Monroe Township Schools
Curriculum Management System
Precalculus
Grade 11
June 2009
* For adoption by all regular education programsBoard Approved: July 2009
as specified and for adoption or adaptation by
all Special Education Programs in accordance
with Board of Education Policy # 2220.
Table of Contents
Monroe Township Schools Administration and Board of Education MembersPage 3
AcknowledgmentsPage 4
District Mission Statement and GoalsPage 5
Introduction/Philosophy/Educational GoalsPages 6
National and State StandardsPage 7
Scope and SequencePage 8-9
Goals/Essential Questions/Objectives/InstructionalTools/ActivitiesPages 10-35
BenchmarksPage 36
AddendumPages 37 -
MONROETOWNSHIPSCHOOL DISTRICT
ADMINISTRATION
Dr. Kenneth Hamilton, Superintendent
Mr. Jeff Gorman, Assistant Superintendent
BOARD OF EDUCATION
Ms. Amy Antelis, President
Ms. Kathy Kolupanowich, Vice President
Mr. Marvin Braverman
Mr. Ken Chiarella
Mr. Lew Kaufman
Mr. Mark Klein
Mr. John Leary
Ms. Kathy Leonard
Mr. Ira Tessler
JAMESBURG REPRESENTATIVE
Ms. Patrice Faraone
Student Board Members
Ms. Nidhi Bhatt
Ms. Reena Dholakia
Acknowledgments
The following individuals are acknowledged for their assistance in the preparation of this Curriculum Management System:
Writers Names: Beth Goldstein
Supervisor Name: Robert O’Donnell, Supervisor of Mathematics & Educational Technology
Technology Staff: Al Pulsinelli
Reggie Washington
Secretarial Staff:Debbie Gialanella
Geri Manfre
Gail Nemeth
Monroe Township Schools
Mission and Goals
Mission
The mission of the MonroeTownshipSchool District, a unique multi-generational community, is to collaboratively develop and facilitate programs that pursue educational excellence and foster character, responsibility, and life-long learning in a safe, stimulating, and challenging environment to empower all individuals to become productive citizens of a dynamic, global society.
Goals
To have an environment that is conducive to learning for all individuals.
To have learning opportunities that are challenging and comprehensive in order to stimulate the intellectual, physical, social and emotional development of the learner.
To procure and manage a variety of resources to meet the needs of all learners.
To have inviting up-to-date, multifunctional facilities that both accommodate the community and are utilized to maximum potential.
To have a system of communication that will effectively connect all facets of the community with the MonroeTownshipSchool District.
To have a staff that is highly qualified, motivated, and stable and that is held accountable to deliver a safe, outstanding, and superior education to all individuals.
INTRODUCTION, PHILOSOPHY OF EDUCATION, AND EDUCATIONAL GOALS
PhilosophyMonroe Township Schools are committed to providing all students with a quality education resulting in life-long learners who can succeed in a global society. The mathematics program, grades K - 12, is predicated on that belief and is guided by the following six principles as stated by the National Council of Teachers of Mathematics (NCTM) in the Principles and Standards for School Mathematics, 2000. First, a mathematics education requires equity. All students will be given worthwhile opportunities and strong support to meet high mathematical expectations. Second, a coherent mathematics curriculum will effectively organize, integrate, and articulate important mathematical ideas across the grades. Third, effective mathematics teaching requires the following: a) knowing and understanding mathematics, students as learners, and pedagogical strategies, b) having a challenging and supportive classroom environment and c) continually reflecting on and refining instructional practice. Fourth, students must learn mathematics with understanding. A student's prior experiences and knowledge will actively build new knowledge. Fifth, assessment should support the learning of important mathematics and provide useful information to both teachers and students. Lastly, technology enhances mathematics learning, supports effective mathematics teaching, and influences what mathematics is taught.
As students begin their mathematics education in Monroe Township, classroom instruction will reflect the best thinking of the day. Children will engage in a wide variety of learning activities designed to develop their ability to reason and solve complex problems. Calculators, computers, manipulatives, technology, and the Internet will be used as tools to enhance learning and assist in problem solving. Group work, projects, literature, and interdisciplinary activities will make mathematics more meaningful and aid understanding. Classroom instruction will be designed to meet the learning needs of all children and will reflect a variety of learning styles.
In this changing world those who have a good understanding of mathematics will have many opportunities and doors open to them throughout their lives. Mathematics is not for the select few but rather is for everyone. Monroe Township Schools are committed to providing all students with the opportunity and the support necessary to learn significant mathematics with depth and understanding. This curriculum guide is designed to be a resource for staff members and to provide guidance in the planning, delivery, and assessment of mathematics instruction.
Educational Goals
Precalculus is the fourth course in the regular college preparatory sequence. This course applies the skills obtained in Algebra I, Geometry and Algebra II. The topics covered include exponents, logarithms, trigonometric functions and identities, solving trigonometric equations, applications involving triangles, inverse trigonometric functions, trigonometric addition formulas, advanced graphing techniques, polar coordinates and complex numbers, sequences and series, and limits.
This course is intended to prepare students for a post secondary education. It emphasizes higher-level mathematical thinking necessary to pursue the study of calculus.
New Jersey State Department of Education
Core Curriculum Content Standards
A note about Mathematics Standards and Cumulative Progress Indicators.
The New Jersey Core Curriculum Content Standards for Mathematics were revised in 2004. The Cumulative Progress Indicators (CPI's) referenced in this curriculum guide refer to these new standards and may be found in the Curriculum folder on the district servers. A complete copy of the new Core Curriculum Content Standards for Mathematics may also be found at:
Precalculus
Scope and Sequence
Quarter IBig Idea: Functions
- Transformations of functions
b. Symmetry in the x-axis, y-axis, and origin
c. Periodic Functions
d. Translations of y=f(x) to y-k=f(x-h)
e. Vertical (y=cf(x)) and horizontal (y=f(cx)) stretching or shrinking of y=f(x). / Big Idea: Trigonometric Functions
- Introduction to Trigonometric Functions
b. Arc length and area of a sector
c. Evaluating trigonometric expressions
d. Graphs of trigonometric functions
Big Idea: Trigonometric Equations and Applications
- Trigonometric Equations and Applications
b. Vertical and horizontal stretching and shrinking of sine and cosine functions
c. Simplifying trigonometric expressions and proving trigonometric identities
d. Trigonometric equations
Quarter II
Big Idea: Triangle Trigonometry
- Triangle Trigonometry
b. Area of a triangle
c. Law of Sines
d. Law of Cosines / Big Idea: Trigonometric Addition Formulas
- Trigonometric Addition Formulas
b. Double angle formulas
c. Trigonometric equations
Quarter III
Big Idea: Polar Coordinates and Complex Numbers
- Polar Coordinates and Complex Numbers
b. Conversions of rectangular and polar coordinates
c. Graphs of polar functions
d. Conversions of complex numbers between rectangular and polar form
e. Product of two complex numbers in polar form
f. De Moivre’s theorem
g. Roots of complex numbers / Big Idea: Sequences and Series
- Sequences and series
b. nth term of a geometric sequence
c. Recursive definitions
d. Sum of finite arithmetic and geometric series
e. Sum of infinite geometric series
f. Sigma notation
Quarter IV
Big Idea: Limits
- Limits
b. Limits of functions that approach a real number
c. Graphs of rational functions / Big Idea: Exponents and Logarithms
- Exponents and Logarithms
b. Compound interest formula and formula for interest compounded continually
c. Evaluate logarithmic expressions (change of base formula)
d. Expand and condense logarithmic expressions
e. Logarithmic and exponential equations
Suggested blocks of Instruction / Curriculum Management System
Grade Level/Subject:
Grade 11/Precalculus / Big Idea:Functions
Goal 1:The student will be able to stretch, shrink, reflect, or translate the graph of a function, and determine the inverse of a function, if it exists.
Objectives / Cluster Concepts /
Cumulative Progress Indicators (CPI's)
The student will be able to: / Essential Questions
Sample Conceptual Understandings / Instructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment Model
1.1.To sketch the reflection of a graph in the -axis and-axis. (4.2.12B.1; 4.2.12B.3)
1.2.To write the inverse of an equation and sketch the graph of the reflection in the line . (4.2.12B.1; 4.3.12B.3)
1.3.To determine if the graph of an equation has symmetry in the -axis, the -axis, the line, and the origin. (4.2.12B.1; 4.3.12B.4)
1.4.To determine if a function is periodic. (4.3.12B.1)
1.5.To evaluate a function using the fundamental period. (4.3.12B.2)
1.6.To determine the period and amplitude of a periodic function. (4.3.12B.2)
1.7.To understand the effect of and sketch the graph of by vertically stretching or shrinking the graph of . (4.2.12B.1; 4.3.12B.3)
1.8.To understand the effect of and sketch the graph of by horizontally stretching or shrinking the graph of . (4.2.12B.1; 4.3.12B.3)
1.9.To understand the effect of and and sketch the graph of the equation by translating the graph of horizontally units and vertically units.
1.10.To determine if two functions are inverse functions by applying the definition.
1.11.To apply the Horizontal Line Test to determine if a function has an inverse. / Essential Questions:
- What changes in an equation produces the reflection of its graph in the -axis, the -axis, and the line ?
- How do you tell whether the graph of an equation has symmetry in the the -axis, the -axis, the line , and the origin?
- Given the graph of , what effect does have on the graph of and ?
- Given the graph of , what effect does and have on the graph of ?
- How can the vertical-line test be used to justify the horizontal-line test?
- If the equation is changed to:
b. , then the graph of is unchanged when and reflected in the -axis when .
c. , then the graph of is reflected in the -axis.
d. , then the graph of is reflected in the line .
e. , then the graph of is stretched vertically.
f. , then the graph of is
shrunk vertically.
g. , then the graph of is
shrunk horizontally.
h. , then the graph of is
stretched horizontally.
i. , then the graph of is translated units horizontally and units vertically.
- A graph is symmetric in the -axis if is on the graph whenever is. An equation of a graph is symmetric in the -axis if an equivalent equation results after substituting for .
- A graph is symmetric in the -axis if is on the graph whenever is. An equation of a graph is symmetric in the -axis if an equivalent equation results after substituting for .
- A graph is symmetric in the line if is on the graph whenever is. An equation of a graph is symmetric in the line if an equivalent equation results after interchanging and .
- A graph is symmetric in the origin if is on the graph whenever is. An equation of a graph is symmetric in the origin if an equivalent equation results after substituting for and for .
- Two functions and are inverse functions if:
2. for all in the domain of .
- The Horizontal Line Test: If the graph of the function is such that no horizontal line intersects the graph in more than one point, thenis one-to-one and has an inverse.
Depending upon the needs of the class, the assessment questions may be answered in the form of essays, quizzes, mobiles, PowerPoint, oral reports, booklets, or other formats of measurement used by the teacher.
Resources:
Precalculus with Limits A Graphing Approach, Fifth Edition, Larson et al; Houghton Mifflin, 2008
Learning Activity:
- In the following activity, how does a change in the equation result in the reflection of its graph in some line? (analysis)
2. Graph and . Graph and . How are the graphs of and related?
3. Graph and . Graph and . How are the graphs of and related?
4. Graph and . Graph and . How is the graph of an equation affected when you interchange the variables in the equation?
- Given the graph of , sketch the graphs of , , and using different colored crayons.
Suggested blocks of Instruction / Curriculum Management System
Grade Level/Subject:
Grade 11/Precalculus / Big Idea:Trigonometric Functions
Goal 2:The student will be able to evaluate and graph trigonometric functions.
Objectives / Cluster Concepts /
Cumulative Progress Indicators (CPI's)
The student will be able to: / Essential Questions
Sample Conceptual Understandings / Instructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment Model
2.1.To convert degree measures of angles to radians.
2.2.To convert radian measures of angles to degrees.
2.3.To determine coterminal angles.
2.4.To determine the arc length and area of a sector of a circle with central angles in either degrees or radians.
2.5.To use the definitions of sine and cosine to evaluate these functions. (4.3.12D.1)
2.6.To use reference angles, calculators or tables, and special angles to evaluate sine and cosine functions. (4.3.12D.1)
2.7.To sketch the graph of sine and cosine functions. (4.3.12B.2)
2.8.To use reference angles, calculators or tables, and special angles to evaluate tangent, cotangent, secant, and cosecant functions. (4.3.12D.1)
2.9.To sketch the graphs of tangent, cotangent, secant, and cosecant functions. (4.3.12B.2)
2.10.To sketch the graph of the inverse of the sine, cosine, and tangent functions, and determine the domain and range. (4.3.12B.2)
2.11.To evaluate the inverse of sine, cosine, and tangent functions with and without a calculator or table. (4.3.12D.1) / Essential Questions:
- What are radians and how are they related to degrees?
- Explain the process of evaluating a trigonometric function using reference angles and the unit circle.
- How do the values on the unit circle correlate to the rectangular graph of a trigonometric function?
- Why is it necessary to restrict the domain in order to discuss inverse trigonometric functions?
- , where is the measure of the central angle, in radians, is the arc length, and is the length of the radius.
- To convert each degree measure to radians, multiply by .
- To convert each radian measure to degrees, multiply by .
- The following formulas are used for the arc length and area of a sector with central angle :
b. If is in radians, then and .
- ,
- ,
- ,
- ,
- The signs of the trigonometric functions (sine and cosecant, cosine and secant, and tangent and cotangent) in the four quadrants can be summarized by the following phrase: All Students Take Calculus.
- In order to evaluate a trigonometric expression,
2. Determine if it is positive, negative, or zero.
3. Determine the reference angle.
4. Determine the exact value, if possible, in simplest radical form. / Sample Assessment Questions:
- Convert each angle to radians in terms of .
b)
c)
d)
- Convert each angle to degrees.
f)
g)
h)
- Give one positive and one negative coterminal angle for each angle below. Use the given form of the angle.
j).
- A sector of a circle has central angle 1.2 radians and radius 6cm.
l)Find its area.
- Find the value of each expression leave answers in simplest radical form. Show reference angle statement when necessary.
n)cos
o)tan
p)sec
q)cos ()
r)sin 3
s)csc
t)cot
u)sec
2)Graphing Project (See Addendum)
Students will complete the following tasks and present their graphs in a neat and accurate presentation.
a)Completea table of exact values for all special angles and quadrantal angles .
b)Graph each of the 6 trigonometric
functions on a separate graph. Include an accurate scale and asymptotes where appropriate.
Suggested blocks of Instruction / Curriculum Management System
Grade Level/Subject:
Grade 11/Precalculus / Big Idea:Trigonometric Equations and Applications
Goal 3:The student will be able to stretch, shrink, and translate sine and cosine functions, simplify trigonometric expressions, and solve trigonometric equations.
Objectives / Cluster Concepts /
Cumulative Progress Indicators (CPI's)
The student will be able to: / Essential Questions
Sample Conceptual Understandings / Instructional Tools / Materials / Technology / Resources / Learning Activities / Interdisciplinary Activities / Assessment Model
3.1.To solve and apply simple trigonometric equations. (4.3.12D.2)
3.2.To determine the slope and equation of a line given the angle of inclination and the coordinates of a point on the line, and determine the angle of inclination given the equation of a line or information about the line. (4.2.12C.1)
3.3.To stretch and shrink the graphs of sine and cosine functions.(4.2.12B.1)
3.4.To determine the period and amplitude of and . (4.3.12B.2)
3.5.To determine the amplitude and period, and write the equation of sine and cosinecurves. (4.3.12B.2)
3.6.To solve equations of the form and . (4.3.12D.2)
3.7.To determine the amplitude, period, axis of wave, and sketch the graph of translated sine and cosine functions. (4.3.12B.3; 4.3.12B.4)
3.8.To determine the amplitude, period, axis of wave, and write the equation of the graph of translated sine and cosine functions. (4.2.12B.1; 4.3.12B.2)
3.9.To simplify trigonometric expressions using the reciprocal relationships, relationships with negatives, Pythagorean relationships, and cofunction relationships of trigonometric functions. (4.3.12D.1)
3.10.To prove trigonometric identities using the reciprocal relationships, relationships with negatives, Pythagorean relationships, and cofunction relationships of trigonometric functions. (4.2.12A.4)
3.11.To use trigonometric identities or graphing calculator to solve more difficult trigonometric equations. (4.3.12D.2) / Essential Questions:
- How does a change in amplitude or period affect the graph of a Sine or Cosine curve?
- Explain the effect of A, B, h, and k on the graph of a sine or Cosine curve using the equations
- Explain how to find all possible solutions to simple trigonometric equations over a given domain.
- How can the graph of a trigonometric function be used to anticipate the number of solutions to a trigonometric equation?
- Could a single curve be described using both a Sine function and a Cosine function? Why?