Download Pre-Calculus - Glen Ridge Public Schools

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Course Title: Pre- Calculus CP
Subject: Mathematics
Grade Level: 11/12
Duration: Full Year
Prerequisite: Algebra II grade of “B” or higher
Elective or Required: Elective
Mathematics Mission Statement
Since Mathematical and Computational thinking are an integral part of our lives and 21 st Century
learning, 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 will
emphasize thinking skills through a balance of computation, intuition, common sense, logic,
analysis and technology.
Students will be engaged and challenged in a developmentally appropriate, student-centered
learning environment. Students will communicate mathematical ideas effectively and apply those
ideas by using manipulatives, computational skills, mathematical models and technology in order
to solve practical problems.
To achieve these goals, students will be taught a standards-based curriculum that is aligned with
the National Common Core Standards in Mathematics and the New Jersey Core Curriculum
Content Standards in Technology and 21st Century Life and Careers.
Course Description:
Pre- Calculus is the course students take between Algebra II and Calculus. It
focuses primarily on trigonometry but it also covers the other major functions
students need in Calculus. It begins with a study of permutations and
combinations just prior to the trigonometry topics. Vectors, sequences and
series, and a review of functions round out the final topics of the course. PreCalculus prepares the students for both high school Calculus as well as college
Calculus I.
Author: Cluny Tierney
Date Submitted: Summer 2012
Course Name Pre- Calculus
Topic/Unit: Permutations and Combinations; Venn Diagrams
Approximate # Of Weeks: 4
Essential Questions:
What is the difference between a permutation and a combination?
What are real world examples of permutations and combinations?
What are the steps on the calculator to solving permutation and combination
problems involving large numbers?
What is probability?
How do you use a Venn Diagram to illustrate word problems?
NJCCS: S-CP 1, 2, 3, 7, 8, 9
Upon completion of this unit students will be able to:
 Solve problems that require them to use their knowledge of permutations and
combinations.
 Use their calculators to solve problems that involve large numbers.
 Find the probability of events that are calculated with the permutation and
combination formulas.
 Draw Venn diagrams to illustrate situations and use that Venn Diagram to answer
questions.
Interdisciplinary Standards (njcccs.org)
 9.1 21st Century Life and Career Skills
 8.1 Computer and Information Literacy
 8.2 Technology Education
 5.1 Science Practice
Activities – include 21st Century Technologies:
 Students will learn to use of Ti-83 plus permutation and combination functions.
 Introductory activity for calculating number of ways people can stand in a line.
 Students will discuss and discover how to calculate permutations and
combinations by completing problems sets.
 Students will take notes on instructor’s lessons.
 Smartboard lessons with students/teacher activities for Venn Diagrams.
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Students will complete given classwork and homework assignment.
Students will discuss their solutions to classwork and homework assignments.
Enrichment Activities:

Students will research and find examples of Venn Diagrams in other
disciplines.
Methods of Assessments/Evaluation:
 Pair/ Share
 Games involving movement
 Revisit Essential Questions
 Unit test
 Multi- media Presentations
 Self Assessments
 Think/Pair/Share
 Homework
 Classwork
 Independent work
 Observation
 Weekly Assessments
Resources/Including Online Resources
 Online Textbook Information:
 Teacher Webpage
 Class notes/ Worksheets
Course Name Pre- Calculus
Topic/Unit: Trigonometric Functions: Unit Circle Approach
Approximate # Of Weeks: 6 weeks
Essential Questions:
What is the equation for the unit circle?
Can you use the equation for the unit circle to find points on the unit circle and
determine if a point lies on the unit circle?
Can you label to unit circle as discussed in class?
What are the terminal points and the reference numbers for values on the unit
circle?
How can you relate the trig functions to the unit circle?
What are the special trig functions’ values?
What are the even and odd properties for the six trig functions?
What are the fundamental trigonometric identities and how are they used?
Can you graph the trig functions and apply the transformations to graphing them?
Can you write the equation of a trigonometric graph?
Can you find the inverse of a trig function and state the appropriate domain and
range restrictions?
Can you describe how simple harmonic motion is seen in real world phenomena?
NJCCS: A-CED 2, 3, 4. A-REI 1. F-IF 1, 2, 3, 4. F-BF 3, 4. F-TF 1, 2, 3, 4, 5, 6, 7, 8.
G-SRT 7, 8. G-GPE 1, 4.
Upon completion of this unit students will be able to:
 Apply the unit circle to solving a variety of problems.
 Use the fundamental trig identities to problem solve.
 Determine if a trig function is even, odd or neither.
 Graph trig equations and apply all the transformations to them.
 Write equations of given trig graphs.
 Define the restricted domain for each of the six basic trig functions.
 Find the inverse of a trig function.
Interdisciplinary Standards (njcccs.org)
 9.1 21st Century Life and Career Skills
 8.2 Technology Education
 5.1 Science Practice
Activities – include 21st Century Technologies:
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Students will discover the breakdown of the unit circle with the teacher’s
guidance.
Students will practice drawing their own unit circle repeatedly until complete
mastery.
Students will take notes on instructor’s lessons.
Smartboard activities and lessons on the trigonometry and relation to the unit
circle.
Students will complete classwork and homework problems.
Students will discuss solutions to classwork and homework problems.
Enrichment Activities:

Trig Bee
Methods of Assessments/Evaluation:
 Exit Slips
 Pair/Share
 Games involving movement
 Unit test
 Self Assessments
 Weekly Assessments
 Homework
 Classwork
 Independent Work
 Observation
Resources/Including Online Resources
 Online Textbook Information:
 Teacher Webpage
 Textbook Chapter 5
Course Name Pre- Calculus
Topic/Unit: Trigonometric Functions: Right Triangle Approach
Approximate # Of Weeks: 6 weeks
Essential Questions:
Can you convert from radians to degrees and degrees to radians?
Can you calculate the area of a sector and length of an arc of a circle?
What are the two types of circular motion and how do you find each?
What are the six trig functions and how do they relate to right triangles?
What are the values of the six trig functions for each of the special angle values in
degrees?
What does it mean to solve a right triangle?
How do you describe the angle of elevation and angle of depression?
What is a reference angle?
How do you find the area of a triangle given two sides and the included angle?
What are the Law of Sines and the Law of Cosines?
NJCCS: A-CED 2, 3, 4. A-REI 1. F-IF 1, 2, 3, 4. F-BF 3, 4. F-TF 1, 2, 5, 6, 7, 8. GSRT 7, 8, 9, 10, 11.
Upon completion of this unit students will be able to:
 Convert between degrees and radians.
 Solve all problems involving circles and linear and angular motion.
 Draw a parallel between degrees and radians.
 Solve any right triangle.
 Use the Pythagorean identities and the basic trig functions to problems solve.
 Calculate inverse trig function problems in terms of degrees.
 Solve problems involving angles of depression and angles of elevation.
 Use the Law of Sines and the Law of Cosines to solve triangles and problem
solve.
Interdisciplinary Standards (njcccs.org)
 9.1 21st Century Life and Career Skills
 8.2 Technology Education
 5.1 Science Practice
Activities – include 21st Century Technologies:
 Smartboard Lessons to demonstrate circular motion.
 Using ti-83 plus graphing calculator to problem solve.
 Students will complete more complicated problems in groups.
 There will be a teacher led discussion of proving identities.
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Students will problem solve in class and for homework.
Students will discuss solutions to their problems.
Enrichment Activities:

Comparing the different speeds and rpms of tires of different sizes.
Methods of Assessments/Evaluation:
 Exit Slips
 Pair/Share
 Games involving movement
 Unit test
 Self Assessments
 Weekly Assessments
 Homework
 Classwork
 Independent Work
 Observation
Resources/Including Online Resources
 Online Textbook Information:
 Teacher Webpage
 Textbook Chapter 6
Course Name Pre- Calculus
Topic/Unit: Analytic Trigonometry
Approximate # Of Weeks: 6 weeks
Essential Questions:
How do you use the following trig identities in verifying other identities:
Pythagorean, Reciprocal, Quotient, Sum/Difference, Double Angle and Half Angle?
How do you solve a trigonometric equation both algebraically and graphically?
What is the significance of the restricted to domain when solving trig equations?
When there is no restriction on the domain, how many solutions are there for trig
equations and why?
NJCCS: A-CED 2, 3. A-REI 1. F-IF 1, 2, 3, 4. F-BF 3, 4. F-TF 1, 2, 5, 6, 7, 8, 9.
Upon completion of this unit students will be able to:
 Use the trig identities to simply expressions and prove identities.
 Use the addition and subtraction formulas for finding exact values of expressions,
to simplify expressions and to prove identities.
 Use the double angle, half angle and product to sum formulas to verify identities.
 Solve basic trig equations both algebraically and graphically.
 Solve more complicated trig equations in the restricted interval [0, 2π).
 Use a calculator to solve equations that cannot be solved by hand.
Interdisciplinary Standards (njcccs.org)
 9.1 21st Century Life and Career Skills
 8.2 Technology Education
 5.1 Science Practice
Activities – include 21st Century Technologies:
 Smartboard Lessons will be used to introduce the topics.
 Use of ti-83 plus graphing calculator for graphing and solving equations that
cannot be done by hand.
 Students will complete problems both in class and for homework.
 Students will discuss their findings and problems.
Enrichment Activities:

Finding all the possible solutions to trig equations.
Methods of Assessments/Evaluation:
 Pair/Share
 Unit test
 Self Assessments
 Weekly Assessments
 Homework
 Classwork
 Independent Work
 Observation
Resources/Including Online Resources
 Online Textbook Information:
 Teacher Webpage
 Textbook Chapter 7
Course Name Pre- Calculus
Topic/Unit: Vectors
Approximate # Of Weeks: 2 weeks
Essential Questions:
What is the magnitude of a vector and how can it be represented?
How do vectors relate to the Pythagorean Theorem?
What are the algebraic operations on vectors and how are they used?
What are the properties of vectors?
What are examples of vectors seen out of the classroom?
How do you represent the components of vectors?
How do you calculate the work done by a force moving along a vector?
NJCCS: N-VM 1, 2, 3, 4, 5.
Upon completion of this unit students will be able to:
 Sketch vectors.
 Find the magnitude of vectors.
 Use properties of vectors to calculate new vectors.
 Find the components of vectors.
 Find the angle formed by two vectors.
 Find the work done by a force in a moving object.
Interdisciplinary Standards (njcccs.org)
 9.1 21st Century Life and Career Skills
 9.1 Career Awareness, Exploration, and Preparation
 8.2 Technology Education
 5.1 Science Practice
Activities – include 21st Century Technologies:
 Students will take notes on Smartboard lessons given by teacher.
 Students will work independently to solve problems.
 Students will discuss solutions to problems.
Enrichment Activities:

How are vectors seen around us every day?
Methods of Assessments/Evaluation:
 Pair/Share
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Self Assessments
Weekly Assessments
Homework
Classwork
Independent Work
Observation
Resources/Including Online Resources
 Online Textbook Information:
 Teacher Webpage
 Textbook Sections 9.1, 9.2
Course Name
Topic/Unit: Sequences and Series
Approximate # Of Weeks: 5 weeks
Essential Questions:
What is the difference between a sequence and a series?
How do you determine if a sequence is arithmetic, geometric or neither?
What is the downfall of defining a sequence recursively?
What are the examples of sequences seen in nature?
When does an infinite geometric series diverge?
NJCCS: F-IF 3. F-BF 1a, 2. A-APR 5.
Upon completion of this unit students will be able to:
 Identify sequences as arithmetic, geometric or neither.
 Find the sum of a sequence.
 Use Sigma Notation.
 Write definitions for finding the nth term of a sequence both recursively and
explicitly.
 Calculate the sum of an infinite geometric series if it exists.
Interdisciplinary Standards (njcccs.org)
 9.1 21st Century Life and Career Skills
 9.1 Career Awareness, Exploration, and Preparation
 8.2 Technology Education
 5.1 Science Practice
Activities – include 21st Century Technologies:
 Smartboard Lessons to relay notes to students.
 Using ti-83 plus graphing calculator to calculate sums.
 Students will complete problems requiring them to use topics discussed in class.
 Students will discuss solutions to problems.
Enrichment Activities:

Fibonacci Series Activity
Methods of Assessments/Evaluation:
 Pair/Share
 Self Assessments
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Weekly Assessments
Homework
Classwork
Independent Work
Observation
Resources/Including Online Resources
 Online Textbook Information:
 Teacher Webpage
 Textbook Chapter 12
Course Name Pre- Calculus
Topic/Unit: Functions
Approximate # Of Weeks: 5 weeks
Essential Questions:
What is a function?
How do you calculate the average rate of change of a function over a given interval?
What is a one- to- one function and what is its significance?
What are the basic polynomial functions and what do their graphs look like?
What properties does a ration function’s graph have?
What is the difference between an exponential function and a logarithmic function?
Can you graph all the different types of functions discussed?
NJCCS: A-SSE 3, 4, 5. A-APR 2, 3, 4. A-REI 1, 2, 3, 4, 5, 6, 7, 10, 11, 12. F-IF 1, 2,
3, 4, 5, 6, 7, 8, 9. F- BF 1a, 1c, 3, 4, 5. F-LE 1, 2, 3, 4, 5
Upon completion of this unit students will be able to:
 Graph polynomial, rational, exponential, and logarithmic functions.
 Solve for x in a variety of situations.
 Use transformations to graph functions.
Interdisciplinary Standards (njcccs.org)
 9.1 21st Century Life and Career Skills
 9.1 Career Awareness, Exploration, and Preparation
 8.2 Technology Education
 5.1 Science Practice
Activities – include 21st Century Technologies:
 Smartboard Lessons
 Using the Ti-83 plus graphing calculator to solve equations.
 Graphing functions activity.
 Students will work independently to solve and graph functions.
 Students will discuss solutions in small groups and as a class.
Enrichment Activities:

Line of best fit activity.
Methods of Assessments/Evaluation:
 Pair/Share
 Self Assessments
 Weekly Assessments
 Homework
 Classwork
 Independent Work
 Observation
Resources/Including Online Resources
 Online Textbook Information:
 Teacher Webpage
 Textbook Chapter 2, Sections 3.1, 3.2, 3.7, Chapter 4