and range of a function
the vertical line test.
the different transformations that can be
performed on a graph
the composition of two functions.
the inverse of a function.
the definition of a one-to-one function
what the correlation of a set of data refers
to
the line of best fit
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
•
complete do now activities
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
B: STUDENTS WILL UNDERSTAND THAT:
•
What is the inverse of a function and how
do you represent it graphically and
algebraically?
•
•
How do you write equations to model realworld data?
•
•
•
•
•
•
•
the nature of the slope of a line
determines whether the line is moving in
a positive, negative, zero, or no direction.
there are multiple ways to display a
function
the vertical line test is used to determine
if a graph is a function
the domain represents the x-values and
the range represents the y-values
the horizontal line test is used to
determine whether a function has an
inverse function
a transformation changes and maintains
certain properties of a graph
two distinct functions can be composed
some functions have inverses and some
functions do not
2
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
•
a set of data may have positive, negative,
or no correlation
C: STUDENTS WILL BE ABLE TO:
•
determine the slope of a line
•
write the equation of a line in slopeintercept, point-slope, and standard forms.
•
identify the domain and range of a
function
•
determine if a set of points is a function by
applying the vertical line test
•
determine if a function has an inverse
using the horizontal line test
•
transform a graph
•
compose two functions
•
represent a function in multiple ways
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 1
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
3
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 2: Polynomial and Rational Functions
BRIEF SUMMARY OF UNIT: This unit focuses on polynomial and rational functions. The students will be able to identify key characteristics and create
graphs of quadratic and other polynomial functions. They will investigate asymptotes, intercepts and holes as they graph a rational function.
LINK TO CONTENT STANDARDS:
High School: Functions » Linear, Quadratic, & Exponential Models*
Construct and compare linear, quadratic, and exponential models and solve problems.
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors
over equal intervals.
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or
(more generally) as a polynomial function.
ct
4. For exponential models, express as a logarithm the solution to ab = d where a, c, and d are numbers and the base b is 2, 10, or e;
evaluate the logarithm using technology.
Interpret expressions for functions in terms of the situation they model.
5. Interpret the parameters in a linear or exponential function in terms of a context.
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
•
What are the important defining
characteristics and representations of a
polynomial and rational functions?
How are these functions used to model
real-world problems?
GUIDING QUESTIONS:
•
How do you sketch graphs and write
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
•
•
•
the definition of a polynomial function
what it means if a graph is continuous
the definition of a zero of the function
the Intermediate Value Theorem.
the definition of a complex number
the Fundamental Theorem of Algebra
the definition of a vertical and a
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
•
complete do now activities
•
display their work on the board
4
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
equations of quadratic functions?
•
How do you sketch the graphs of
polynomial functions?
•
How do you find the rational and real
zeros of polynomials?
•
How do you perform operations with
complex numbers?
•
How do you find the domain and
asymptotes of a rational function?
•
How do you sketch the graph of a rational
function?
•
How do you use scatter plots to model data
and make predictions?
•
horizontal asymptote
a quadratic equation can be used to better
model a scatter plot
B: STUDENTS WILL UNDERSTAND THAT:
•
•
•
•
•
•
•
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
quadratic functions can be used to model
real-world problems
the graph of a quadratic function is a
parabola
the degree of a polynomial function plays
an important role in determining other
characteristics of the polynomial
the Intermediate Value Theorem
concerns the existence of real zeros of a
polynomial function
i represents the square root of negative
one
the Fundamental Theorem of Algebra is
used to determine the number of zeros of
a polynomial function
a scatter plot can be used to give you an
idea of which type of model will best fit a
set of data
C: STUDENTS WILL BE ABLE TO:
•
write a quadratic function is standard form
and use the results to sketch the graph
•
find the minimum and maximum value of
a quadratic function in real-life
applications
•
use transformations to sketch graphs of
polynomial functions
•
use the Leading Coefficient Test to
determine the end behavior of graphs of
polynomial functions
•
apply the Intermediate Value Theorem to
help locate the zeros of a polynomial
5
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
function
•
add, subtract, and multiply complex
numbers
•
find all zeros of polynomial functions
•
find horizontal and vertical asymptotes of
graphs of rational functions
•
find the domain of a rational function
•
analyze and sketch the graph of a rational
function
•
classify scatter plots
•
use scatter plots and the graphing
calculator to find quadratic models for data
•
choose a model that best fits a set of data
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 2
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
6
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 3: Exponential and Logarithmic Functions
BRIEF SUMMARY OF UNIT: This unit focuses on exponential and logarithmic functions. The students will be able to write, graph, and recognize the basic
characteristics of exponential and logarithmic functions. They will investigate how they can use exponential and logarithmic models to solve real-world
problems including compound interest and growth and decay.
LINK TO CONTENT STANDARDS:
High School: Functions » Linear, Quadratic, & Exponential Models*
Construct and compare linear, quadratic, and exponential models and solve problems.
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors
over equal intervals.
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or
(more generally) as a polynomial function.
ct
4. For exponential models, express as a logarithm the solution to ab = d where a, c, and d are numbers and the base b is 2, 10, or e;
evaluate the logarithm using technology.
Interpret expressions for functions in terms of the situation they model.
5. Interpret the parameters in a linear or exponential function in terms of a context.
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
•
What are the important defining
characteristics and representations of
exponential and logarithmic functions?
How are these functions used to model
real-world problems?
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
GUIDING QUESTIONS:
the definition of an exponential function
the definition of a logarithmic functions
the difference between a common
logarithm and a natural logarithm
the characteristics of an exponential and
a logarithmic graph
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
•
complete do now activities
7
COURSE TITLE: Pre-Calculus Honors
•
How do you write and graph exponential
functions?
•
How do you recognize, evaluate, and
graph logarithmic functions?
•
How do you rewrite logarithmic
expressions in order to simplify or
evaluate them?
•
How do you solve exponential and
logarithmic equations?
•
How do you use exponents and logarithms
to model a variety of situations?
•
How do you fit nonlinear models to data?
GRADE(S): 11-12
•
•
•
•
the properties of logarithms
the properties of exponents
the techniques used to solve exponential
and logarithmic equations
the five most common types of models
involving exponential and logarithmic
functions
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
B: STUDENTS WILL UNDERSTAND THAT:
•
•
•
•
•
•
exponential and logarithmic functions
are called transcendental functions
exponential and logarithmic functions
are inverses
the two most frequently used logarithms
are of base 10 and base 10
the change-of-base formula is used to
evaluate logarithms with other bases
exponential and logarithmic functions
are used to model real-world problems
a scatter plot can be used to give you an
idea of which type of model will best fit a
set of data
C: STUDENTS WILL BE ABLE TO:
•
recognize and evaluate exponential
function and logarithmic functions
•
graph exponential and logarithmic
functions
•
use properties of logarithms to expand or
condense logarithmic expressions
•
solve exponential and logarithmic
equations
•
recognize the five most common types of
models involving exponential or
logarithmic functions
•
use exponential and logarithmic models to
8
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
solve real-life problems
•
use scatter plots and a graphing calculator
to find models for data
•
choose a model that best fits a set of data
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 3
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
9
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 4: Trigonometric Functions
BRIEF SUMMARY OF UNIT: This unit focuses on trigonometric functions. The students will be able to recognize, write, and graph the basic trigonometric
functions. They will investigate the basic characteristics of the trigonometric functions, their reciprocals, and their inverses. Students will then use the
trigonometric rations to solve problems in a variety of contexts.
LINK TO CONTENT STANDARDS:
High School: Functions » Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
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.
3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express
the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.
4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★
6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse
to be constructed.
7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret
them in terms of the context.★
Prove and apply trigonometric identities.
2
2
8. Prove the Pythagorean identity sin (θ) + cos (θ) = 1 and use it to calculate trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
•
What are the important defining
characteristics and representations of
trigonometric functions?
How are these functions used to model
real-world problems?
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
the definition of trigonometry
the difference between degrees and
radians
the unit circle
the six trigonometric ratios
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
10
COURSE TITLE: Pre-Calculus Honors
GUIDING QUESTIONS:
•
How do you describe angles and angular
movement?
•
How do you evaluate trigonometric
functions by using the unit circle?
•
How do you use trigonometry to find
unknown side lengths and angles in right
triangles?
GRADE(S): 11-12
•
•
•
•
•
the definition of periodic
the characteristics of the sine and cosine
graph
the definition of a reference angle
the domain and range of sine, cosine, and
tangent
the inverse of sine and cosine
B: STUDENTS WILL UNDERSTAND THAT:
•
How do you evaluate trigonometric
functions of any angle?
•
•
How do you sketch the graphs of sine and
cosine functions?
•
•
•
How do you sketch the graphs of other
trigonometric functions?
•
•
How do you evaluate and graph the
inverses of trigonometric functions?
•
•
How do you use trigonometric functions to
solve real-life problems?
•
•
•
•
complete do now activities
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
there are two ways to measure angles, in
degrees and in radians
there are six trigonometric functions
functions that behave in a repetitive
manner are called periodic
the Fundamental Trigonometric
Identities are relationships between the
six basic trigonometric functions
the sign of a trigonometric function is
based upon the quadrant it lies in
the amplitude represents the distance
between the maximum and minimum
values of the function
in order to find the inverse of the
trigonometric functions sine, cosine, and
tangent the domain must be restricted
trigonometric functions are used to solve
real-world problems
C: STUDENTS WILL BE ABLE TO:
•
describe angles
•
convert between radian and degree
measure
•
evaluate trigonometric functions using the
unit circle
•
use the domain and period to evaluate sine
and cosine functions
•
use a calculator to evaluate trigonometric
11
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
functions
•
apply the Fundamental Identities
•
evaluate trigonometric functions of any
angle
•
use reference angles to evaluate
trigonometric functions
•
sketch the basic graphs of sine, cosine, and
tangent
•
identify the domain, range, amplitude, and
period of the graphs of the trigonometric
functions
•
sketch translations of sine and cosine
•
evaluate the inverses of sine, cosine, and
tangent
•
evaluate compositions of trigonometric
functions
•
solve real-life problems involving right
triangles
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 4
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
12
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 5: Analytic Trigonometry
BRIEF SUMMARY OF UNIT: This unit focuses on the methods used to simplify expressions and solve equations using trigonometric identities. The
students will be able to rewrite and verify trigonometric identities. They will solve trigonometric equations of varying degrees and rewrite trigonometric
expressions that contain functions of multiple or half-angles.
LINK TO CONTENT STANDARDS:
High School: Functions » Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
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.
3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express
the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.
4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★
6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse
to be constructed.
7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret
them in terms of the context.★
Prove and apply trigonometric identities.
2
2
8. Prove the Pythagorean identity sin (θ) + cos (θ) = 1 and use it to calculate trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
How are the Fundamental Identities used
to explore and verify other trigonometric
identities?
GUIDING QUESTIONS:
•
How do you rewrite trigonometric
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
the Fundamental Identities
the guidelines for verifying trigonometric
identities
the Sum and Difference Formulas
the Multiple-Angle Formulas
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
13
COURSE TITLE: Pre-Calculus Honors
expressions in order to simplify and
evaluate trigonometric functions?
•
•
How do you verify a trigonometric
identity?
How do you solve a trigonometric
equation?
•
How do you simplify expressions that
contain sums or differences of angles?
•
How do you rewrite trigonometric
expressions that contain functions of
multiple or half-angles?
GRADE(S): 11-12
•
•
the Half-Angle Formulas
the Product-to-Sum Formulas and the
Sum-to-Product Formulas
B: STUDENTS WILL UNDERSTAND THAT:
•
•
•
•
the Fundamental Trigonometric
Identities can be used to evaluate
trigonometric functions, simplify
trigonometric expressions, and solve
trigonometric equations
there is usually more than one way to
verify a trigonometric identity
the Sum and Difference Formulas can be
used to find exact values of trigonometric
functions involving sums or differences
of special angles
the Multiple-Angle, Half-Angle, Productto-Sum, and Sum-to-Product formulas
are used to rewrite trigonometric
functions
•
complete do now activities
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
C: STUDENTS WILL BE ABLE TO:
•
recognize and write the fundamental
trigonometric identities
•
use the fundamental trigonometric
identities to evaluate, simplify, and rewrite
trigonometric expressions
•
verify trigonometric identities
•
solve trigonometric equations
•
use sum and difference formulas to
evaluate trigonometric expressions and
solve trigonometric equations
•
use multiple-angle, half-angle, product-tosum, and sum-to-product formulas to
rewrite and evaluate trigonometric
functions
14
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 5
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
15
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 6: Additional Topics in Trigonometry
BRIEF SUMMARY OF UNIT: This unit focuses on the trigonometry used on oblique triangles. Students will be able to find the side lengths, angles, and area
of oblique triangles by applying the Law of Sines and the Law of Cosines.
LINK TO CONTENT STANDARDS:
High School: Functions » Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
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.
3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express
the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any real number.
4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Model periodic phenomena with trigonometric functions.
5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★
6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse
to be constructed.
7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret
them in terms of the context.★
Prove and apply trigonometric identities.
2
2
8. Prove the Pythagorean identity sin (θ) + cos (θ) = 1 and use it to calculate trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
How are the Fundamental Identities used
to explore and verify other trigonometric
identities?
GUIDING QUESTIONS:
•
How do you use trigonometry to solve and
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
•
the definition of an oblique triangle
the Law of Sines
the Law of Cosines
the area of an oblique triangle
Heron’s Area Formula
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
•
complete do now activities
16
COURSE TITLE: Pre-Calculus Honors
find the areas of oblique triangles?
GRADE(S): 11-12
B: STUDENTS WILL UNDERSTAND THAT:
•
•
•
•
•
there are four cases of oblique
triangles(AAS or ASA, SSA, SSS, SAS)
to solve an oblique triangle you need to
know the measure of at least one side and
the measure of any two other parts of the
triangle
the Law of Sines is used to solve two
cases of oblique triangles
the Law of Cosines is used to solve two
cases of oblique triangles
Heron’s Formula is used to find the area
of a triangle when the 3 side lengths are
given
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
C: STUDENTS WILL BE ABLE TO:
•
use the Law of Sines to solve oblique
triangles (AAS or ASA and SSA)
•
use the Law of Cosines to solve oblique
triangles (SSS and SAS)
•
find the area of oblique triangles
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 6
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
17
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 9: Topics in Analytic Geometry
BRIEF SUMMARY OF UNIT: This unit focuses on the conic sections. Students will be able classify a conic section by its equations in general forms. They
will also be able to graph circles, parabolas, ellipses, and hyperbolas.
LINK TO CONTENT STANDARDS:
High School: Geometry » Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of
a circle given by an equation.
2. Derive the equation of a parabola given a focus and directrix.
3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
What are conic sections and how are they
applied to the real-world?
GUIDING QUESTIONS:
•
How do you recognize each conic section
and solve problems involving parabolas?
•
How do you solve problems involving
ellipses?
•
How do you solve problems involving
hyperbolas?
•
How do you classify a conic section on the
basis of its general equation?
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
•
•
•
•
•
the definition of a circle
the standard form of a circle
the definition of a parabola
the standard form of a parabola
the definition of a ellipse
the standard form of an ellipse
the definition of a hyperbola
the standard form of a hyperbola
general form of the conics
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
•
complete do now activities
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
B: STUDENTS WILL UNDERSTAND THAT:
•
•
•
•
a conic is the intersection of a plane and
a double-napped cone
conics are defined as a collection of
points satisfying a certain geometric
property
characteristics of conics can be found
using the standard form
conics can be classified when they are
18
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
written in general form
C: STUDENTS WILL BE ABLE TO:
•
write equations of circles in standard form
•
write equations of parabolas in standard
form
•
write equations of ellipses in standard
form
•
write equations of hyperbolas in standard
form
•
classify conics from their general
equations
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 9
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
19
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 11: Limits and an Introduction to Calculus
BRIEF SUMMARY OF UNIT: This unit focuses on the introduction to calculus using the concept of tangent lines and the area of a region. Students will be
exposed to limits and derivatives. They will use these concepts to find the area of a region bounded by a function.
LINK TO CONTENT STANDARDS:
High School: Functions » Building Functions
1. Write a function that describes a relationship between two quantities.★
Determine an explicit expression, a recursive process, or steps for calculation from a context.
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling
body by adding a constant function to a decaying exponential, and relate these functions to the model.
(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a
weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
What does the study of calculus involve?
•
How is the concept of a limit used in realworld applications?
GUIDING QUESTIONS:
•
•
How do you find and interpret the limit of
a function for a certain value of x?
How do you evaluate limits that cannot be
solved through the use of direct
substitution?
•
How do you find the slope of a graph at
any single point?
•
How do find the limits of functions at
infinity?
•
How do you approximate and find the
exact areas of plane regions defined by
functions?
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
•
•
•
•
•
•
the definition of a limit
the properties of limits
the direct substitution method
the limits of polynomial and ration al
functions
various techniques for evaluating a limit
the definition of the tangent line to a
graph
the definition of a derivative
limits at infinity
summation formulas and properties
the area of a plane region
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
•
complete do now activities
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
B: STUDENTS WILL UNDERSTAND THAT:
•
•
•
limits are the building blocks of calculus
some limits are quantities that can be
approached but not reached
many limits are quantities that do exist on
20
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
•
•
•
•
•
the function
limits can be evaluated using direct
substitution
limits can be found both graphically and
numerically
calculus is a branch of mathematics that
studies rates of change of functions
finding the slope f a graph at point is the
same as finding the slope of the tangent
line at the point
to find the area under a curve, use must
use the limit of a summation
C: STUDENTS WILL BE ABLE TO:
•
determine whether limits of functions exist
•
use properties of limits and direct
substitution to evaluate limits
•
use various techniques to evaluate limits
•
approximate limits of functions
graphically and numerically
•
use a tangent line to approximate the slope
of a graph at a point
•
find the derivatives of functions
•
use derivatives to find the slopes of graphs
•
evaluate limits of functions at infinity
•
use rectangles to approximate areas of
plane regions
•
use limits of summations to find areas of
plane regions
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 11
Instructional DVDs
21
COURSE TITLE: Pre-Calculus Honors
•
•
•
•
•
•
•
•
GRADE(S): 11-12
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
22
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 7: Linear Systems and Matrices
BRIEF SUMMARY OF UNIT: This unit focuses on the solutions to system s of equations. Students will use various techniques, including graphing,
substitution, elimination, and matrices to solve a system of equations. They will be introduced to several real-world applications that involve the use of systems
of equations.
LINK TO CONTENT STANDARDS:
High School: Number & Quantity » Vector & Matrix Quantities
Perform operations on matrices and use matrices in applications.
6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
8. (+) Add, subtract, and multiply matrices of appropriate dimensions.
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.
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.
12. (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
High School: Algebra » Reasoning with Equations & Inequalities
Solve systems of equations.
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.
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find
2
2
the points of intersection between the line y = –3x and the circle x + y = 3.
8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.
9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
23
COURSE TITLE: Pre-Calculus Honors
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
How can systems of equations be used in
the real world?
GUIDING QUESTIONS:
•
How do you use substitution and graphing
to solve systems of equations?
•
How do you use elimination to solve
systems of equations?
•
How do you solve systems of linear
equations in more than two variables?
•
How do you use matrices to solve systems
of equations?
GRADE(S): 11-12
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
•
•
•
•
•
B: STUDENTS WILL UNDERSTAND THAT:
How do you perform operations on
matrices?
•
•
How do you find and use the inverse of a
square matrix?
•
•
How do you find the determinant of a
square matrix?
•
•
How do you use matrices to solve systems
of equations?
•
•
the definition of a system of equations
the method of substitution, graphing, and
elimination
points of intersection
the graphical interpretation of twovariable systems
definition of a matrix
matrix addition and scalar multiplication
matrix multiplication
the inverse of a matrix
the definition of a determinant
•
•
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
•
complete do now activities
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
a solution to a system is an ordered pair
that satisfies each equation in the system
there are various methods to solve a
system of equations
the break-even point refers to the point of
intersection of the cost and revenue
curves
the correlation between the number of
solutions and graphically interpretation
of a system
matrices can be added, subtracted, and
multiplied if they are the same size
matrices can be used to solve a system of
equations
C: STUDENTS WILL BE ABLE TO:
•
solve a system of equations using
substitution, graphing, and elimination
•
use systems of equations to model and
solve real-life problems
24
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
•
graphically interpret the number of
solutions of a system of linear equations in
two variables
•
write matrices and identify their orders
•
use matrices to solve systems of linear
equations
•
add, subtract, multiply two matrices
•
find the inverse of a matrix
•
find the determinant of a square matrix
•
use determinants to find the areas of
triangles
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 7
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
25
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
UNIT NUMBER AND TITLE: Unit 8: Sequence, Series, and Probability
BRIEF SUMMARY OF UNIT: This unit focuses on the concepts of sequences and series. Students will analyze arithmetic and geometric sequences and
series. They will expand binomials using the Binomial Theorem and Pascal’s Triangle.
LINK TO CONTENT STANDARDS:
High School: Statistics & Probability » Conditional Probability & the Rules of Probability
Understand independence and conditional probability and use them to interpret data
1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,
intersections, or complements of other events (“or,” “and,” “not”).
2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities,
and use this characterization to determine if they are independent.
3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional
probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example,
compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability model
6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms
of the model.
9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Algebra » Arithmetic with Polynomials & Rational Expressions
Use polynomial identities to solve problems.
4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity
2
2 2
2
2 2
2
(x + y ) = (x – y ) + (2xy) can be used to generate Pythagorean triples.
n
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y) in powers of x and y for a positive integer n, where x and y are any
numbers, with coefficients determined for example by Pascal’s Triangle.
26
COURSE TITLE: Pre-Calculus Honors
ESSENTIAL QUESTIONS THAT WILL
FOCUS TEACHING AND LEARNING:
•
How can systems of equations be used in
the real world?
GUIDING QUESTIONS:
•
How do you represent a sequence of
numbers or the sum of a sequence?
•
How do you find the nth term of an
arithmetic sequence?
•
How do you find terms and sums of
geometric sequences?
•
How do you find the expansion of a
binomial?
•
How do you count the number of ways in
which an event can occur?
•
How do you find the probability that a
series of events will occur?
GRADE(S): 11-12
ESSENTIAL KNOWLEDGE, SKILLS, AND
ENDURING UNDERSTANDINGS:
A: STUDENTS WILL KNOW:
•
•
•
•
•
•
•
•
•
•
the definition of a sequence
the difference between an infinite
sequence and a finite sequence
summation notation
the properties of sums
definition of a series
the difference between an arithmetic
sequence and a geometric sequence
the Binomial Theorem
Pascal’s Triangle
the Fundamental Counting Principle
the difference between a permutation and
a combination
ASSESSMENT (EVIDENCE OF
KNOWLEDGE AND UNDERSTAND?)
STUDENTS WILL:
•
answer the essential question
•
complete exercises with and without a
graphing calculator
•
complete do now activities
•
display their work on the board
•
engage in mathematical discussions
•
demonstrate knowledge, skill, and
understanding on a chapter test
•
participate in an evaluation of each other’s
homework
B: STUDENTS WILL UNDERSTAND THAT:
•
•
•
•
•
•
•
•
a sequence is a function whose domain is
the set of positive integers
sequences are written using subscript
notation
summation notation involves using the
upper case Greek letter sigma
an arithmetic sequence has a common
difference
a geometric sequence has a common
ratio
the Binomial Theorem is used to
calculate binomial coefficients
Pascal’s Triangle is an array that helps to
calculate binomial coefficients
the difference between permutations and
combinations is order
C: STUDENTS WILL BE ABLE TO:
•
use sequence notation to write the terms of
27
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
sequences
•
use summation notation to write sums
•
find sums of infinite series
•
recognize, write, and find the nth terms of
arithmetic sequences
•
recognize, write, and find the nth terms of
geometric sequences
•
find sums of infinite geometric series
•
use the Binomial Theorem to calculate
binomial coefficients
•
use Pascal’s Triangle to calculate binomial
coefficients
•
solve simple counting problems
•
use permutations and combinations to
solve counting problems
•
find probabilities of mutually exclusive
and independent events
SUGGESTED SEQUENCE OF LEARNING ACTIVITIES, INCLUDING THE USE OF TECHNOLOGY AND OTHER RESOURCES:
•
•
•
•
•
•
•
•
•
•
•
•
Do not do Section 8.4
Vocabulary and examples in Notetaking Guide
Animation Pre-Calculus for Chapter 8
Instructional DVDs
HM mathSpace CD-Rom
Real-World Application Problems
Graphing Calculator Explorations
Digital Lessons
Digital Art and Figures
Answer Transparency Masters
www.CalcChat.com
Make a Decision exercises
28
COURSE TITLE: Pre-Calculus Honors
GRADE(S): 11-12
29