NEXT GENERATION

MATH I

(Textbook)

AuthorsMariel T. Atregenio earned her master’s degree in Mathematics and bachelor’s degree in Secondary Education major in Mathematics, magna cum laude, from the University of the Philippines (UP)–Diliman. She has taught Mathematics at Malayan High School of Science, Claret School of Quezon City, and Francisco Homes College. In college, Ms. Atregenio was a presidential scholar, a UP College of Education Academic Excellence awardee, and a consistent recipient of scholarship awards.

Katrina Grace Q. Sumagit earned her master’s degree in Mathematics and bachelor’s degree in Secondary Education major in Mathematics and minor in Statistics from UP–Diliman. She is taking up her doctorate in Mathematics also at UP–Diliman. She taught grade school and high school Mathematics at the University of the Philippines Integrated School (UPIS) from 2006 to 2008. Ms. Sumagit has conducted seminar-workshops for high school and grade school Mathematics teachers for Gurong Pahinungod (GP) and Ateneo Center for Educational Development (ACED). Currently, she is a full-time faculty at the Ateneo de Manila Grade School.

Consultant-ReviewerLorelei B. Ladao-Saren obtained her master’s degree in Mathematics, with high distinction, from the De La Salle University (DLSU)–Dasmariñas and her bachelor’s degree in Statistics from UP–Diliman. She is presently pursuing her doctorate in Mathematics Education at the Philippine Normal University. Ms. Ladao-Saren was a former director for Research, Publication, and Community Extension Services at World Citi Colleges. She has also taught Mathematics at Asia Pacific College, Southville Foreign University, and DLSU–Dasmariñas. She currently teaches Mathematics at DLSU–College of St. Benilde and at the Graduate School of Rizal Technological University.

Next Generation Math ITextbook

Philippine Copyright 2011 by DIWA LEARNING SYSTEMS INCAll rights reserved. Printed in the Philippines

Editorial, design, and layout by University Press of First Asia

No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical, including photocopying, recording, or any information storage and retrieval systems, without permission in writing from the copyright owner.

Exclusively distributed by DIWA LEARNING SYSTEMS INC 4/F SEDCCO 1 Bldg. 120 Thailand corner Legazpi Streets Legaspi Village, 1229 Makati City, Philippines Tel. No.: (632) 893-8501 * Fax: (632) 817-8700

ISBN 978-971-46-0180-2

Preface

The Next Generation Math series covers topics and competencies that are aligned with the Basic Education Curriculum (BEC) and the Engineering and Science Education Program (ESEP) of the Department of Education. It is composed of different mathematics disciplines: elementary algebra in first year; intermediate algebra in second year; geometry in third year; and advanced algebra, trigonometry, statistics, and calculus in fourth year. It tries to cover numerous important topics that will satisfy the needs of different groups of learners.

The series supports the constructivist approach to teaching and learning process. Lessons are presented through meaningful activities which are designed to provide you an opportunity to make different connections between concrete situations and mathematics. The activities are designed to develop your skills in problem solving, critical thinking, decision making, and creative thinking through exchange of ideas and your own discovery. Each book in this series provides opportunities for you to discuss, explore, and construct mathematical ideas and interpret new information and knowledge at a different perspective. You will also be able to structure and evaluate your own conjectures and apply previously acquired knowledge and skills.

The series has the following salient features:

• Lessons are inquiry based, enriched with applicable technologies, and integrated with science and real-life applications.

• Emphasis on the development of higher-order thinking skills is evident in the illustrative examples and exercises provided in every lesson. To enhance your mathematics skills, the degree of difficulty of the problems ranges from simple to more challenging ones.

• Exercises include research work to emphasize the importance of research as a tool in satisfying the quest for knowledge and acquiring valuable insights about certain topics.

• Historical notes, application of mathematical ideas in future careers, and pieces of trivia are presented in each chapter.

It is with a sincere desire to provide a useful tool in enhancing appreciation and better understanding of mathematics that the Next Generation Math series was conceptualized.

Table of Contents

Unit I Number Theory, Real Number System, and Measurement

Chapter 1 Whole NumbersLesson 1 Addition and Subtraction of Whole Numbers ................................................ 2 Lesson 2 Multiplication and Division of Whole Numbers .............................................. 8Lesson 3 Order of Operations ..................................................................................... 13IT Matters ................................................................................................................... 18

Chapter 2 Number TheoryLesson 1 Divisibility Rules ........................................................................................ 20Lesson 2 Prime and Composite Numbers, and Prime Factorization ............................ 24Lesson 3 Greatest Common Factor and Least Common Multiple ................................ 28IT Matters ................................................................................................................... 32

Chapter 3 SetsLesson 1 Set Notations and Kinds of Sets .................................................................. 34Lesson 2 Set Relations ............................................................................................... 38Lesson 3 Set Operations ............................................................................................ 43IT Matters ................................................................................................................... 50

Chapter 4 IntegersLesson 1 The Signed Numbers .................................................................................. 51Lesson 2 Addition and Subtraction of Integers .......................................................... 57Lesson 3 Multiplication and Division of Integers ........................................................ 63IT Matters ................................................................................................................... 68

Chapter 5 Rational and Irrational NumbersLesson 1 Squares and Square Roots .......................................................................... 70Lesson 2 Rational and Irrational Numbers ................................................................ 74Lesson 3 Addition and Subtraction of Rational Numbers ........................................... 78Lesson 4 Multiplication and Division of Rational Numbers ........................................ 84IT Matters ................................................................................................................... 91

Chapter 6 MeasurementLesson 1 Units of Length, Mass, and Capacity .......................................................... 93Lesson 2 Units of Time and Temperature ................................................................ 104Lesson 3 Angles ...................................................................................................... 109IT Matters ......................................................................................................................... 114

Unit II Algebraic Expressions

Chapter 7 Fundamentals of Algebraic ExpressionsLesson 1 Simplifying Numerical Expressions ........................................................... 116Lesson 2 Evaluating and Simplifying Algebraic Expressions .................................... 122Lesson 3 Translating Verbal Phrases into Algebraic Expressions ............................. 128Lesson 4 Exponents and Scientific Notation ............................................................ 134IT Matters ................................................................................................................. 141

Chapter 8 PolynomialsLesson 1 Introduction to Polynomials ...................................................................... 144Lesson 2 Adding and Subtracting Polynomials ........................................................ 150Lesson 3 Multiplying Polynomials ........................................................................... 155Lesson 4 Factoring Polynomials ............................................................................... 161Lesson 5 Dividing Polynomials ................................................................................ 168IT Matters ................................................................................................................. 178

Chapter 9 Rational ExpressionsLesson 1 Simplifying Rational Expressions ............................................................... 179Lesson 2 Multiplying and Dividing Rational Expressions .......................................... 185Lesson 3 Adding and Subtracting Rational Expressions ........................................... 192IT Matters ................................................................................................................. 200

Unit III Equations and Inequalities

Chapter 10 Linear Equations in One Variable and Literal EquationsLesson 1 Solving Linear Equations in One Variable ................................................. 203Lesson 2 Solving Linear Equations in One Variable Involving Absolute Value .......... 212Lesson 3 Solving Literal Equations .......................................................................... 219IT Matters ................................................................................................................. 224

Chapter 11 Linear Inequalities in One VariableLesson 1 Solving Linear Inequalities in One Variable ............................................... 225Lesson 2 Solving Linear Inequalities in One Variable Involving Absolute Value ........ 233IT Matters ................................................................................................................. 240

Unit IV Systems of Linear Equations and Linear Inequalities in Two Variables

Chapter 12 Linear Equations and Their GraphsLesson 1 The Cartesian Plane ................................................................................. 243Lesson 2 Graphing Linear Equations ...................................................................... 253Lesson 3 Point-slope and Slope-intercept Forms of Equations of Lines .................... 269Lesson 4 Parallel and Perpendicular Lines .............................................................. 278IT Matters ................................................................................................................. 284

Chapter 13 Systems of Linear Equations and Linear Inequalities in Two Variables

Lesson 1 Solving Systems of Linear Equations in Two Variables by Graphing ........... 285Lesson 2 Solving Systems of Linear Equations in Two Variables by Substitution ..... 292Lesson 3 Solving Systems of Linear Equations in Two Variables by the Addition

or Elimination Method .............................................................................. 299Lesson 4 Graphing Linear Inequalities in Two Variables ........................................... 307IT Matters ................................................................................................................. 318

Glossary ................................................................................................................. 321Bibliography ................................................................................................................. 326Index ................................................................................................................. 329

Number Theory,Real Number System, and Measurement

Unit I

Mathematicsisafieldofstudythatdealswithnumbersandsymbols.Itisanessentialpartofaperson’severydaylifeexperiences.Asyouprogressinyourstudyofmathematics,youwillfindoutandappreciateitsmanyusefulapplicationsinyourlife.

Inthisunit,youwilllearnabouttherealnumbersystemandmeasurement.Thisunitisdividedintosixchapters.InChapter1,youwillstudyaboutwholenumbersandreviewthebasicmathematicaloperationsonwholenumbers.Numbertheory,whichincludesdivisibilityrules,primenumbers,compositenumbers,primefactorization,greatestcommonfactor,andleastcommonmultiple,willbetackledinChapter2.Chapter3includeslessonsonsetsandsetnotations,kindsofsets,setrelations,andsetoperations.Chapter4isaboutintegerswhereyouwilllearnmoreaboutthesignednumbersandthebasicmathematicaloperationsinvolvingintegers.

Chapter 5 discusses rational and irrational numbers. Topics, such as squares, squareroots,andoperationsinvolvingrationalandirrationalnumberswillbecoveredinthischapter.InChapter6,youwilllearnaboutmeasurementandangles.

2 Next Generation Math I

Chapter 1

Lesson 1

WHOLE NUMBERS

LearningObjectives• Add,subtract,multiply,anddividewholenumbers• Identify,illustrate,andusethebasicpropertiesofadditionandmultiplicationof

wholenumbers• Interpretgraphsandtables• Performseriesofoperations• Illustratewordproblems• Solvewordproblemsinvolvingthefourbasicoperations• Useacalculatortosolvewordproblems

Addition and Subtraction of Whole Numbers

ThegraphbelowshowstheelectricbillsoftheReyesfamilyforsixmonths.Studythegraphandanswerthequestionsonthenextpage.

January0

500

1 000

1 500

2 000

2 500

3 000

February March April May June

Month

ElectricBill(p

esos

)

Reyes Family’s Electric Bills

Power Up

Number Theory, Real Number System, and Measurement 3

1. Howmuchisthetotalelectricbillofthefamilyforsixmonths? 2. HowmuchlessistheelectricbillofthefamilyinMaythaninApril? 3. Howmuchmoreistheelectricbillofthefamilyforthelastthreemonthscomparedto

theirelectricbillforthefirstthreemonths?

• Additionistheprocessofcombiningtwoormorenumbers.Thenumbersthatareaddedarecalledaddends.Theresultoranswerinadditioniscalledthesumortotal.

• Basicpropertiesofadditionofwholenumbers

1. ClosureProperty:Thesumoftwoormorewholenumbersisalsoawholenumber.

Foranywholenumbersx,y,andz:x+y=z.

2. CommutativeProperty:Theorderoftheaddendsdoesnotchangethesum.

Foranywholenumbersxandy:x+y=y+x.

3. AssociativeProperty:Thewaytheaddendsaregroupeddoesnotchangethesum.

Foranywholenumbersx,y,andz:(x+y)+z=x+(y+z).

4. IdentityProperty:Thesumofanynonzerowholenumberandzeroisalwaysthatnonzeronumber.Zero(0)istheidentityelementforaddition.

Foranynonzerowholenumberx:x+0=x.

• Subtractionisfindingthemissingaddends.Thenumberfromwhichyousubtractiscalledthesubtrahend.Thenumberthatissubtractedfromthesubtrahendiscalledtheminuend.Theresultoranswerinsubtractioniscalledthedifference.

Example1:Findthemissingnumbern.

a. 1357+2064+3809=n

b. 2706–1354=n

c. 2509+1346+n=4807

d. 7500–n=6453

e. n–3029=1579

Solution:

a. b. c.

n=1352 n=952 n=7230

Walk Through

13572 0643 8097 230

+

2 50913463 855

4 8073 855

952+ −

2 70613541352

−

� Next Generation Math I

d. e.

n=1047 n=4608

Example2:Findthemissingnumbernandidentifythebasicpropertyofadditionofwholenumbersused.

a. 1305+2604+3987=1305+n+2604b. 1732+0=nc. (2048+3609)+4507=n+(3609+4507)d. (5902+4302)+3018=3018+(5902+n)e. 1407+1302=n+1407+1302

Solution:a. n=3987;CommutativePropertyb. n=1732;IdentityPropertyc. n=2048;AssociativePropertyd. n=4302;CommutativePropertye. n=0;IdentityProperty

Example3: Useacalculatortosolveeachproblem.

a. Mrs.Martinezsold145orangesonMonday,102orangesonTuesday,and213orangesonWednesday.Howmanyorangesdidshesellaltogetherforthreedays?

b. Johnhad100collectiblecards.Hegave23cardstoMarkand32cardstoLuke.Hethensoldtherestofhiscollectiblecards.Howmanycollectiblecardsdidhesell?

Solution:

a. Letx=totalnumberoforangessoldforthreedays. x=145+102+213 =460

Mrs.Martinezsold460orangesinallforthreedays.

b. Lety=numberofcollectiblecardssold. y=100–(23+32) =100–55 =45

Johnsold45collectiblecards.

7 5006 4531047

−1 5793 0294 608

+

Number Theory, Real Number System, and Measurement �

Example4:Useacalculatortosolvetheproblem.

Thereare5318nonfictionbooksinthelibrary.Thereare2405fewerfictionbooksthannonfictionbooks.Howmanynonfictionandfictionbooksarethereinthelibrary?

Solution:

Letz=numberoffictionbooksinthelibrary.z=5318–2405 =2913

Thereare2913fictionbooksinthelibrary.

Letw=totalnumberofnonfictionandfictionbooksinthelibrary.w=5318+2913 =8231

Thereare8231nonfictionandfictionbooksinthelibrary.

Example5:Katrinahas40stamps.Gracehas18morestampsthanKatrina.Maehas12morestampsthanGrace.Howmanystampsdotheyhavealtogether?

Solution:Letx=numberofstampsGracehas.x=40+18 =58Gracehas58stamps.

Lety=numberofstampsMaehas.y=58+12 =70Maehas70stamps.

Letz=totalnumberofstampsKatrina,Grace,andMaehave.z=40+58+70 =168

Katrina,Grace,andMaehave168stampsaltogether.

6 Next Generation Math I

I. Encircletheletterthatcorrespondstothecorrectvalueofn.

1. 1006+2090+3135=n

a. 3096 c. 5225 b. 4141 d. 6231 2. 3901–2046–1308=n

a. 547 c. 1855 b. 738 d. 2593 3. 4203−3013+2192=n

a. 1190 c. 5024 b. 3382 d. 7216

4. 5310+2345–3875=n

a. 2965 c. 7655 b. 3780 d. 11530

5. 6385–(4784–2129)=n

a. 1601 c. 3730 b. 2655 d. 4256

6. 6724–(3384+1120)=n

a. 2220 c. 4504 b. 3340 d. 5604

7. 2189+n=4305

a. 2016 c. 6484 b. 2116 d. 6494

8. n+3709+1247=9656

a. 4700 c. 5700 b. 4956 d. 5947 9. 4142–n=2984

a. 1058 c. 7126 b. 1158 d. 7026

Move Up

Number Theory, Real Number System, and Measurement �

10. n–3792=2071

a. 1721 c. 5863

b. 2721 d. 6863

II. Identifythebasicpropertyofadditionusedineachequation.

1. 1286+714=714+1286

2. (579+312)+608=579+(312+608)

3. 459+367+0=459+367

4. 802+(0+315)=802+315

5. (3218+3201)+7042=7042+(3218+3201)

6. 2907+(1704+3416)=(2907+1704)+3416

7. 0+2379=2379

8. (3560+0)+5077=(0+3560)+5077

9. 1025+7035+0=1025+0+7035

10. (1306+4509)+(2064+0)=(1306+4509)+2064

III. Readandanalyzeeachproblemcarefully.Useacalculatortosolveeachproblem.

1. MannedApolloflightstothemoonusedthegiantSaturnVrocket.Therockethadthreestages.Thefirststageproduced7500000pounds(lb)ofthrust(power),thesecondstageproduced6500000lbofthrustlessthanthefirststage,andthethirdstageproduced800000lbofthrustlessthanthesecondstage.Howmanypoundsofthrustwasproducedinall?

2. Mrs.Santiagobaked1500muffins.Shesold360muffinsonMonday.Shesold143moremuffinsonTuesdaythanonMondayand132fewermuffinsonWednesdaythanonTuesday.Howmanymuffinsdidshehaveleft?

3. Raphael,George,andManuelhaveatotalof1000icecreamsticks.Raphaelhas432icecreamsticks.Georgehas124fewericecreamsticksthanRaphael.HowmanyicecreamsticksdoesManuelhave?