EL SEGUNDO UNIFIED
EL SEGUNDO HIGH SCHOOL
COURSE OF STUDY
Course Title: Pre-Calculus Honors and Pre-Calculus
Department: Mathematics
Grade Levels: 11-12
COURSE DESCRIPTION
PRE-CALCULUS :
Pre-Calculus is a discipline that combines many of the trigonometric, geometric, and algebraic techniques needed for the preparation of the study of Calculus, and strengthens conceptual understanding and mathematical reasoning when solving problems. This course takes a functional point of view to these topics. First semester concentrates on algebraic and transcendental functions and probability, and second semester is trigonometry, linear algebra, and sequences and series. Trigonometry is a discipline that utilizes the techniques of both the algebra and geometry that students have previously learned. The trigonometric functions studied are defined geometrically, rather than in terms of algebraic equations. Students will also use trigonometry in polar and parametric equations and conics. Facility with these functions as well as being able to prove basic identities regarding them is especially important for students intending to study Calculus, more advanced mathematics, Physics and other sciences, and engineering in college. Linear algebra includes matrix manipulation, and vectors. Sequences and series involves proof by mathematical induction as well as finding the terms in geometric and arithmetic sequences.
PRE-CALCULUS HONORS: This course covers the same material in Pre-Calculus; however the expectations for mastery are at a higher level, and the projects are more involved.
Length: One Year
Prerequisite for Enrollment: Pre-Calculus: 90% or better in Algebra 2, 80% or better in Algebra 2 Honors; Score 80% or better on placement exam. Pre-Calculus Honors: 95% or better in Algebra 2 or 85% or better in Algebra 2 Honors; Score 90% or better on placement exam.
Recommendation for Enrollment: Teacher recommendation and a strong desire on the part of the student to continue a rigorous study of mathematics
Type of Course: College preparatory (UC/CSU) and meets math graduation requirement.
COURSE OUTLINE & STANDARDS
ALGEBRA 2 REVIEW: (Integration of Core Knowledge; Critical
Thinking/Problem Solving)
ELEMENTARY, POLYNOMIAL, AND RATIONAL FUNCTIONS
1. The student will analyze elementary, polynomial, and rational functions
algebraically, and graphically. Students will find the domain, range, real
and complex zeros, vertical, horizontal and oblique asymptotes, the interval where the function is increasing and decreasing, and the location of minimums and maximums.
2. The student will use translations to move functions in the coordinate plane
3. Students will simplify and solve complex rational equations.
4. Students will, given an applied problem, identify data, graph, and choose both
the appropriate algebraic model and method of solution.
5. Students know the statement of and can apply the Fundamental Theorem of
Algebra. Students can apply partial fractions to a rational expression.
TRANSCENDENATAL FUNCTIONS (Review of Algebra 2 Content)
1. The student will graph, identify the domain, range, and asymptotes of
exponential and logarithmic functions
2. The student will use exponential, logarithmic and logistics functions to solve
nonlinear modeling problems.
3. The student will solve exponential and logarithmic equations.
Trigonometry
Trigonometry
uses the techniques that students have previously learned from the study of
algebra and geometry. The trigonometric functions studied are defined
geometrically rather than in terms of algebraic equations. Facility with these
functions as well as the ability to prove basic identities regarding them is
especially important for students intending to study calculus, more advanced
mathematics, physics and other sciences, and engineering in college.
( Trigonometry and Math Analysis requires a student to integrate core knowledge, think critically, communicate effectively, and further develop personal and social skills.)
1.0
Students understand the notion of angle and how to measure
it, in both degrees and radians. They can convert between degrees and radians.
2.0
Students know the definition of sine and cosine as y-
and x-coordinates
of points on the unit circle and are familiar with the graphs of the sine and
cosine functions.
3.0
Students know the identity cos 2
(x) + sin 2
(x) = 1:
3.1 Students prove that this identity is equivalent to the
Pythagorean theorem
(i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity).
3.2 Students prove other trigonometric identities and
simplify others by using the identity
cos2
(x) + sin2 (x) = 1. For
example, students use this identity to prove that sec2 (x) = tan 2 (x) + 1.
4.0
Students graph functions of the form f(t)
= A sin (Bt
+ C) or
f(t)
= A cos
(Bt + C)
and interpret A,
B, and C
in terms of amplitude, frequency, period, and phase
shift.
5.0
Students know the definitions of the tangent and cotangent
functions and can
graph them.
6.0
Students know the definitions of the secant and cosecant
functions and can graph them.
7.0
Students know that the tangent of the angle that a line makes
with the x-axis is equal to the slope of the
line.
8.0
Students know the definitions of the inverse trigonometric
functions and can
graph the functions.
9.0
Students compute, by hand, the values of the trigonometric
functions and the
inverse trigonometric functions at
various standard points.
10.0
Students demonstrate an understanding of the addition
formulas for sines and cosines and their proofs and
can use those formulas to prove and/or simplify other trigonometric identities.
11.0
Students demonstrate an understanding of half-angle and
double-angle formulas for sines and cosines and can
use those formulas to prove and/or simplify other trigonometric identities.
12.0
Students use trigonometry to determine unknown sides or
angles in right
triangles.
13.0
Students know the law of sines and
the law of cosines and apply those laws to solve problems.
14.0
Students determine the area of a triangle, given one angle
and the two adjacent sides.
15.0
Students are familiar with polar coordinates. In particular,
they can determine polar coordinates of a point given in rectangular
coordinates and vice versa.
16.0
Students represent equations given in rectangular coordinates
in terms of polar coordinates.
17.0
Students are familiar with complex numbers. They can
represent a complex
number in polar form and know how to
multiply complex numbers in their polar
form.
18.0
Students know DeMoivre’s theorem
and can give nth roots of a complex number given
in polar form.
19.0
Students are adept at using trigonometry in a variety of
applications and word problems.
Mathematical
Analysis
This
discipline combines many of the trigonometric, geometric, and algebraic
techniques needed to prepare students for the study of calculus and strengthens
their conceptual understanding of problems and mathematical reasoning in
solving problems. These standards take a functional point of view toward those
topics. The most significant new concept is that of limits. Mathematical
analysis is often combined with a course in trigonometry or perhaps with one in
linear algebra to make a yearlong pre-calculus course.
1.0
Students are familiar with, and can apply, polar coordinates
and vectors in the plane. In particular, they can translate between polar and
rectangular coordinates and can interpret polar coordinates and vectors
graphically.
2.0
Students are adept at the arithmetic of complex numbers. They
can use the trigonometric form of complex numbers and understand that a
function of a complex variable can be viewed as a function of two real
variables. They know the proof of DeMoivre’s theorem.
3.0
Students can give proofs of various formulas by using the
technique of mathematical induction.
4.0
Students know the statement of, and can apply, the
fundamental theorem of
algebra.
5.0
Students are familiar with conic sections, both analytically
and geometrically:
5.1 Students can take a quadratic equation in two variables;
put it in standard form by completing the square and using rotations and
translations, if necessary; determine what type of conic section the equation
represents; and determine its geometric components (foci, asymptotes, and so
forth).
5.2 Students can take a geometric description of a conic
section—for example, the locus of points whose sum of its distances from (1, 0)
and (-1, 0) is 6—and derive a quadratic equation representing it.
6.0
Students find the roots and poles of a rational function and
can graph the function and locate its asymptotes.
7.0
Students demonstrate an understanding of functions and
equations defined
parametrically and can graph them.
8.0
Students are familiar with the notion of the limit of a
sequence and the limit of a function as the independent variable approaches a
number or infinity. They
determine whether certain sequences
converge or diverge.
INSTRUCTIONAL METHODS
A. Lecture and Guided Practice
B. Investigations
C. Manipulatives
D. Class work and Homework
E. Individual Work
F. Group Work
G. Projects
EVALUATION/GRADING OF STUDENT WORK
A. Quizzes and Chapter Tests
B. Semester Exam
C. Comprehensive final exam
D. Homework and classroom participation
E. Projects
F. Written Essays
INSTRUCTIONAL MATERIALS
A. Text: BEFORE CALCULUS 3 by Leithold, c. 1994.
B. Graphing
Calculator (TI-89)