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Glen Ridge Public Schools –Mathematics Curriculum

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Course Title: Pre Calculus Honors

Subject: Mathematics

Grade Level: 11th

Duration: Full Year

Prerequisite: Algebra II Honors grade of “B” or higher; Teacher recommendation

Elective or Required: Elective

Mathematics Mission Statement

Since Mathematical and Computational thinking are an integral part of our lives and 21st 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 Honors is a comprehensive course that covers an array of topics. These topics are based in functions where their graphs, equations, and solutions are deeply explored. The emphasized functions are trigonometric, polynomial, exponential, logarithmic and rational. There is a study of sequences and series and the course concludes with a preview of Calculus. This course moves at a quick pace to prepare students for AP Calculus.

Author: Cluny Tierney

Date Submitted: Summer 2012

Course Name: Pre- Calculus Honors

Topic/Unit: Permutations and Combinations; Venn Diagrams

Approximate # Of Weeks: 3 Weeks

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 ()

• 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.

• Students will complete given classwork and homework assignment.

• Students will discuss their solutions to classwork and homework assignments.

Enrichment Activities:

• Students write their own word problems that require the use of Venn Diagrams.

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 Honors

Topic/Unit: Trigonometric Functions: Unit Circle Approach

Approximate # Of Weeks: 5 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 apply what you know about trig graphs and functions to modeling simple harmonic motion?

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.

• Find the inverse of a trig function.

• Graph the inverse of trig functions.

• Problem solve situations involving simple harmonic motion.

Interdisciplinary Standards ()

• 9.1 21st Century Life and Career Skills

• 8.2 Technology Education

• 5.1 Science Practice

Activities – include 21st Century Technologies:

• 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 Honors

Topic/Unit: Trigonometric Functions: Right Triangle Approach

Approximate # Of Weeks: 4 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. G-SRT 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 ()

• 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.

• Students will problem solve in class and for homework.

• Students will discuss solutions to their problems.

Enrichment Activities:

• Use the circle formulas to calculate distances between cities on the Earth.

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 Honors

Topic/Unit: Analytic Trigonometry

Approximate # Of Weeks: 5 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 ()

• 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:

• Activity on Traveling and Standing Waves

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

• Worksheets

Course Name Pre- Calculus Honors

Topic/Unit: Polar Coordinates

Approximate # Of Weeks: 3 weeks

Essential Questions:

What is the relationship between Polar and Rectangular Coordinates?

What are the variables in polar equations and what do they represent?

How do the graphs of polar equations differ from the graphs of rectangular equations?

What is the significance of DeMoivre’s Theorem and why is it helpful?

How can you find all the roots for complex numbers?

NJCCS: N-CN 1, 3, 4, 5, 6

Upon completion of this unit students will be able to:

• Convert points and equations back and forth between Polar and Rectangular forms.

• Graph Polar Equations.

• Use DeMoivre’s Theorem in problem solving.

• Calculate all the roots for any given complex number.

Interdisciplinary Standards ()

• 9.1 21st Century Life and Career Skills

• 8.2 Technology Education

• 5.1 Science Practice

Activities – include 21st Century Technologies:

• Matching graphs/equations

• Graphing on Polar Axis

• Using ti-83 plus graphing calculator to graph Polar Equations

• Students will work in groups and independently to solve problems.

• Students will discuss solutions as a class.

Enrichment Activities:

• Research on Mathematicians: DeMoivre, Agnesi, Galilei.

Methods of Assessments/Evaluation:

• 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 8

• Worksheets

Course Name Pre- Calculus Honors

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?

What is the dot product and what does it represent?

What are orthogonal vectors and how do you determine if two vectors are orthogonal?

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.

• Calculate the dot product of two vectors.

• Find the angle formed by two vectors.

• Calculate the component of a vector along another vector.

• Find the work done by a force in a moving object.

Interdisciplinary Standards ()

• 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.

• Labs involving vectors: the way the wind blowing affects a boat.

Enrichment Activities:

• Explore Vector Fields in the Plane

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 Sections 9.1, 9.2

Course Name Pre- Calculus Honors

Topic/Unit: Matrices

Approximate # Of Weeks: 2 weeks

Essential Questions:

What are the necessary properties of matrices that are going to get added or subtracted?

What is different about multiplying matrices from multiplying other quantities?

How are matrices used in real world applications?

How can matrices be used to represent vectors?

NJCCS: N-VM 6, 7, 8, 9, 10, 11, 12.

Upon completion of this unit students will be able to:

• Perform basic matrix addition, subtraction, scalar multiplication and multiplication or determine that such operations are impossible.

• Use matrices to represent data.

• Find the inverse of a matrix and use it to solve matrix equations.

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Interdisciplinary Standards ()

• 9.1 21st Century Life and Career Skills

• 9.3 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 to multiply matrices.

• Communication matrices activity.

• Students will problem solve collaboratively and independently.

• Students will discuss findings as a class with the teacher leading the discussion.

Enrichment Activities:

• Explore specific careers and examples where matrices are used.

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 Sections 10.4, 10.5

• Worksheets

Course Name Pre- Calculus Honors

Topic/Unit: Sequences and Series

Approximate # Of Weeks: 4 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?

How does the Binomial Theorem relate to combinations we discussed in the beginning of the year?

What are the steps to proving a statement by Mathematical Induction?

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

Interdisciplinary Standards ()

• 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.

• Pascals Triangle Activity

• Students will complete problems requiring them to use topics discussed in class.

• Students will discuss solutions to problems.

• Students will work together and independently on proving statements using Mathematical Induction.

Enrichment Activities:

Fibonacci Series 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 12

Course Name Pre- Calculus Honors

Topic/Unit: Functions

Approximate # Of Weeks: 4 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 given x values.

• Use transformations to graph functions.

Interdisciplinary Standards ()

• 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:

• Applications of logarithms to earthquake 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

Course Name Pre- Calculus Honors

Topic/Unit: Limits: A Preview of Calculus

Approximate # Of Weeks: 3 weeks

Essential Questions:

What is the tangent line problem?

How do we find the slope of a line tangent to a curve?

What is a limit?

What are the types of limits that do not exist?

How does continuity play into the evaluation of limits?

NJCCS: A-SSE 3a. A- CED 1, 2, 3. A- REI 1, 2, 3, 4, 10, 11.

Upon completion of this unit students will be able to:

• Find the slope of a line tangent to a curve at a given point and in general.

• Evaluate limits both graphically and algebraically.

• Identify limits that do not exist.

• Explain why a limit does not exist.

• Define a derivative.

Interdisciplinary Standards ()

• 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

• Calculus in Motion demonstrations

• Ti-83 plus graphing calculator activities

• Students will work independently and in small groups to complete tangent line problem activities.

• Students will work independently to evaluate limits.

• Students will discuss solutions to problems.

Enrichment Activities:

• Applications of derivatives to linear motion.

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 13

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