PRECALCULUS
COURSE OF STUDY
POMPTON LAKES PUBLIC SCHOOLS
JUNE 2010
Submitted by
The Mathematics Department
Dr. Terrance Brennan, Superintendent
Vincent Przybylinski, Principal
Anthony Mattera, Vice Principal
Frances J. Macdonald Mathematics Supervisor K-12
Mary Curran, Board of Ed President
Mr. Ray Keating, III, Board of Ed Vice President
Board Members
Mr. William Baig, Mr. Joel Bernstock, Mrs. Catherine Brolsma,
Mrs. Joyce Colfax, Mr. Scott Croonquist, Mr. Tom Salus, Mrs. Stephanie Shaw
Honors Precalculus
2
I.
RATIONAL
This year long course is a preparation for the continuation of mathematics
courses such as Honors Calculus, and the attainment of success on the SATs
and the New Jersey HSPA.
II.
DESCRIPTION
This year long course is thoroughly prepares for college level courses. Stress is
places on pre-calculus topics such as functions and limits. Circular functions and
trigonometry are developed with empasis on proof and application. There is
comprehensive study of algebraic, exponential and logarithmic functions, linear
sequences and series, mathematical induction, the derived function and
graphing. The TI-84Plus calculator is an integral part of the curriculum.
III.
THE CORE CURRICULUM CONTENT STANDARDS
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
All students will develop the ability to pose and solve mathematical
problems in mathematics, other disciplines, and everyday
experiences.
All students will communicate mathematically through written, oral,
symbolic, and visual forms of expression.
All students will connect mathematics to other learning by
understanding the interrelationships of mathematical ideas and the
roles that mathematics and mathematical modeling play in other
disciplines and in life.
All students will develop reasoning ability and will become selfreliant, independent mathematical thinkers.
All students will regularly and routinely use calculators, computers,
manipulatives, and other tools to enhance mathematical thinking,
understanding and power.
All students will develop number sense and an ability to represent
numbers in a variety of forms and use numbers in diverse
situations.
All students will develop spatial sense and an ability to represent
geometric properties and relationships to solve problems in
mathematics and in everyday life.
All students will understand, select, and apply various methods of
performing numerical operations.
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.
All students will use a variety of estimation strategies and recognize
situations in which estimation is appropriate.
All students will develop an understanding of patterns,
relationships, and functions and will use them to represent and
explain real world phenomena.
All students will develop an understanding of statistics and
probability and will use them to describe sets of data, model
situations, and support appropriate inferences and arguments.
All students will develop algebraic concepts and processes and will
3
4.14
4.15
4.16
IV.
use them to represent and analyze relationships among variable
quantities and to solve problems.
All students will apply the concepts and methods of discrete
mathematics to model and explore a variety of practical situations.
All students will develop an understanding of the conceptual
building blocks of calculus and will use them to model and analyze
natural phenomena.
All students will demonstrate high levels of mathematical thought
through experiences which extend beyond traditional computation,
algebra, and geometry.
STANDARD 9.1 (Career and Technical Education)
All students will develop career awareness and planning, employment skills, and
foundational knowledge necessary for success in the workplace.
Strands and Cumulative Progress Indicators
Building knowledge and skills gained in preceding grades, by the end of Grade
12, students will:
A.
Career Awareness/Preparation
1.
Re-evaluate personal interests, ability, and skills through various
measures including self assessments.
2.
Evaluate academic and career skills needed in various career
clusters.
3.
Analyze factors that can impact an individual’s career
4.
Review and update their career plan and include plan in portfolio.
5.
Research current advances in technology that apply to a sector
occupational career cluster.
B.
Employment skills
1.
Assess personal qualities that are needed to obtain and retain a job
related to career clusters.
2.
Communicate and comprehend written and verbal thoughts, ideas,
directions and information relative educational and occupational
settings.
3.
Select and utilize appropriate technology in the design and
implementation of teacher-approved projects relevant to
occupational and /or higher educational settings
4.
Evaluate the following academic and career skills as they relate to
home, school, community, and employment.
Communication
Punctuality
Time management
Organization
Decision making
Goal setting
4
5.
Resources allocation
Fair and equitable competition
Safety
Employment application
Teamwork
Demonstrate teamwork and leadership skills that include student
participation in real world applications of career and technical
educational skills.
All students electing further study in career and technical education will
also: participate in a structural learning experience that demonstrates
interpersonal communication, teamwork, and leadership skills.
V.
UNITS
A.
LINEAR RELATIONS AND FUNCTIONS
Time line 28days CCCS4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.9,4.10,4.13,4.14,4.15
1.
Objectives
a.
Determine whether a given relation is a function.
b.
Identify the domain and range of any relation.
c.
Perform operations with functions.
d.
Find composite functions.
e.
Find and recognize inverse functions.
f.
Find zeros of linear functions.
g.
Graph linear equations and inequalities.
h.
Find the distance between two points.
i.
Find the slope of a line through two points.
j.
Prove geometric theorems involving slope, distance, and
midpoints analytically.
k.
Write linear equations using slope-intercept form.
l.
Write linear equations using point-slope form.
m.
Write equations of parallel and perpendicular lines.
2.
Content
a.
Functions and relations
b.
Composite and inverse functions
c.
Graphing linear equations and inequalities using the
graphing calculators
d.
Linear and inequalities
e.
Distance and slope
f.
Forms of linear equations
g.
Parallel and perpendicular lines
3.
Assessments
a.
Write a function f(x).
b.
If g (x) = 2x2 - 1, does (f + g) (x) = (g + f) (x)? Justify your
answer.
c.
Does (f – g) (x) = (g – f) (x)? Justify your answer.
5
d.
e.
f.
Does (f ∙ g) (x) = (g ∙ f) (x)? Justify your answer.
Does ( f ÷ g )( x ) = ( g ÷ f )( x )
What can you conclude about the commutatively of adding,
subtracting, multiplying, or dividing two functions?
B.
SYSTEMS OF EQUATIONS AND INEQUALITIES
Time line 20days CCCS4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.9,4.10,4.13,4.14,4.15
1.
Objectives
a.
Solve systems of equations graphically.
b.
Solve systems of equations algebraically.
c.
Add, subtract, and multiply matrices.
d.
Evaluate determinants.
e.
Find inverses of matrices.
f.
Solve systems of equations by using inverses of matrices.
g.
Solve systems of equations by using matrices.
h.
Find the maximum of minimum value of a function defined or
a polygonal convex set.
2.
Content
a.
Graphing calculators and systems of linear equations
b.
Systems of equations
c.
Graphing calculators and matrices
d.
Operations on matrices
e.
Systems of equations and matrices
f.
Graphing calculators and systems of inequalities
g.
Systems of inequalities
3.
Assessment
Solve the system
2x+ 5y+ 8z = 5, -2x+ 9y+ 8z = 5, and 4x+ 6y -4z = 3
C.
THE NATURE OF GRAPHS
Time line 30days CCCS4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.9,4.10,4.13,4.14,4.15
1.
Objectives
a.
Identify symmetrical graphs.
b.
Use symmetry to complete a graph.
c.
Identify an odd function and an even function.
d.
Identify the graphs of simple polynomial functions, absolutevalue functions, and step functions.
e.
Sketch the graphs of these functions.
f.
Determine the inverse of a relation or function.
g.
Graph a function and its inverse.
h.
Determine horizontal, vertical, and slant asymptotes.
i.
Graph rational functions.
j.
Find the critical points of the graph of a polynomial function
and determine if each is a minimum, maximum, or point of
inflection.
k
Determine continuity or discontinuity of functions.
6
l.
Identify the end behavior of graphs.
Content
a.
Graphing calculators and polynomial functions, radical
functions, rational functions, critical points, continuity and
end behavior
b.
Symmetry
c.
Polynomial functions
d.
Family graphs
e.
Inverse functions and relations
f.
Rational functions and asymptotes
g.
Graphs of inequalities
h.
Tangent to a curve
i.
Critical points of polynomial functions
j.
Continuity and end behavior
3. Assessment
What are the vertical and horizontal asymptotes for the function
f(x)= x-2___
x2+4x+3
2.
D.
POLYNOMIAL AND RATIONAL FUNCTIONS
Time line 26 days CCS4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.9,4.10,4.13,4.14,4.15
1.
Objectives
a.
Determine roots of polynomial equations.
b.
Apply the fundamental theorem of algebra.
c.
Solve quadratic equations.
d.
Use the discriminant to describe the roots of quadratic
equations.
e.
Graph quadratic equations.
f.
Identify all possible rational roots of a polynomial equation.
g.
Determine the number of positive and negative real zeros a
polynomial function has.
h.
Approximate the real zeros of a polynomial function.
i.
Graph polynomial functions.
j.
Find the least common denominator of rational expressions.
k.
Solve rational equations and inequalities.
l.
Solve radical equations and inequalities.
2.
Content
a.
Graphing calculators and quadratic functions and location of
zeros of polynomial functions
b.
Quadratic equations and inequalities
c.
Zeros of a function
d.
Rational equations
e.
Radical equations and inequalities
3.
Assessment
Given the function f(x) = 6x5 + 2x4 – 5x3 – 4x2 + x – 4, answer the
following:
7
a.
b.
c.
d.
e.
f.
g.
How many positive real zeros are possible? Explain.
How many negative real zeros are possible? Explain.
What are the possible rational zeros? Explain.
Is it possible that there are real zeros? Explain.
Are there any real zeros greater than 2? Explain.
What term could you add to the above polynomial to
increase the number of possible positive real zeros by one?
Does the term you added increase the number of possible
negative real zeros? How do you know?
Write a polynomial equation. Then describe its roots.
E.
THE TRIGONOMETRIC FUNCTIONS
Time line 21 daysCCCS4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.9,4.13,4.14,4.15
1.
Objectives
a.
Change from radians to degree measure and vice versa.
b.
Find angles that are coterminal with a given angle.
c.
Find the reference angle for a given angle.
e.
Find the length of an arc given the measure of the central
angle.
f.
Find the values of the six trigonometric functions of an angle
in standard position given a point on its terminal side.
g.
Find exact values for the six trigonometric functions of
special angles.
h.
Find decimal approximations for the values of the six
trigonometric functions of any angle.
i.
Solve right triangles.
j.
Determine whether a triangle has zero, one, or two
solutions.
k.
Solve triangles by using the law of sines.
l.
Solve triangles by using the law of cosines.
2.
Content
a.
Angles measurement
b.
Central angles and arcs
c.
Circular functions
d.
Trigonometric functions of special angles
e.
Right triangles
f.
Law of sines
g.
Law of cosines
3.
Assessment
In triangle DLM D= Dock L= Lighthouse M = Marine
DL = 4.5 miles and DM = 8.1 miles and <LDM = 32 º
a.
How far is the Lighthouse to the Marina?
b.
What is the angle between the route from the lighthouse to t
Marina?
F.
GRAPHS OF THE TRIGONOMETRIC FUNCTIONS
8
Time line 10 daysCCCS4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.9,4.13,4.14,4.15
1.
Objectives
a.
Use the graphs of the trigonometric functions.
b.
Find the amplitude, period, and phase shift for a
trigonometric functions.
c.
Write equations of trigonometric functions given the
amplitude, period, and phase shift.
d.
Graph various functions.
2.
Content
a.
Graphing calculators and trigonometric functions
b.
Graphs of trigonometric functions
c.
Amplitude, period and phase shift
3.
Assessment
a.
Explain what is meant by a sine function with an amplitude
of 3. Draw a graph in your explanation.
b.
Explain what is meant by a cosine function with a period of
π. Draw a graph in your explanation.
c.
Explain what is meant by a tangent function with a phase
shift of – π/4. Draw a graph in your explanation.
d.
Choose an amplitude, a period, and a phase shift for a sine
function. Write the equation for these attributes and graph
it.
G.
3.
TRIGONOMETRIC IDENTITIES AND EQUATIONS
Time line 12 daysCCCS4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.9,4.13,4.14,4.15
1.
Objectives
a.
Identify and use reciprocal identities, quotient identities,
Pythagorean identities, and symmetry identities.
b.
Use the basic trigonometric identities to verify other
identities.
c.
Find numerical values of trigonometric functions.
d.
Use the sum and difference identities for sine, cosine, and
tangent functions.
e.
Use the double-and half-angle identities for the sine, cosine,
and tangent functions.
f.
Solve trigonometric equations.
g.
Find the distance from a point to a line.
2.
Content
a.
Graphing calculator and verifying trig identities
b.
Basic trigonometric identities
c.
Verification of trig identities
d.
Sum and difference identities
e.
Double and half angle identities
f.
Trigonometric equations
Assessment
a.
Verify that (cos θ)/ (1- sin θ) – (1 + sin θ)/ ( cos θ) = 0 is an
9
b.
c.
H.
identity.
Why is it usually easier to transform the more complicated
side of the equation into the simpler side rather than the
other way around?
Is the following method for verifying an identity correct?
Why or why not? If not, write a correct verification.
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Time line 32 days CCCS 4.2,4,4.5,4.6,4.8,4.13,4.14,4.15
1.
Objectives
a.
Use the properties of exponents
b.
Evaluate and simplify expressions containing rational
exponents
c.
Evaluate expressions with irrational exponents
d.
Graph exponential functions
e.
Graph exponential inequalities
f.
Use the exponential function y=ex
g.
Evaluate expressions involving logarithms
h.
Solve equations involving logarithms
i.
Graph logarithmic functions
j.
Find common logarithms and antilogarithms of numbers
k.
Use common logarithms to compute powers and roots
l.
Solve exponential and logarithmic equations
m.
Find natural logarithms of numbers
n.
Solve equations using natural logarithms
o.
Apply equations using natural logarithms to solving word
problems
2.
Content
a.
3.
Graphing calculator and graphs of exponential and
logarithmic functions
b.
Rational exponents
c.
Exponential functions
d.
The number e
e.
Logarithmic functions
f.
Common logarithms
g.
Exponential and logarithmic equations
h.
Natural logarithms
i.
Applications in word problems
Assessments
a.
A 1991 report estimated that there were 640 salmon in
certain river. If the population is decreasing exponentially at
a rate of 4.3% per year, what is the expected population in
2002?
10
b.
Find the balance in an account at the end of 12 years if
$4000 is invested at an interest rate of 9% that is
compounded continuously.
I.
COMPLEX NUMBERS
Time line 5 days CCCS 4.2,4.4,4.6,4.8,4.13,4.14,
1.
Objectives
a.
Add, subtract, multiply, and divide complex numbers in
rectangular form.
b
Find the product and quotient of complex numbers in polar
form.
2
Content
a.
Rectangular equations
b.
Complex numbers
c.
Products and quotients of complex numbers in polar form
3.
Assessment
Write the equations of a quadratic functions with roots of 2+ i
J.
CALCULATORS
1.
Objective
a.
Use a calculator to perform basic 4 functions
b.
Know how to use the calculator to solve problems on HSPT
c.
Know how to use the TI 83 Plus and the TI 84 Plus graphing
calculator to solve problems
2.
Content
a.
Demonstration of calculator use
b.
Demonstration of TI 83 Plus and the TI 84 Plus Graphing
Calculators
c.
Key strokes
VI.
EVALUATIONS
A.
Tests
B.
Quizzes
C.
Semester exams
D.
Homework
E.
Classwork
F.
Research projects
G.
Writing assignments
H.
Assessments which require students to:
1.
Use calculator functions to perform basic computation
2.
Check manual computation with calculators
3.
Determine appropriate use of calculator
VII.
BENCHMARKS
A.
(Semester I Exam)
1.
Functions and relations
11
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12
13
14
15
16.
17.
18.
19.
20.
21.
22.
23.
24.
25
26.
27.
B.
Composite and inverse functions
Graphing linear equations and inequalities using the
graphing calculators
Linear and inequalities
Distance and slope
Forms of linear equations
Parallel and perpendicular lines
Graphing calculators and systems of linear equations
Systems of equations
Graphing calculators and matrices
Operations on matrices
Systems of equations and matrices
Graphing calculators and systems of inequalities
Systems of inequalities
Graphing calculator and graphs of exponential and
logarithmic functions
Rational exponents
Exponential functions
The number e
Logarithmic functions
Common logarithms
Exponential and logarithmic equations
Natural logarithms
Applications in word problems
Parabola
Demonstration of calculator use
Demonstration of TI 83 Plus and the TI 84 Plus Graphing
Calculators
Key strokes
(Semester II Exam)
1.
Graphing calculators and polynomial functions, radical
functions, rational functions, critical points, continuity and
end behavior
2.
Symmetry
3.
Polynomial functions
4.
Family graphs
5.
Inverse functions and relations
6.
Rational functions and asymptotes
7.
Graphs of inequalities
8.
Critical points of polynomial functions
9.
Continuity and end behavior
10.
Graphing calculators and quadratic functions and location of
zeros of polynomial functions
11
Quadratic equations and inequalities
12
Zeros of a function
12
13.
14.
15.
16.
17.
18.
19.
20.
21.
22
23
24
25
26
27
Rational equations
Radical equations and inequalities
Angles measurement
Central angles and arcs
Circular functions
Trigonometric functions of special angles
Right triangles
Law of Sines
Law of Cosines
Graphing calculators and trigonometric functions
Graphs of trigonometric functions
Amplitude, period and phase shift
Demonstration of calculator use
Demonstration of TI 83 Plus and the TI 84 Plus Graphing
Calculators
Key strokes
VIII.
AFFIRMATIVE ACTION
Evidence of:
A-1 minorities and females incorporated into plans
A-2 human relations concepts being taught
A-3 teaching plans to change ethnic and racial stereotypes
IX.
BIBLIOGRAPHY
Holiday.B.,Cuevas, G., McClure, M., Carter,J., Marks, D., Advanced
Mathematical Concepts:Precalculus with Applications . Columbus, Ohio,
Glencoe/McGraw-Hill, 2006.
.