pre-calculus 2007- complete

Board Approval: August 21, 2007

Curriculum Committee

Barbara Popp, Supervisor of Curriculum and Instruction Rebecca Fosbre Jennifer Griffiths Jillian Goldstein Carol McGinley

Rosemary Perrotti Jennifer Pisano Karen Sweeney

Members of the Board of Education: Louis Petzinger, President

Andrew Zangara, Vice President Herman Brunn

Gary Cortelyou Michael Impellizeri

George Jakelski Frank Jurewicz

Ken Lessing Ned Panfile

Dr. Donald Burkhardt, Superintendent

Richard D. Reilly, Board Secretary

Pre-Calculus 2007

Mission/Vision Statement

Pre-Calculus Curriculum 2007 2

The mission of the Manville School District is to create for our community a progressive learning environment, which imparts knowledge and the love of learning, inspires respect for all people, instills pride, and embraces change. All students of the Manville School District will achieve excellence in mathematics as measured by regular formative and summative assessment, by engaging in inquiry-centered learning, which is based on the New Jersey Core Curriculum Content Standards. Each student will experience success in basic mathematics and higher level thinking through active hands-on learning, problem solving strategies, and appropriate tools and technology. All students leaving the Manville School District will pursue a post-secondary education with a desire for life-long learning.

Course Sequence K-12


MANVILLE SCHOOL DISTRICT K-12 MATHEMATICS COURSE SEQUENCES

K-3 4-5 6-8 9 10 11 12

Geometry

Honors

Algebra 2 Honors Pre-Calculus Calculus

Grade 6 Pre-Algebra

Part 1 Geometry Algebra 2 Algebra 2 Pre-Calculus

Grade 7 Pre-Algebra

Part 2

Pre-Algebra Part 2 Honors

Geometry Probability and Statistics

Probability and Statistics

Comprehensive Algebra

Probability and Statistics

Math Applications

Math Applications

Math K, 1, 2, 3 Math 4, 5

Grade 8 Algebra 1

Algebra Honors Foundations of

Math 1 Foundations of



Math 4

Unit Design


MANVILLE SCHOOL DISTRICT

Precalculus UNIT 1 – Prerequisites for Precalculus

Stage 1- Desired Results

Established Goals: 4.3.A-D; 4.5.A-F Understandings: Students will understand that…

• sets of numbers can be arranged in a hierarchical fashion

• a rigorous structure for representing numbers and sets of numbers exists

• two sets of numbers interact with each other

• you can visually represent the interaction between two sets of numbers

Essential Questions: • In what ways can all of the mathematics you

have learned thus far in your academic career be used in pre-calculus?

• What other types of linear relationships exist?

• What methods and tools are available to solve equations and inequalities?

• What constitutes a linear relationship?

Students will know… • how the various subsets of the real

number system relate to each other • how to deal with linear equations

and inequalities in various forms and representations

• how to solve various equations and inequalities in a variety of ways

• how to deal with complex numbers and the unique situation they present

Students will be able to… • work with the real numbers and its various

subsets • work with the Cartesian coordinate system

to represent various sets of numbers • use increments to calculate slopes; write an

equation and sketch a graph given specific information

• solve equations and inequalities by various methods

• work with the complex number system and its various representations

Stage 2- Assessment Evidence

Performance Tasks: • Lesson P.5 – Effective Reach of a Ladder • Lesson P.7 – Projectile Motion

Key Criteria • Use specific/general rubric


Other Evidence • Open-ended assessments • Self and peer assessments • Communicators • ACTIVote quizzes • Teacher created materials • Formative assessments • Projects • Web-quests • Standardized-test scores

Stage 3- Learning Plan

Learning Activities: • Begin with an entry question to hook students (prior knowledge) • Introduce and discuss essential questions (revisit throughout the unit) • ‘Do Probs’ / ‘Do Nows’ / ‘Problem of the Day’ • Introduce Learning Expectation(s) • Complete applicable lessons from textbook, Precalculus: Graphical, Numerical, Algebraic

by Pearson Prentice Hall o Chapter P, Lessons P.1 through P.7

• Use Promethean Board presentations to enhance learning • Utilization of varying questioning techniques to develop concept, explore concept, or

reinforce concept • Independent/Cooperative learning explorations • Evoke student participation both from their seat and at the board (use of

ACTIVotes/ACTIVslate) • Establish a set of general strategies for student independence and self-evaluation • Graphing calculator activities • United Streaming Math Video



Precalculus UNIT 2 – Functions and Graphs


Established Goals: 4.3.B&C; 4.5.A-F Understandings: Students will understand that…

• what a function is • what makes a function special • a function can be defined • how a function can be used • there is a specific “language” used

for specifying functions

Essential Questions: • How can functions be used to model many

things that happen in the real world? • How can functions be changed just as things

in the real world change?

Students will know… • that there are a variety of ways to

specify and define functions • that the two key concepts of “limit”

and “continuity” are tightly coupled • there are twelve basic functional

models that cover virtually any situation

• functions can be combined • functions can be modified • functions can be used to build

models of real world situations

Students will be able to… • build numerical, algebraic, and graphical

models for functions • define a function using the proper notation • understand and explain the concepts of limit

and continuity • recognize and work with the “twelve basic

functions” • combine functions to produce other

functions • define functions using parametric methods • transform functions using graphical

methods • build functional models from a variety of

information provided


Performance Tasks: • Lesson 1.4 – The Shrinking Snowball • Lesson 1.7 – Investment Returns





Learning Activities: • Begin with an entry question to hook students (prior knowledge) • Introduce and discuss essential questions (revisit throughout the unit) • ‘Do Probs’ / ‘Do Nows’ / ‘Problem of the Day’ • Introduce Learning Expectation(s) • Complete applicable lesson from textbook, Precalculus: Graphical, Numerical, Algebraic

by Pearson Prentice Hall o Chapter 1, Lessons 1.1 through 1.7 o Chapter 10, Lessons 10.1 through 10.3






Precalculus UNIT 3 – Polynomial, Power and Rational Functions



• some models have real solutions, while others will not

• some models may have multiple solutions, but some of these may not be appropriate

• the graph of a function can provide significant insights into the nature of solutions for the function

• what it means to “model” a situation

Essential Questions • What is a “solution” of a model? • How many solutions can exist? Explain. • Do we care about all of the solutions? Why

or why not? • When is a general picture worth more than a

lot of specific computation? • How can more than one model be possible

for a given real world situation?

Students will know… • that various types of functions can

be used to model different types of situations

• what the Fundamental Theorem of Algebra states about a given polynomial function

• that some functions have no real solutions, but all functions have complex solutions

Students will be able to… • work with and solve linear, quadratic, and

higher degree functions • determine how many solutions will exist for

a given function • apply both polynomial division and

synthetic division to find solutions for a function

• locate the real zeros (if any) of a function of varying degree

• locate the complex zeros of a function of varying degrees



Performance Tasks: • Lesson 2.2 – Windmill Power Output • Lesson 2.4 – Product Demand Estimation • Lesson 2.7 – Bike and Car Trip Analysis





by Pearson Prentice Hall o Chapter 2, Lessons 2.1 through 2.8






Precalculus

UNIT 4 – Exponential, Logistic, and Logarithmic Functions



• the algebraic functions (those with rational exponents) are expanded and enhanced by the transcendental functions (those with variable exponents)

• inverse functions provide a key problem-solving tool since they “undo” transformations

• the definitions will frequently provide the best starting point for understanding key mathematical concepts

Essential Questions: • What happens when a population grows or

decays in an unrestricted fashion? • What happens when a population grows or

decays in a restricted fashion? • How do you solve a problem when the

variable appears in the exponent? • What is meant when someone says “Time is

money”?

Students will know… • that exponential and logarithmic

functions are inverses of each other • that growth and decay, both with

and without restraint, can be modeled using exponential functions

• that logarithms provide an essential problem solving tool

• that financial models can demonstrate the effect of time on various investments

Students will be able to… • understand and work with the Natural Base

e to solve a variety of problems • apply exponential and logistic functions,

and interpret their graphs • build growth and decay models • understand and utilize logarithms as inverse

functions to solve problems • build models to simulate varied financial

transactions over time



Performance Tasks: • Lesson 3.2 – Deer Population • Lesson 3.4 – Earthquake Intensity • Lesson 3.5 – Newton’s Law of Cooling • Lesson 3.6 – IRA Account Deposits











Precalculus

UNIT 5 – Trigonometry



• all trigonometry is based upon right triangles

• multiple approaches may exist to solve a given problem, but all paths should lead to the same answer

• many natural phenomena can be modeled using trigonometric functions

Essential Questions: • What is trigonometry all about, and why is

it important? • What trigonometric relationships exist? • How do you apply trigonometric

relationships? • Can you work backwards? Do inverse

relationships exist?

Students will know… • that angles can be measures in

terms of both degrees and radians • that trigonometric functions can be

either even or odd • that trigonometric functions repeat

over time • that various methods can be used to

find the values of trigonometric functions

• that, with appropriate restrictions on the domain, the inverse of a trigonometric function can be found

Students will be able to… • convert between radians and degrees, and

find arc length • identify the periodicity and even-odd

properties of trigonometry functions • find values of trigonometric functions • generate the graphs of the trigonometric

functions and explore various transformations upon these graphs

• use the inverse trigonometric functions to solve problems


Performance Tasks: • Lesson 4.4 – Temperature Modeling • Lesson 4.8 – Building Height Determination






by Pearson Prentice Hall o Chapter 4, Lessons 4.1 through 4.8 o Chapter 5, Lessons 5.1 through 5.6 o Chapter 6, Lessons 6.1 through 6.6






Precalculus UNIT 6 – Systems and Matrices



• some systems of equations have solutions and some do not

• frequently, the real world concern is not “what is the solution?” but rather “does the solution exist?”

• real world systems are often modeled by multiple and multi-variate equations

• choosing the best solution from a list of possibilities is a science


Essential Questions: • How do you deal with two (or more)

“things” that happen at the same time? • How can matrices make my life better? • How do you identify the “best” solution

from a set of options?

Students will know… • that some systems of linear

equations have no solutions, some have one solution, and some have many solutions

• that matrices provide a powerful tool to capture and manipulate coefficients from equations

• that inverse matrices can greatly simplify some problem solving situations

• that linear programming is a problem solving tool used with decision making in management science

Students will be able to… • solve systems of equations using both

graphical and algebraic methods • perform matrix operations such as addition,

subtraction, and multiplication • utilize inverse matrices as problem solving

tools • apply the technique of partial fraction

decomposition to solve certain rational functions

• apply the techniques used with systems of equations to solve systems of inequalities

• find the “optimal” solution to a system of inequalities



Performance Tasks: • Lesson 7.2 – Construction Cost Estimation • Lesson 7.3 – Investment Diversification • Lesson 7.5 – Diet Planning











Precalculus UNIT 7 – Discrete Mathematics



• their first experiences with mathematics – counting – is an example of discrete mathematics

• mathematical induction provides the basis for an accepted form of valid proof

• it is important to know the differences between a “population” and a “sample” when dealing with statistics


Essential Questions: • What is “discrete” mathematics, and why is

it important? • How does discrete mathematics differ from

other forms of mathematics? • What is the difference between inductive

reasoning and deductive reasoning? • What is the difference between probability

and statistics? • What is a “normal” distribution?


Students will know… • that some situations lend

themselves to discrete rather than continuous mathematics

• that permutations and combinations are related, but different, concepts

• how to determine the probabilities of simple, compound and conditional events

• that some sequences and series have limits, and some do not

• that only one type of reasoning, inductive or deductive, forms the basis of a valid mathematical proof

• the terminology and techniques of statistics

Students will be able to… • understand the differences between discrete

mathematics and continuous mathematics • understand and apply permutations and

combinations • understand and apply the Binomial

Theorem • determine and represent the probabilities of

different events • find limits, if they exist, of arithmetic and

geometric sequences • find the limit or convergence of a series • apply the processes of mathematical

induction to problem solving • understand and apply basic statistical

concepts • work with and understand the

characteristics of normal distributions


Performance Tasks: • Lesson 9.3 – Correct Answers for Exam Questions • Lesson 9.8 – Population Heights





Precalculus UNIT 8 – Introduction to Calculus



• position and velocity are inextricably linked

• the concept of the derivative allows you to work from position to determine velocity

• not all functions have derivatives • functions that have derivatives may

not have them at every point • the definitions will frequently

provide the best starting point for understanding key mathematical concepts

• the concept of definite integral • the concept of an indefinite integral

Essential Questions: • What is the relationship between position

and velocity? Explain. • Explain why not all functions have

derivatives. • How does the concept of the integral allow

you to work from position to determine distanced traveled?

Students will know… • that average and instantaneous

velocity are two distinct concepts • that there is a continuum of ever

stronger concepts for a function at a given point: the limit exists, continuity exists, the derivative exists

• that the derivative of a function is defined as a limit of a quotient

• that the integral of a function over an interval is defined as the limit of the area of a set of rectangles

Students will be able to… • compute average and instantaneous velocity

for an object in motion • compute the derivative of a function • compute distance given a velocity • compute a definite integral



Performance Tasks: • Lesson 10.1 – Speed of a Falling Object • Lesson 10.2 – Rocket Launch









Resources and Technology



RESOURCES AND TECHNOLOGY

Precalculus

Resources

Precalculus: Graphical, Numerical, Algebraic, by Demana, Waits, Foley and Kennedy (7th

edition): Pearson Prentice Hall ©2007 – Student Text (ISBN 0-321-35694-4) Precalculus: Graphical, Numerical, Algebraic, by Demana, Waits, Foley and Kennedy (7th

edition): Pearson Prentice Hall ©2007 – Annotated Teacher Edition (ISBN 0-321-37423-1) Precalculus: Graphical, Numerical, Algebraic, by Demana, Waits, Foley and Kennedy (7th

edition): Pearson Prentice Hall ©2007 – Resource Manual (ISBN 0-321-36995-5) Precalculus: Graphical, Numerical, Algebraic, by Demana, Waits, Foley and Kennedy (7th

edition): Pearson Prentice Hall ©2007 – Solutions Manual (ISBN 0-321-35693-9) Precalculus: Graphical, Numerical, Algebraic, by Demana, Waits, Foley and Kennedy (7th

edition): Pearson Prentice Hall ©2007 – Tests and Quizzes (ISBN 0-321-35692-0) Precalculus: Graphical, Numerical, Algebraic, by Demana, Waits, Foley and Kennedy (7th

edition): Pearson Prentice Hall ©2007 – Graphing Calculator Manual (ISBN 0-321-37000-7) Math Manipulatives Communicators

Technology

Pearson Prentice Hall Website – www.phschool.com Pearson Prentice Hall – Video Lectures on CD-ROM (ISBN 0-321-41058-0) Pearson Prentice Hall – Presentation Express CD-ROM (ISBN 0-321-36999-8) Pearson Prentice Hall – Teacher Express CD-ROM (ISBN 0-321-41293-1) Pearson Prentice Hall – Student Express CD-ROM (ISBN 0-321-40995-7) Pearson Prentice Hall – TestGen with QuizMaster CD-ROM (ISBN 0-321-36996-3) Pearson Prentice Hall – MathXL® www.mathxl.com Pearson Prentice Hall – InterAct Math Tutorial www.interactmath.com Promethean Board – www.prometheanplanet.com

• ACTIVstudio Primary • ACTIVstudio Professional • ACTIVslate • ACTIVote

Texas Instruments graphing calculators – TI-84 Plus United Streaming- www.unitedstreaming.com

Homework Policy


MANVILLE SCHOOL DISTRICT HOMEWORK POLICY

Mathematics is a very important component of the students learning experience. The Manville Board of Education has a Homework/Make-Up Work Policy, File Code 6154. A portion of the daily, allotted homework time MUST be in mathematics. Recommendations are as follows:

• Grades K-1: 5 minutes minimum • Grades 2-3: 10 minutes minimum • Grades 4-5: 20 minutes minimum • Grades 6-8: 30 minutes minimum • Grades 9-12: 40 minutes minimum

MANVILLE BOARD OF EDUCATION File Code: 6154 Manville, New Jersey 08835-1542 Policy

HOMEWORK/MAKEUP WORK

The board of education believes that homework relevant to material presented in

class provides an opportunity to broaden, deepen or reinforce the pupil’s knowledge and skills. Homework provides the time and space for pupils to reflect on what they have learned by thinking, reading, writing, and problem-solving skills essential for success in college and in life.

Teachers have the responsibility of using discretion in deciding the number and length of homework assignments and must also monitor pupil homework. The board encourages the use of interrelated major homework assignments such as term papers, themes, and creative art projects. Homework shall not be used for punitive reasons.

Pupils absent for any reason are responsible for making up assignments, class work and tests within a reasonable length of time. In most cases, a reasonable length of time shall be the same number of days as the days missed.

Parents are partners in the education of pupils and are responsible for homework completion by their children by providing space and time, and by monitoring the workload and communicating with the teachers. Manville board of education guidelines for the amount of time that students should spend doing homework each day are as follows:

• Grades K-1: 15 minutes • Grades 2-3: 30 minutes • Grade 4: 45 minutes • Grade 5: 1 Hour • Grade 6: 1.25 Hours • Grades 7-8: 1.5 Hours • Grades 9-10: 2.0-2.5 Hours • Grades 11-12: 2.5 - 3 Hours


Date: November 17, 1992 Revised: December 14, 1993 HOMEWORK/MAKEUP WORK (continued) File Code: 6154 Legal References:

N.J.S.A. 18A:11-1 General mandatory powers and duties N.J.S.A. 18A:36-14 Religious holidays; absence of pupils on; effect N.S.A.C. 6:8-9 Approved public elementary and secondary school

summer sessions

Cross References: 1320 Participation in out-of-school community activities 1322 Contests for pupils 5020 Role of parents/guardians 5113 Absences and excuses 5124 Reporting to parents/guardians 6145 Co-curricular activities 6153 Field trips 6174 Summer school

NJ Core Curriculum Content Standards

http://www.nj.gov/education/cccs/s4_math.pdf

pre-calculus 2007- complete

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