GLEN RIDGE PUBLIC SCHOOLS
Curriculum Guide
Course Title: MATH ANALYSIS/PRE-CALCULUS HONORS
Subject: Mathematics
Grade Level: 11
Department/School: Mathematics/Glen Ridge High School
Duration: Full Year
Number of Credits: 5
Prerequisite: Algebra II Honors Grade of “B” or higher
Elective or Required: Elective
Author:
Date Submitted: Summer 2007
Course Description
Math Analysis Honors
Math Analysis Honors represents a complete course in topics that are required for the course in calculus. It is also designed to develop an attitude in a structured approach to problem solving. The main body of the course is composed of the study of elementary functions. There is a strong emphasis on trigonometric topics. Other math concepts that are presented either as adjunct tools or topics of interest include D’Moives’ Theorem, matrices and determinants, sequences and series, permutations, combinations and probability.
GLEN RIDGE PUBLIC SCHOOLS
MATHEMATICS
Mathematics
and Computer Science are an integral part of our lives. Students must be actively involved in their
mathematics education with problem solving being an essential part of the
curriculum. The mathematics and computer
science curricula should emphasize thinking skills through a balance of
computation, intuition, common sense, logic, analysis and technology. Students will be engaged and challenged in a
student-centered learning environment that is developmentally appropriate. Students will communicate mathematical ideas
effectively by applying hands-on manipulatives, basic computational skills,
mathematical models, and technology in order to solve practical problems.
The Mathematics Standards
consist of five statements, which describe what is essential to excellent
mathematics education, and presents a view of mathematics teaching and learning
that integrates the processes of mathematical activity, the content of
mathematics, and the learning environment in the classroom. The following standards were adopted by the
New Jersey State Board of Education.
This course will cover the
following Core Curriculum Standards:
STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS) ALL
STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL OPERA-TIONS
AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS.
Descriptive Statement: Numbers and
arithmetic operations are what most of the general public think about when they
think of mathematics; and, even though other areas like geometry, algebra, and
data analysis have become increasingly important in recent years, numbers and
operations remain at the heart of mathematical teaching and learning. Facility with numbers, the ability to choose
the appropriate types of numbers and the appropriate operations for a given
situation, and the ability to perform those operations as well as to estimate
their results, are all skills that are essential for modern day life.
Number Sense
Numerical Operations
Estimation
STANDARD 4.2 (GEOMETRY AND MEASUREMENT) ALL STUDENTS
WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES,
RELATION-SHIPS, AND MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA.
Descriptive Statement: Spatial sense
is an intuitive feel for shape and space.
Geometry and measurement both involve describing the shapes we see all
around us in art, nature and the things we make. Spatial sense, geometric modeling, and
measurement can help us to describe and interpret our physical environment and
to solve problems.
Geometric Properties
Transforming Shapes
Coordinate Geometry
Units of Measurement
Measuring Geometric Objects
STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL
REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES AND SOLVE
PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.
Descriptive Statement: Algebra is a
symbolic language used to express mathematical relationships. Students need to understand how quantities
are related to one another, and how algebra can be used to concisely express
and analyze those relationships. Modern
technology provides tools for supplementing the traditional focus on algebraic
procedures, such as solving equations, with a more visual perspective, with
graphs of equations displayed on a screen.
Students can then focus on understanding the relationship between the
equation and the graph, and on what the graph represents in a real-live
situation.
Patterns
Functions and Relationships
Modeling
Procedures
STANDARD 4.4 (DATA ANALYSIS, PROBABILITY, AND DISCRETE
MATHEMATICS) ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND
TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL
USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS AND ANALYZE AND DRAW APPROPRIATE
INFERENCES FROM DATA.
Descriptive Statement: Data analysis,
probability, and discrete mathematics are important interrelated areas of
applied mathematics. Each provides
students with powerful mathematical perspectives on everyday phenomena and with
important examples of how mathematics is used in the modern world. Two important areas of discrete mathematics
are addressed in this standard: a third area, iterations and recursion, is
addressed in Standard 4.3 (Patterns and Algebra).
Data Analysis
Probability
Discrete Mathematics – Systematic Listing and Counting
Discrete Mathematics – Vertex-Edge Graphs and
Algorithms
STANDARD 4.5 (MATHEMATICAL PROCESSES) ALL STUDENTS
WILL USE MATHEMATICAL PROCESSES OF PROBLEM SOLVING, COMMUNICATION, CONNECTIONS,
REASONING, REPRESENTATIONS, AND TECHNOLOGY TO SOLVE PROBLEMS AND COMMUNICATE
MATHEMATICAL IDEAS.
Descriptive Statement: The
mathematical processes described here highlight ways of acquiring and using the
content knowledge and skills delineated in the first four mathematical
standards.
Problem Solving Reasoning
Communication Representations
Connections Technology
Curriculum Description
The students have studied
Units I-V during the previous school year in Algebra II Honors. Teachers may choose to give sections as the
summer assignment, may choose to pretest, teach or skip according to students’
ability.
UNIT I: FUNDAMENTALS
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1.
Find the domain
and range for rational functions.
2.
Simplify rational
expressions.
3.
Multiply and
divide rational functions.
4.
Add and subtract
rational functions.
5.
Simplify a
compound fraction.
6.
Model with equations.
7.
Solve linear and
quadratic inequalities.
8.
Solve absolute
value inequalities.
9.
Graph points,
lines, parabolas on the Coordinate Plane.
10.
Derive and apply
the Distance Formula and the
11.
Define and find
the x and y intercepts.
12.
Graph the equation
of a circle.
13.
Find the symmetry
of a graph.
14.
Use a graphing
calculator to solve equations and inequalities graphically.
15.
Write the
equations of lines.
Approximate duration: 10-12 days
UNIT II: FUNCTIONS
CCCS: 4.1, 4.2, 4.3, 4.4, 4.5
Objectives:
Students will be able to:
1.
Define and use a
function.
2.
Graph a function.
3.
Determine if a
graph is a function.
4.
Identify
intervals of increase and decrease from the graph of a function.
5.
Calculate the
Average Rate of Change of a function.
6.
Transform a
function around the Coordinate Plane.
7.
Calculate the
maximum or minimum of a quadratic function.
8.
Model with a
function.
9.
Combine
functions.
10.
Identify if a
function is one to one or not and find an inverse if it exists.
Approximate duration: 10-12 days
UNIT III: POLYNOMIAL AND RATIONAL FUNCTIONS
CCCS: 4.1, 4.2, 4.3,
4.5
Objectives:
Students will be able to:
1.
Graph polynomial
functions by finding the zeros.
2.
Graph rational
functions.
3.
Find the vertical
and horizontal asymptotes of a rational function algebraically.
Approximate duration:
4-5 days
UNIT IV: EXPONENTIAL AND LOGARITHMIC FUNCTIONS
CCCS: 4.1, 4.2, 4.3
Objectives:
Students will be able to:
1.
Define, evaluate
and graph exponential functions.
2.
Define, evaluate
and graph logarithmic functions.
3.
Use the laws of
logs to simplify and expand log expressions.
4.
Solve exponential
and logarithmic equations.
5.
Model with
exponential and logarithmic functions.
Approximate duration: 4-7 days
UNIT V: TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS
CCCS: 4.1, 4.2, 4.3
Objectives:
Students will be able to:
1.
Draw, label and
use the unit circle.
2.
Use the terminal
points from the unit circle to find the six trig functions of each point.
3.
Graph the six
trig functions and their relations.
4.
Write an equation
from a trig graph.
Approximate duration:
5-7 days
UNIT VI: TRIGONOMETRIC FUNCTIONS OF ANGLES
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1.
Define radian.
2.
Calculate and
convert angles from degrees to rads and rads to degrees.
3.
Calculate linear
and angular speeds.
4.
Find the arc
length and the area of a sector.
5.
Use right
triangle trig to calculate the unknown sides and angles of right triangles.
6.
Define and apply
the trigonometric function of angles.
7.
Define and use
the law of sines to find the unknown sides and angles of non-right triangles.
8.
Define and use
the law of cosines to find the unknown sides and angles of non-right triangles.
Approximate duration:
25-28 days
UNIT VII: ANALYTIC TRIGONOMETRY
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1.
Define and apply
the fundamental trig identities to simplifying expressions and proving
identities.
2.
Use the addition
and subtraction formulas to find the exact values of trig expressions and to
prove identities.
3.
Use the double
angle and half angle formulas to find the exact values of trig expressions and
to prove identities.
4.
Define, graph and
use the inverse trig functions.
5.
Solve
trigonometric equations both algebraically and graphically.
Approximate duration: 20-24 days
UNIT VIII: POLAR COORDINATES AND VECTORS
CCCS: 4.1, 4.2, 4.3
Objectives:
Students will be able to:
1.
Define polar
coordinates and convert rectangular coordinates to polar coordinates.
2.
To graph polar
equations.
3.
Represent complex
numbers, geometrically, and write complex numbers in polar form.
4.
Use and
understand DeMoivre’s Theorem.
Approximate duration: 14-16 days
UNIT IX: SEQUENCES AND SERIES
CCCS: 4.1, 4.3
Objectives:
Students will be able to:
1.
Identify if a
sequence or series is geometric, arithmetic, or neither.
2.
Write equations
that will give the nth term of a sequence.
3.
Write equations
that will give the sum of the first n terms of a series.
4.
Use the Binomial
Theorem.
Approximate duration:
15-18 days
UNIT X: SYSTEMS AND MATRICES
CCCS: 4.1, 4.3, 4.4, 4.5
Objectives:
Students will be able to:
1.
Represent systems
of equations as matrices.
2.
Solve matrices by
performing matrix operations.
3.
Draw a diagram of
a communication matrix, and make a communication matrix of a diagram.
4.
Use transition
matrices to predict future production outcomes.
Approximate duration: 14-16 days
UNIT XI: PROBABILITY
CCCS: 4.1, 4.3, 4.4, 4.5
Objectives:
Students will be able to:
1.
Solve permutation
and combination problems.
2.
Define emperical
and theoretic probabilities and know how to use each to predict outcomes of
events.
3.
Find sample
spaces of experiments.
4.
Determine if
events are dependent or independent.
Approximate duration: 20-24 days
UNIT XII: LIMITS: A PREVIEW OF CALCULUS
CCCS: 4.1, 4.2, 4.3, 4.5
Objectives:
Students will be able to:
1.
Evaluate basic
limits.
2.
Understand the
basis for the Tangent Line Problem.
3.
Determine from a
graph if a function is continuous or not.
4.
Graph piecewise
functions and understand their connection to continuity and limits.
Approximate duration: 10-14 days
List of texts,
resources, and/or literature:
TEXTBOOK: Stewart, Redlin, Watson, Precalculus, Thomson Brooks/Cole, 2006.