FANTASTIC FACTORING!!!FANTASTIC FACTORING!!!

Greatest Common Factor

Difference of Squares

Perfect Square Trinomial

Leading Coefficient of One

Leading Coefficient Not One

All Five Types

Appendices

Teacher’s Guide

Don’t let factoring blow

you away!

This presentation is to be used for educational purposes only.

EXIT

TEACHER’S GUIDETEACHER’S GUIDE

Introduction

Aim

Rationale

Learners

Prerequisites

Subject Matter

Goals and Objectives

Instructional Plan

Materials

Assessment

IntroductionIntroduction

Factoring…students often find it hard to Factoring…students often find it hard to learn and teachers often find it hard to teach! learn and teachers often find it hard to teach! Students wonder how they can factor Students wonder how they can factor polynomials when each one seems to be a polynomials when each one seems to be a different style than the last. Teachers wonder different style than the last. Teachers wonder how they can find enough time to teach it how they can find enough time to teach it thoroughly when they have so many other thoroughly when they have so many other topics to cover. topics to cover.

AimAim

The purpose of this curriculum web is to The purpose of this curriculum web is to provide a resource for students and teachers provide a resource for students and teachers where students can learn and practice where students can learn and practice factoring the most common types of factoring the most common types of binomials and trinomials.binomials and trinomials.

RationaleRationale

Factoring is often one of the mathematical Factoring is often one of the mathematical skills that students find the most difficult to skills that students find the most difficult to learn, along with graphing and proofs. learn, along with graphing and proofs. Factoring can also be a source of frustration Factoring can also be a source of frustration for teachers who think students should be for teachers who think students should be learning the concepts faster or who find learning the concepts faster or who find themselves re-teaching factoring in courses themselves re-teaching factoring in courses they feel they should not have to spend time they feel they should not have to spend time doing. doing.

Rationale Continued…Rationale Continued…

Sometimes students wonder why they have to Sometimes students wonder why they have to learn factoring. There are at least four reasons learn factoring. There are at least four reasons learning to factor polynomials is important.learning to factor polynomials is important.

First, factoring is a skill that is needed to First, factoring is a skill that is needed to simplify rational expressions, locate holes in simplify rational expressions, locate holes in graphs, and decompose fractions. It can be graphs, and decompose fractions. It can be used to solve some quadratic equations. used to solve some quadratic equations.

Rationale Continued…Rationale Continued… Second, factoring is found throughout the high Second, factoring is found throughout the high

school math curriculum. Factoring occurs in school math curriculum. Factoring occurs in Geometry and Precalculus but factoring is found Geometry and Precalculus but factoring is found predominantly in Algebra I and Algebra II.predominantly in Algebra I and Algebra II.

Third, as students practice these factoring skills, Third, as students practice these factoring skills, they will also practice the very important they will also practice the very important algebraic skill of multiplying polynomials which algebraic skill of multiplying polynomials which involves other fundamental skills such as working involves other fundamental skills such as working with exponents, combining like terms, and with exponents, combining like terms, and multiplying positive and negative numbers.multiplying positive and negative numbers.

Rationale Continued…Rationale Continued…

Fourth, factoring and skills used in factoring Fourth, factoring and skills used in factoring are among the academic standards for state are among the academic standards for state testing. For example, the Ohio Academic testing. For example, the Ohio Academic Content Standards for Mathematics lists three Content Standards for Mathematics lists three educational objectives covered by factoring: educational objectives covered by factoring: “Solve quadratic equations…by factoring,” “Solve quadratic equations…by factoring,” “Add, subtract, multiply and divide monomials “Add, subtract, multiply and divide monomials and polynomials…” and “Simplify rational and polynomials…” and “Simplify rational expressions by eliminating common factors…” expressions by eliminating common factors…”

Rationale Continued…Rationale Continued…

This curriculum web can be a resource This curriculum web can be a resource for teachers to provide for their students for teachers to provide for their students when they do not have the personal time when they do not have the personal time to review how to factor the five different to review how to factor the five different types of polynomials presented. It can types of polynomials presented. It can also be a resource parents can provide also be a resource parents can provide their children who are struggling with their children who are struggling with factoring. It can even be a resource for factoring. It can even be a resource for parents themselves to learn factoring and parents themselves to learn factoring and explain it to their children. explain it to their children.

Rationale Continued…Rationale Continued… This curriculum web can enable students This curriculum web can enable students

to realize that most factoring problems to realize that most factoring problems are not unique but fall into general are not unique but fall into general categories. Students can realize that categories. Students can realize that factoring polynomials does employ factoring polynomials does employ algorithms (step-by-step procedures to algorithms (step-by-step procedures to solve them). Lastly, students can solve them). Lastly, students can overcome their fear of failure, start to overcome their fear of failure, start to succeed at factoring, and gain confidence succeed at factoring, and gain confidence in approaching factoring problems.in approaching factoring problems.

LearnersLearners

This curriculum web is designed to be used This curriculum web is designed to be used by high school students who have previously by high school students who have previously been taught how to factor binomials and been taught how to factor binomials and trinomials. Although it could be used to trinomials. Although it could be used to initially learn how to factor, it is designed to initially learn how to factor, it is designed to provide explanations and practice for those provide explanations and practice for those who have been exposed to factoring.who have been exposed to factoring.

PrerequisitesPrerequisites

Currently taking Algebra I or higherCurrently taking Algebra I or higher Previous exposure to factoring binomials and Previous exposure to factoring binomials and

trinomialstrinomials Interest in learning more details about Interest in learning more details about

factoring and practicing different problem factoring and practicing different problem stylesstyles

Subject MatterSubject Matter

Algebra I and IIAlgebra I and II Factoring binomials and trinomialsFactoring binomials and trinomials Multiplying polynomials to check factoringMultiplying polynomials to check factoring

Goals and ObjectivesGoals and Objectives

Students will learn to factor five different Students will learn to factor five different styles of polynomials:styles of polynomials:

Greatest Common FactorGreatest Common Factor Difference of SquaresDifference of Squares Perfect Square TrinomialPerfect Square Trinomial Trinomials with a leading coefficient of oneTrinomials with a leading coefficient of one Trinomials with a leading coefficient of an Trinomials with a leading coefficient of an

integer other than oneinteger other than one

Instructional PlanInstructional Plan

This curriculum web is designed to be used by This curriculum web is designed to be used by students without the help of a teacher. Students students without the help of a teacher. Students can access the curriculum web and follow the can access the curriculum web and follow the instructions. Students will find:instructions. Students will find: Detailed explanations of problemsDetailed explanations of problems Practice quizzes on each factoring stylePractice quizzes on each factoring style Helpful resources for further explorationHelpful resources for further exploration

MaterialsMaterials

An Internet-linked computer and web An Internet-linked computer and web browserbrowser

Paper and pencil to perform calculations and Paper and pencil to perform calculations and check work before submitting answerscheck work before submitting answers

Hand-held calculator is optional and online Hand-held calculator is optional and online calculator links are providedcalculator links are provided

AssessmentAssessment Learning that occurs through this curriculum web Learning that occurs through this curriculum web

can be assessed by the online quizzes that can be can be assessed by the online quizzes that can be taken and the results e-mailed to the teacher, taken and the results e-mailed to the teacher, parent, or student. parent, or student.

Students can do a self-assessment by using a Students can do a self-assessment by using a rubric to examine their improving skill level.rubric to examine their improving skill level.

EVALUATION: The curriculum web itself can be EVALUATION: The curriculum web itself can be evaluated by using a link to a feedback form in the evaluated by using a link to a feedback form in the Appendices.Appendices.

AppendicesAppendices

ResourcesResources Ask Dr. MathAsk Dr. Math Does the CAREER I want use MATH?!Does the CAREER I want use MATH?! More Factoring PracticeMore Factoring Practice

Leading Coefficient of OneLeading Coefficient of One Leading Coefficient Not One Leading Coefficient Not One

Online Calculator Online Calculator

AppendicesAppendices Glossary (in your language!)Glossary (in your language!)

BinomialBinomial – two terms – two terms CoefficientCoefficient – number in front of the letter – number in front of the letter DegreeDegree - highest exponent - highest exponent FactoringFactoring – factors that can be multiplied to get – factors that can be multiplied to get

the original problemthe original problem Greatest Common FactorGreatest Common Factor – what all the terms – what all the terms

have in commonhave in common Perfect SquarePerfect Square – has the same two factors – has the same two factors TrinomialTrinomial – three terms – three terms

AppendicesAppendices

Have you ever used a product and had an idea Have you ever used a product and had an idea you KNOW would make it better? Well here’s you KNOW would make it better? Well here’s your chance to make your voice heard! (Fill your chance to make your voice heard! (Fill out this short and simple four-question out this short and simple four-question evaluation.)evaluation.)

EVALUATIONEVALUATION

FantasticFactoring

GreatestCommon

Factor

DifferenceOf

Squares

PerfectSquare

Trinomial

LeadingCoefficient

of One

LeadingCoefficientNot One

AllFive

Types

Explanation

OnlinePracticeQuizzes

Explanation

OnlinePracticeQuizzes

Appendices

Resources

Glossary

Explanation

OnlinePracticeQuizzes

Explanation

OnlinePracticeQuizzes

OnlinePracticeWebsite

Explanation

OnlinePracticeQuizzes

Teacher’sGuide

Evaluation

Teacher’s Guide

Introduction

Aim

Rationale

Learners

Prerequisites

Subject Matter

Goals and Objectives

Instructional Plan

Materials

Assessment

AppendicesAppendicesIf you would like to see if your If you would like to see if your factoring skills are improving you can factoring skills are improving you can click on the link below to rank click on the link below to rank yourself on a scale from 0 to 5.yourself on a scale from 0 to 5.

SELF-ASSESSMENTSELF-ASSESSMENT

GREATEST COMMON FACTORGREATEST COMMON FACTOR

DetailedExample

Practice Quizzes

Easier than facing a

300 lb GORILLA!

DIFFERENCE OF SQUARESDIFFERENCE OF SQUARES

DetailedExample

Practice Quizzes

Go for the

GOLD!!!

PERFECT SQUARE TRINOMIALPERFECT SQUARE TRINOMIAL

DetailedExample

Practice Quizzes

I love trinomials!!!

LEADING COEFFICIENTLEADING COEFFICIENTOF ONEOF ONE

DetailedExample

Practice Quizzes

Your future’s so bright you

have to wear

shades!!!

LEADING COEFFICIENTLEADING COEFFICIENTNOT ONENOT ONE

DetailedExample

Practices Quizzes

Come on sleepy head!!!

ALL FIVE TYPESALL FIVE TYPES

Practice Quizzes

Don’t be a chicken!!!

Greatest Common FactorGreatest Common Factor

Factor 3ty3 + 36ty2

– Largest number that goes into 3 and 36? Answer: 3– Highest power of t in both? Answer: t– Highest power of y in both? Answer: y2

Overall Answer: 3ty2 (y + 12)

Difference of SquaresDifference of Squares Factor xFactor x22 – 36 – 36

– What times what would equal xWhat times what would equal x22? Answer: x? Answer: x– What times what would equal 36? Answer: 6What times what would equal 36? Answer: 6– Put x and 6 together, one with a plus and one with Put x and 6 together, one with a plus and one with

a minus.a minus.

Final answer: (x + 6)(x - 6)Final answer: (x + 6)(x - 6)

Perfect Square TrinomialPerfect Square Trinomial

Factor xFactor x22 – 10x + 25 – 10x + 25

– What times what would equal xWhat times what would equal x22? Answer: x? Answer: x– What times what would equal 25? Answer: 5What times what would equal 25? Answer: 5– Put x and 5 together using the same signs as thePut x and 5 together using the same signs as the

negative 10x.negative 10x.

Final Answer: (x - 5)(x - 5)Final Answer: (x - 5)(x - 5)

Leading Coefficient of OneLeading Coefficient of One

Factor xFactor x22 - 2x - 48 - 2x - 48

– List factors of 48. Answer: 1&48, 2&24, 4&12, List factors of 48. Answer: 1&48, 2&24, 4&12, 6&86&8

– Choose pair that gets 2. Answer: 6&8Choose pair that gets 2. Answer: 6&8– Get signs from - 48. Answer: one + and one –Get signs from - 48. Answer: one + and one –– Put bigger number with the same sign as middle Put bigger number with the same sign as middle

term of - 2xterm of - 2x

Final Answer: (x – 8)(x + 6)Final Answer: (x – 8)(x + 6)

Leading Coefficient Not OneLeading Coefficient Not One

Factor 8xFactor 8x22 - 46x + 30 - 46x + 30– Find 8 times 30. Answer: 240Find 8 times 30. Answer: 240

– List factors of 240. Answer: 1&240, 2&120, List factors of 240. Answer: 1&240, 2&120, 3&80, 4&60, 5&48, 6&40, 8&30, 12&203&80, 4&60, 5&48, 6&40, 8&30, 12&20

– Choose the pair that gets 46. Answer: 6&40Choose the pair that gets 46. Answer: 6&40

– Write - 46x as - 6x and - 40x. Write - 46x as - 6x and - 40x. Answer: 8xAnswer: 8x22 - 40x - 6x + 30 - 40x - 6x + 30

– Continued on next page…Continued on next page…

Leading Coefficient Not OneLeading Coefficient Not One

Problem Continued…Problem Continued…> 8x> 8x22 - 40x - 6x + 30 - 40x - 6x + 30

> Factor first two terms. Answer: 8x (x – 5)> Factor first two terms. Answer: 8x (x – 5)

> Factor last two terms. Answer: - 6 (x – 5)> Factor last two terms. Answer: - 6 (x – 5)

> Put 8x and - 6 together. Answer (8x – 6)> Put 8x and - 6 together. Answer (8x – 6)

> Write (x – 5) only once.> Write (x – 5) only once.

Overall Answer: (8x – 6)(x – 5)Overall Answer: (8x – 6)(x – 5)

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