ICE-EM Mathematics 4ed Year 5 – uncorrected sample pages by cambridge_aused

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FourthEdition2026 2019181716151413121110987654321

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5DProperfractions,improperfractionsandmixednumbers124

5EAddingandsubtractingfractionswiththesamedenominator131

5FAddingandsubtractingfractionswithdifferentdenominators134

5GReviewquestions–Demonstrateyourmastery

ICE-EMMathematicsFourthEdition isaseriesoftextbooksforstudentsinYears5to10throughout AustraliawhostudytheAustralianCurriculumV9.0anditsstatevariations.

DevelopedbytheAustralianMathematicalSciencesInstitute(AMSI),the ICE-EMMathematicsFourth Edition serieswasdevelopedinrecognitionoftheimportanceofmathematicsinmodernsocietyandthe needtoenhancethemathematicalcapabilitiesofAustralianstudents.Studentswhousetheserieswillhavea strongfoundationforfurtherstudy.

Highlightsofthe ICE-EMMathematicsFourthEdition seriesinclude:

• updatedandrevisedcontenttoprovidecomprehensivecoverageoftheAustralianCurriculumV9.0and itsstatevariationsinasingletextbookforeachyearlevel

• anewdesigntoprovidestudentswiththebestpreparationforsuccessinseniorhighschoolsubjectssuch as SpecialistMathematics and MathematicalMethods (MathematicsExtension and Advanced Mathematics inNSW)

• newcontenttohelpconnectmathematicallearningtoFirstNationsPeoples’knowledgeandcultures

• AMSI’sextensiveonlinesupplementarycontentsuchasworkedsolutions,videoexplanationsandthe AMSICalculateteacherandstudentresources

• anInteractiveTextbook:adigitalresourcewherealltextbookmaterialcanbeansweredonline,plus additionalquizzesandfeatures.

Background

TheInternationalCentreofExcellenceforEducationinMathematics(ICE-EM)wasanAustralian GovernmentprogrammanagedbytheAustralianMathematicalSciencesInstitute(AMSI),whichpublished thefirsteditionofthetextbookseriesin2006.TheCentreoriginallypublishedtheseriesaspartofaprogram toimprovemathematicsteachingandlearninginAustralia.In2012,AMSIandCambridgeUniversityPress collaboratedtopublishtheSecondandThirdEditionsoftheseries.TheFourthEditionalignswiththe AustralianCurriculumV9.0andhasbeendevelopedwiththegeneroussupportoftheBHPFoundation.

Theseries

ICE-EMMathematicsFourthEdition seriesprovidesaprogressivedevelopmentfromupperprimaryto middlesecondaryschool.ThewritersoftheseriesaresomeofAustralia’smostoutstandingmathematics teachersandsubjectexperts.Thetextbooksareclearlyandcarefullywrittenandcontainbackground information,examplesandworkedproblems.

TheyaresupplementedbyAMSI’sextensiveonlinetextbookcontent,whichisavailableonlineat www.schools.amsi.org.au.Thiscontentincludes:

• videoexplanationsoftextbookworkedexamples

• workedsolutionsforallexercisequestionsets

• userguideonsolvingtextbookquestionsusingAImathsapps

• AMSICalculateteacherandstudentresources

• algorithmicthinkingcontentandexamples,whichwillhelpdevelopstudents’abilitytosolve mathematicalproblemsusingboththe Scratch and Python programminglanguages.

FirstNationsPeoples’knowledgeandcultures

TheAustralianCurriculum:MathematicsV9.0includesthecross-curriculumpriorityAboriginalandTorres StraitIslanderHistoriesandCulture,sothat ‘studentscanengagewithandvaluethehistoriesandcultures ofAustralianFirstNationsPeoplesinrelationtomathematics.’

The ICE-EMMathematicsFourthEdition textbooksallincludeachapterwhichconnectsmathematical learningtoFirstNationsPeoples’knowledgeandcultures.ThesematerialshavebeenwrittenbyProfessor RowenaBallandDrHongzhangXufromthe MathematicsWithoutBorders programattheAustralian NationalUniversity.Therearequestionsonastronomyandeclipses,songlines,fishingpractices,animal tracking,gameplaying,kinshipstructuresandfiremanagement,whichwillenablestudentsandteachersto learnabouttheculturesofFirstNationsPeoplesinamathematicalcontext.

STEMcareersandmathematicsstudy

Thistextbookhassectionsonsixstudystrands:Number,Algebra,Measurement,Space,Statisticsand Probability.Allthesestrandsarefundamentalbuildingblocksforstudentswhowishtostudyscience, technology,engineeringandmathematics(STEM)atschoolanduniversity.

STEMcareersencompassthenaturalsciences,engineering,computerscience,informationtechnologyand themathematicalsciences.Adegreeinmathematicsisapassportforentryintocareersinvolvingfieldssuch asdatascience,artificialintelligence,machinelearning,cybersecurity,finance,logisticsandoptimisation. AMSI’sMathsAddsCareersGuideisavaluablesourceofinformationonthefullrangeofcareersin mathematics.

IfyouwishtopursueaSTEMcareer,thenitiscriticalthatyoucontinuetostudymathematicsinhighschool. InYears11and12youshouldaimtostudy SpecialistMathematics and/or MathematicalMethods (MathematicsExtension and AdvancedMathematics inNSW),asthesesubjectswillgiveyouthebest possiblepreparationforSTEMandmathsdegreesatuniversity.

Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies Authorbiographies

LeadAuthor

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MichaelEvans

MichaelEvanshasaPhDinMathematicsfromMonashUniversityandaDiplomaofEducationfrom LaTrobeUniversity.HecurrentlyholdsthehonorarypositionofSeniorFellowatAMSI,theUniversityof Melbourne.HewasHeadofMathematicsatScotchCollege,Melbourne,andhasalsotaughtinpublic schools.Hehasbeeninvolvedwithcurriculumdevelopmentatbothstateandnationallevels.Michaelwas awardedanhonoraryDoctorofLawsbyMonashUniversityforhiscontributiontomathematicseducationin 1999,hereceivedtheBernhardNeumannAwardforcontributionstomathematicsenrichmentinAustraliain 2001,andreceivedtheAMSIMedalforDistinguishedServicein2013.

ContributingAuthors

ColinBecker

ColinBeckerworkedasaMathematicsandITLTspecialistatanindependentboys’schoolinAdelaide.Colin haswrittenforprofessionalpublications,presentedatconferencesandschools,andisactivelyinvolvedin mathematicseducation.

SeonaidChio

SeonaidChioisHeadofTeachingandLearningatGrimwadeHouse,MelbourneGrammarSchool.Withover 20yearsofteachingexperienceandmorethanadecadeofleadingteachingandlearning,shebringsa depthofknowledgeincurriculumdesign.Herleadershipspansarangeofschoolsacrossthreedifferent countries,enrichingherapproachwithdiverseeducationalperspectives.Sheispassionateabout empoweringteacherstobuildstudentconfidenceandcuriosity,particularlyinmathematics,through collaborativepractice,explicitteachingandreflectivedialogue,allgroundedinbestpractice.

HowardCole

HowardColewasSeniorMathematicsMasteratSydneyGrammarSchoolEdgecliffPreparatoryformany years.Heoutlinedthewholeprimarycurriculumduringthattime,aswellaswritingandproducingin-school workbooksforYears5and6.Nowretiredfromteaching,hestillmaintainsakeeninterestinmathematics andcurriculumdevelopment.

AndyEdwards

AndyEdwardstaughtinsecondarymathematicsclassroomsfor31yearsinVictoria,Canadaand Queensland.HehasworkedfortheQueenslandCurriculumandAssessmentAuthority,writingmaterialsfor theirassessmentprogramsfromYears3to12,andasatestitemdeveloperforWA’sOLNAprogram.Hehas writtennon-routineproblemsfortheAustralianMathematicsTrustandreceivedaBernardNeumannAward fromtheAustralianMathsTrustforhiswork.

AdrienneEnglish

AdrienneEnglishistheEnrichmentCoordinatoratGrimwadeHouse,MelbourneGrammarSchool,where shehasledgiftededucationandmathematicsenrichmentforover15years.WithaMastersinEducation (GiftedEducation)andmorethan25yearsofexperienceinprimaryteachingandleadershipacross Melbourneindependentschools,Adriennebringsdeepexpertiseincurriculumdesignanddifferentiated instruction.Herpassionformathematicshasdriventhedevelopmentoftargetedprogramsaimedat fosteringbothexcellenceandagrowthmindsetinstudents.AdriennealsoservesasaDirectorontheBoard oftheMathematicalAssociationofVictoria.

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GarthGaudry

ThelateGarthGaudrywasHeadofMathematicsatFlindersUniversitybeforemovingtoUNSW,wherehe becameHeadofSchool.HewastheinauguralDirectorofAMSIbeforehebecametheDirectorofAMSI’s InternationalCentreofExcellenceforEducationinMathematics.Hispreviouspositionsincludemembership oftheSouthAustralianMathematicsSubjectCommitteeandtheEltisCommitteeappointedbytheNSW GovernmenttoenquireintoOutcomesandProfiles.HewasalifememberoftheAustralianMathematical SocietyandEmeritusProfessorofMathematics,UNSW.

JacquiRamagge

JacquiRamaggeisExecutiveDeanofSTEMattheUniversityofSouthAustraliaandisPresidentofthe AustralianCouncilofDeansofScience.Aftergraduatingin1993withaPhDinMathematicsfromthe UniversityofWarwick(UK),sheworkedattheUniversityofNewcastle(Australia),theUniversityof Wollongong,theUniversityofSydneyandDurhamUniversity,UK.ShehasservedontheAustralian ResearchCouncilCollegeofExperts,includingasChairofAustralianLaureateFellowshipsSelectionAdvisory Committee.ShehastaughtmathematicsatalllevelsfromprimaryschooltoPhDcoursesandhaswona teachingaward.ShecontributedtotheVermontMathematicsInitiative(USA)andisafoundingmemberof theAustralianMathematicsTrustPrimaryProblemsCommittee.In2013shereceivedaBHNeumannAward fromtheAustralianMathematicsTrustforhersignificantcontributiontotheenrichmentofmathematics learninginAustralia.

JanineSprakel(formerlyMcIntosh)

JanineSprakelisanexperiencedmathematicseducatorandteachertrainer.Shehasastrongbackgroundin primaryeducationandmathematicspedagogy,withextensiveexperienceindevelopinginnovative educationalresources.Janinehascontributedtothedesignofonlineandcareersmaterialstosupport mathematicseducationandwasawriterfortheAustralianCurriculum.Janinehasdemonstratedleadership andprojectmanagementskillsandfosteredsuccessfulpartnershipswithindustryandgovernmentpartners. ShehasworkedasalecturerinmathematicseducationattheUniversityofMelbourneandhasbeenactively involvedininitiativesaimedatpromotingmathematicsenjoymentandstudyacrossAustralia.Sheis passionateaboutadvancingqualitymathematicseducation,encouraginggenderequalityinSTEMand inspiringlearnersandeducatorstostickwithmathematicstogrowcapacityandcommunity.

AuthorsofFirstNationscurriculumcontent

RowenaBall

RowenaBallisanappliedmathematicianattheMathematicalScienceInstitute,AustralianNational University.HerresearchonIndigenousandnon-Westernmathematicshasshownthatsophisticated mathematicalconceptswereknownandexpressedculturallywithinIndigenoussocieties,openingup possibilitiesfornewmathematicalapproachesto21st-centuryproblems.Sheworkswithscientistsfrom otherdisciplines,includingphysics,chemistryandengineering,tomodelandsolvereal-worldproblems involvingcomplexdynamicsandemergentbehaviour.

HongzhangXu

DrHongzhangXuisanAdjunctResearchFellowattheAustralianNationalUniversity(ANU)andasenior ecohydrologistattheMurray–DarlingBasinAuthority.HehasworkedattheMathematicalSciencesInstitute ANU,asapost-doctoralresearcher,investigatingAboriginalandTorresStraitIslandermathematicsand sciences.Hisworkisbroadlyreadandcitedfrequently,andheregularlyreceivesinvitationstocommenton popularissuesfrommajormediasuchasCNN,ABC,TheConversation,BloombergandNatureNews.

Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements Acknowledgements

Wewishtothanktheteamofwriterswhohavepreparedthenewcontentforthe ICE-EMMathematics FourthEdition series,theCUPeditorsandproductionteam.WealsogratefullyacknowledgetheBHP FoundationfortheirfinancialsupportaspartoftheChooseMATHSproject.

Wehopethatyouenjoyusingthistextbookandthatithelpsyouprogressalongyourownmathematical journey.

MichaelEvansandTimMarchant, AustralianMathematicalSciencesInstitute, September2025

Theauthorandpublisherwishtothankthefollowingsourcesforpermissiontoreproducematerial:

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Everyefforthasbeenmadetotraceandacknowledgecopyright.Thepublisherapologisesforanyaccidental infringementandwelcomesinformationthatwouldredressthissituation.

Tocome

Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource Howtousethisresource

Thetextbook

Thetextbookiswritteninthestyleofa‘conversation’.Thatconversationismeanttotakeavarietyofforms: conversationsbetweentheteacherandstudentsabouttheideasandmethodsastheyaredeveloped; conversationsamongthestudentsthemselvesaboutwhattheyhavedoneandlearnt,andthedifferentways theyhavesolvedproblems;andconversationswithothersathome.Eachchapteraddressesaspecific AustralianCurriculumcontentstrandandcurriculumelements.Theexerciseswithinchapterstakean integratedapproachtotheconceptofproficiencystrands,ratherthanseparatingthem.Studentsare encouragedtodevelopandapplyUnderstanding,Fluency,Problem-solvingandReasoningskillsinevery exercise.

Questiontags

Thequestionsineachchapteraretagged.Thetagsareintendedasaguidetoteachers.Theyshouldbe regardedasawayofencouragingstudentprogress.

These givestudentspracticeusingthebasicideasandmethodsofthesection.Theyshouldgivestudents confidencetogoonsuccessfullytothenextlevel.

These buildonthepreviouslevelandhelpstudentsacquireamorecompletegraspofthemainideasand techniques.Somequestionsrequireinterpretation,usingareadingabilityappropriatetotheagegroup.

For thesequestions,studentsmayneedtoapplyconceptsfromoutsidethesectionorchapter. Problem-solvingskillsandahigherreadingabilityareneeded,andthesequestionsshouldhelpdevelop thoseattributes.

Challengeexercises

TheChallengeexercises,whichareintheprintbookandcanalsocanbedownloadedviatheInteractive Textbook,areavitalpartoftheICE-EMMathematicsresource.Theseareintendedforstudentswith above-averagemathematicalandreadingability.However,thequestionsvaryconsiderablyintheirlevelof difficulty.Studentswhohavemanagedtheharderquestionsintheexercisesreasonablywellshouldbe encouragedtotrytheChallengeexercises.

TheInteractiveTextbookandtheOnlineTeachingSuite

TheInteractiveTextbookistheonlineversionofthetextbookandisaccessedusingthe16-charactercode ontheinsidecoverofthisbook.TheOnlineTeachingSuiteistheteacherversionoftheInteractiveTextbook andcontainsallthesupportmaterialfortheseries,includingtests,curriculumdocumentationandmore.

TheInteractiveTextbookandOnlineTeachingSuitearedeliveredontheCambridgeHOTmathsplatform, providingaccesstoaworld-classLearningManagementSystemfortesting,taskmanagementandreporting.

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TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand TheInteractiveTextbookand TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite TheInteractiveTextbookand theOnlineTeachingSuite theOnlineTeachingSuite

InteractiveTextbook

TheInteractiveTextbookistheonlineversionoftheprinttextbookandcomesincludedwithpurchaseofthe printtextbook.Itisaccessedbyfirstactivatingthecodeontheinsidecover.Itiseasytonavigateandisa valuableaccompanimenttotheprinttextbook.

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Studentscanshowtheirworking

AlltextbookquestionscanbeansweredonlinewithintheInteractiveTextbook.Studentscanshowtheir workingforeachquestionusingeithertheDrawtoolforhandwriting(iftheyareusingadevicewitha touch-screen),theTypetoolforusingtheirkeyboardinconjunctionwiththepop-upsymbolpalette,orby importingafileusingtheUploadtool.

Onceastudenthascompletedanexercise,theycansavetheirworkandsubmitittotheteacher,whocan thenviewthestudent’sworkingandgivefeedbacktothestudent,astheyseeappropriate.

Auto-markedquizzes

TheInteractiveTextbookalsocontainsmaterialnotincludedinthetextbook,suchasashortauto-marked quizforeachsection.Thequizcontains10questionswhichincreaseindifficultyfromquestion1to10and coverallproficiencystrands.Theauto-markedquizzesareagreatwayforstudentstotracktheirprogress throughthecourse.

OnlineTeachingSuite

TheOnlineTeachingSuiteistheteacher’sversionoftheInteractiveTextbook.Muchmorethana‘Teacher Edition’,theOnlineTeachingSuitefeaturesthefollowing:

• Theabilitytoviewstudents’workingandgivefeedback–whenastudenthassubmittedtheirworkonline foranexercise,theteachercanviewthestudent’sworkandcangivefeedbackoneachquestion.

• AccesstoChaptertests,BlacklineMasters,Challengeexercises,curriculumsupportmaterial,andmore.

• ALearningManagementSystemthatcombinestask-managementtools,apowerfultestgenerator,and comprehensivestudentandwhole-classreportingtools.

Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready Gettingready

Usefulskillsforthistopic

• understandingofplacevalueofnumbersto1000000andbeyond

• understandingofeachdigit’spositionwithinanumberanditsplacevalue

• recognisingtheroleofzeroinplacevaluenotation

• recognisingthevalueofanumbercanberepresentedonanumberline

Vocabulary

Numbers

Placevalue

Estimating

Numberline

Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Oddoneout

Whichoneistheoddoneout?Explainwhy.

• 540000

• 54tenthousands

• 4hundredthousandsand140thousand

• 540hundreds

• 54000tens

Digits

Comparing

Rounding

Greaterthan

Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers Wholenumbers

Weusenumberseverydaytodescribepeople,placesandthingsandtorecord amounts.Forexample:

• howmuch paperweneedtocoveratabletoprotectitfrompaint

• howlong untilschoolfinishes

• howmany pencilsthereare

• howmuch waterthereis.

Thischapterlooksatwholenumbers,whicharesometimescalledthe‘counting numbers’.Thewholenumbersarethenumbers0, 1,

, 11,and soon.Thelistofwholenumbersisinfinite–itneverends.

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1A Placevalue

Numbers arewrittenusingthe digits 0, 1, 2, 3, 4, 5, 6, 7, 8and9.Thevalueofadigit changesaccordingtowhereitisplaced.

Thedigits0, 1, 2, 3, 4, 5, 6, 7, 8and9canbeusedtowrite:

• a5-digitnumber,suchas24871

• a6-digitnumber,suchas390513

• a1-digitnumber,suchas3.

Eachplaceinanumberhasaspecialvalue,thisiscalledits placevalue

Forexample,inthenumber2483:

• the2means2thousands

• the4means4hundreds

• the8means8tens

• the3means3ones.

2483

Thevalueofadigitchangesifitisinadifferentplace.Ifwe taketheexample2483fromabove,andchangethedigitsto make3428:

• the3means3thousands

• the4means4hundreds

• the2means2tens

• the8means8ones. 3428

Knowingthepositionofthedigithelpsustoreadlargenumbers.

1millionis1000thousands.1000000 = 1000lotsof1000.

Thenamingstartsagainformillions:wehavewholenumbersofmillions,tensof millionsandhundredsofmillions.

Weread342781956as‘threehundredandforty-twomillion,sevenhundredand eighty-onethousand,ninehundredandfifty-six’.

Example1

Writethevalueofthe3ineachnumber.

Solution

a In43000,the3isinthethousandsplace,soitstandsfor3thousandsor3000.

b In2301,the3isinthehundredsplace,soitstandsfor3hundredsor300.

c In35,the3isinthetensplace,soitstandsfor3tensor30.

d In23,the3isintheonesplace,soitstandsfor3onesor3.

1A Wholeclass LEARNINGTOGETHER

1 Workinpairs.Person1readsparts a, b, e and f toPerson2. Person2writesdowneachnumberastheyhearit.

Thenswaprolesfor c, d, g and h.

2 WhoamI?

Readeach‘WhoamI?’aloudtotheclass.Askstudentstowriteeachnumber intheirbooks.

a Ihave8tensand7ones.WhoamI?

b Ihave9hundreds,4tensand9ones.WhoamI?

c Ihave3thousands,8hundredsand5ones.WhoamI?

d Ihave4hundredsand32ones.WhoamI?

e Ihave12tensand7ones.WhoamI?

f Ihave2hundreds,9thousands,5onesand7tens.WhoamI?

g Ihave7ten-thousands,6hundreds,4ones,5thousandsand3tens. WhoamI?

h Ihave53ones,twothousandsand8hundreds.WhoamI?

APPLYYOURLEARNING 1 Copyandcompletethisplace-valuechart.

2 Writeeachnumberinnumerals.

a threehundredandforty-five

b twenty-sixthousandandseventy-seven

c sixhundredandseventhousand,threehundredandninety-three

d twomillionandfiftythousand

3 Writethevalueofeachhighlighteddigit.

4 Writethesenumbers.

a 63hundreds,4tensand7ones

b 1thousand,47tensand3ones

c 6thousands,5hundredsand21ones

d 127tensand8ones

5 Writethesenumbers.

a 72hundreds,9ten-thousands,6onesand1ten

b 84ones,5thousands,1hundredand3ten-thousands

c 86thousands,9ones,2hundredsand5tens

Thenumberline

Comparingnumbers

A numberline helpsustomakesenseofnumbers.

Tomakeanumberline,drawalineonpaper.Thearrowsshowthatthelinecontinues bothwaysforever.

Anumberlinecanbeusedtoshowanynumber,fromthesmallestuptothelargest numberyoucanthinkof.

Numbersgetlargeraswegototherightonthenumberline.So50islargerthan40 becauseitliesfurthertotheright.

Example2

Showwhere250wouldbeonthisnumberline.

Solution

500ishalfwaybetween0and1000,so250isone-quarterofthedistance.

Example3

Whichnumberislarger:45600or48200?

Solution

Placebothnumbersonanumberline.

45600isbetween45000and46000.48200isbetween48000and49000.

48200islargerbecauseitisfurthertotheright.

Nowthatyoucan compare numbersandunderstand placevalue,youcanmakeand saylargerorsmallernumbers.

Tofindthenumberthatis400morethan1387weincreasethehundredsdigitby4 andweget1787.

Tofindthenumberthatis200lessthan1387wedecreasethehundredsdigitby2and weget1187.

Example4

Writethenumberthatis10000morethan:

Example5

Writethenumberthatis1000lessthan:

1B Wholeclass LEARNINGTOGETHER

1 Useapieceofstringasa0–10000numberline.Writeanumberbetween 0and10000onapieceofpaperandpegittothestring.

2 Ifyourollfive10-sided (0–9) dicetomakea5-digitnumber:

a whatisthelargestpossiblenumberthatcanbemade?

b whatisthesmallestpossible5-digitnumberthatcanbemade?

Addbothofthesenumberstoyourstringnumberlinefromquestion 1

3 Workinagroupof3to6.Oneplayerremovesthepicturecardsfromadeck ofcardsandshufflesthedeck.Eachplayerdraws6cards.Placeyourcards downintheordertheyaredrawn,lefttoright,tomakea6-digitnumber.(An aceisequalto1.)Eachplayercanrearrangetheirnumberonce,bypicking anycardandmovingittotherightoftherowtobecomethelastdigit.The playerwiththelargestnumberwinstheround.

1B Individual APPLYYOURLEARNING

Drawanumberlinewith0and10markedonit.Usealargedottomarkthe numbers2, 3, 5and9.

Drawanumberlinewith0and20markedonit.Usealargedottomarkeachodd numberbetween10and20.

Drawanumberlinewith0and100markedonit.Usealargedottomarkthe numbers10, 20, 30, 40, 50, 60, 70, 80and90onit.

Drawanumberlinewith0and1000markedonit.Usealargedottomarkthe numbers5and625onit.

5 Writethenumberthatis100morethan: 682 a 981 b 1025 c 12092 d

6 Writethenumberthatis100lessthan: 682 a 981 b 1025 c 12092 d

7 Writethenumberthatis1000morethan: 439 a 2733 b 3033 c 19999 d

Writethenumberthatis1000lessthan: 2222 a 11000 b 1043 c 21837 d

9 Writethenumberthatis110lessthan: 1035 a sevenhundred b 2000+64 c 13409 d

10 Writethenumberthatis1010morethan: 14000+808 a 19680 b twothousand,onehundredandninety c 69990 d

11 Orderthenumbersfromlargesttosmallest. 17928198271790190990

12 Place lessthan<orgreaterthan> betweenthefollowingnumbers.

a 10208 10200

b 134680 144680

c 6589 7028

d 6569990 6599690

13 Orderthesenumbersfromsmallesttolargestonanumberline.Recordeachasa wholenumber.

• 6987234

• 6000000+600000+5000+600+30+2

• sixmillion,sevenhundredandfiftythousand,fourhundredandthirty-five

• 6000000+340000+500+89

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1C Rounding

Rounding makesnumberseasiertounderstandandusethroughsimplifyingtheirvalue butkeepingthemclosetotheiroriginalvalue.Thiscanbeusefulineverydaylife,for example,whenthinkingabouthowfarweneedtotravelonatriporhowlongitcould taketogetthere.

Wecanmakean estimate whichwillgiveusanumberthatiscloseenough.

Weusuallyroundnumberstothenearestten,hundred,thousandorlargerplace.We canroundtoanyplacevalue,dependingonwhatweneedthenumbersfor.

Thesymbolforapproximatelyequalslookslikethis ≈.

ThedistancefromMelbournetoBallaratisapproximatelyonehundredkilometres, whichwecanrecordas ≈ 100km.

Turnandtalk

86354peopleattendedtheAFLGrandFinalattheMCG.Roundedtothenearest hundred,thisis86400.Roundedtothenearestthousand,thisis86000.Roundedto thenearesttenthousand,thisis90000.

Discusswithapartnerwhichnumberwouldbemostusefulinthefollowingsituations andexplainyourreasoning.

• Distributingawristbandtoeveryperson

• Decidinghowmanypiestoorder

• Decidinghowmanygatestoopenforpeopletoentersafely

• Decidinghowmuchmerchandisetohaveinstock.

Howdoweround?

Decidewhichplacevalueyouneedtoroundtoandusethenumbertotherightofthis placetoguideyou.Inconsideringwhichtenisnearestwecanlookatthis0to99 numberchart.

Thisclearlyshowsthefirstnumbersinlightblue(40,41,42,43and44)arenearer tofourtenssoroundtoforty,wheretheothernumbersindarkblue(45,46,47,48 and49)areattheendoftherowsotheyroundto50.

Anumberlineisalsousefultovisualiseroundingnumbers.Ifwewantedtoroundto thenearesttenonthisnumberline,wecansee52iscloserto50,while57iscloser to60.Thehalfwaynumberalwaysroundstothehighervalue,so55would roundto60.

Thesameideacanbeusedforroundingtolargerplacevalues.Remember,ifthedigit totherightoftheroundingplaceis0to4,rounddown,andifthedigittotherightof theroundingplaceis5to9,roundup.

• 7345819roundedtothenearesttenis7345820

• 7345819roundedtothenearesthundredis7345800

• 7345819roundedtothenearestthousandis7346000

• 7345819roundedtothenearesttenthousandis7350000

• 7345819roundedtothenearesthundredthousandis7300000

• 7345819roundedtothenearestmillionis7000000.

1C

Wholeclass LEARNINGTOGETHER

Theseactivitiescanbeusedaswhole-classorsmall-groupactivities.Teacherscan limitthetimeorthenumberofentriesthatstudentscomplete.

1 Drawopennumberlinesintoyourworkbookusingthestartandendnumbers givenbelow.Recordwheretheunroundednumbershouldgoonyournumber line.Calculatewhichmultipletheywouldroundto.Remembertofirstaddthe halfwaynumbertothenumberlinetoguideyou.

a Drawanumberlinestartingat90andendingat100.Place94onthis numberlineandroundittothenearest10.

b Drawanumberlinestartingat100andendingat200.Place153onthis numberlineandroundittothenearest100.

c Drawanumberlinestartingat50000andendingat60000.Place54672 onthisnumberlineandroundittothenearest10000.

2 Roundthefollowingnumberstotheplacevaluegiven.

a 245tothenearestten

b 245tothenearesthundred

c 7126tothenearestthousand

d 1234567tothenearesthundredthousand

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1 Roundthefollowingnumberstotheplacevaluegiven.

a 451tothenearesthundred

b 6923tothenearestthousand

c 26450tothenearesttenthousand

d 567906tothenearesthundredthousand

2 Matchthefollowingnumberstotheirclosesttensofthousandsmultiple. 82724 60000 65395 80000 87928 70000 64999 90000

3 Nancysays567roundedtothenearesttenis570.Isshecorrect?Explainwhy orwhynot.

4 Jacobsays459320roundedtothenearestthousandis460000.Ishecorrect? Explainwhyorwhynot.

5 Identifyfivenumbersthatwouldberoundedto6500.

4 Copyandcompletethisplace-valuechart.

a 2306

b 479

c 89210

d 2007

5 Drawanumberlinewith0and100markedonit. Usealargedottomark80, 25and38.

6 WhoamI?

a Ihave1hundred,2tensand6ones.WhoamI?

b Ihave7thousands,2hundredsand8ones.WhoamI?

c Ihave3hundreds,4millions,6thousandsand23ones.WhoamI?

7 Writethenumberthatis100morethan:

8 Writethenumberthatis100lessthan:

9 Writethenumberthatis1000morethan:

10 Writethenumberthatis1000lessthan:

11 Placethecorrectsymbol<and>betweenthefollowingnumbers.

a 22255 5555

b 105782 105680

c 4697094 469714

d 36924 39642

12 Round12346totheplacevaluegiven.

a 12346tothenearesthundred

b 12346tothenearestthousand

c 12346tothenearesttenthousand

13 ChoosethreeormoreAustraliananimalstoinvestigate.Youcanselectthembased onyourinterests.

a Findthefollowingmeasurementsforeach:weight(inkilograms),lengthor height(inmetres).

b Createatabletoorganiseyourfindingsliketheonebelow.Includethenameof theanimal,itsweight,anditslength/height.

c Foreachmeasurement,identifytheplacevalueofeachdigit.Forexample, for1 708:

• 1(1kilogram)–onesplace

• 7(0 7kg)–tenthsplace

• 8(0 008kg) –thousandthsplace

d Writeafewsentences tosummariseyourfindings.Comparetheweights andlengths/heightsofyourchosenanimals.Usephraseslike"greaterthan", "lessthan"and"equalto".

1E Challenge–Ready,set,explore!

Egyptiannumbers

TheancientEgyptiansdidnothaveanalphabetlikeours.Theywrotebycreating pictures–calledhieroglyphs–witheachpicturerepresentingawordorsyllable.

Weusethedigits0, 1, 2, 3, 4, 5, 6, 7, 8, 9towritenumbersfrom0intothemillionsand beyond.ThesearecalledHindu–Arabicnumbers.

TheancientEgyptianshadabase-10mathematicssystem,butwithsomeimportant differencesfromthesystemweusetoday.

TheancientEgyptiansusedthesesevensymbolstowritenumbers.

Eachsymbolcouldbeuseduptoninetimesbeforechangingtothenexthighervalue. Forexample,towrite60,theancientEgyptianswouldwritethesymbolfor 10sixtimes. = 60

Thedigitsfrom1to9wereshownbystrokes. 10isahobble(adeviceattachedtothefrontorbacklegsofcattletostopthemfrom wandering).100isacoilofrope.1000isalotusplant.10000isafinger.100000isa frog.1000000isamanwithhisarmsraised.Numberscouldbewrittenintwoor threerows.

Example6

WritetheseasEgyptiannumbers.

Example7

WritetheseasHindu–Arabicnumbers.

5 Ascribecanwrite sheetsofpapyruseachday.Howmanypagescanscribes writein days?

6 Fatima’sheartbeats timeseveryminute.Howmanytimeswillherheartbeat in minutes?

7 Afteralongrun,Fatima’sheartwasbeatingat beatsperminute.Howmany timeswillitbeatin minutes?

8 WhatisthedifferenceinthenumberofFatima’sheartbeatsbetweenyouranswers toquestion6andquestion7?

Usefulskillsforthistopic

• understandingofplacevalueofnumbersto1000000andbeyond

• applyingadditionandsubtractionfactsto20fluently

• recordingadditionandsubtractionequationsonaverticalalgorithm

• applyingefficientmentalstrategiesforadditionandsubtraction

• estimationandrounding,checkingforthereasonablenessofananswer

Vocabulary

Addition • Sum

Minus

Takeaway

Algorithm • Part-part-whole

Calculate

Partitioning

Whatisyourapproach?

Renaming

Compensation

Rounding

1 Whatstrategycouldyouusetosolvethefollowingequations?

Wouldyouuseamentalstrategyoraverticalalgorithm?

Solveeachoneindependently,thenshareyourstrategyandwhyyouchoseit.

2 WhoamI?

Iamanumber. Iam6morethan25. a Iam9lessthan46. b Iam7morethan144. c Iam7lessthan1000. d Iam99morethan47. e Iam14lessthan51. f Iam55morethan38. g Iam6lessthan322. h Iam112morethan288. i Iam142lessthan972. j

Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction subtraction Additionand subtraction Additionand subtraction Additionand Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction Additionand subtraction subtraction

Weuse addition everydaytoworkoutthe total numberof things–forexample:

• calculating howmanypeopleattendedaconcertinaweek (wedothisbyaddingthedailytotals)

• estimating howmuchmoneyweneedforshoppingtomake surewehaveenoughmoneytopayforeverything.

Weuse subtraction everydaytorecordthe differencebetween two amounts–forexample:

• howmanystudentsarepresentatschool(wedothisbysubtractingthenumber absentfromthenumberofenrolledstudents)

• howmuchchangewereceivewhenwebuyabottleofwater.

2A Mentalstrategiesfor addition

The sum oftwonumbersisthetotalofthosenumberswhentheyareaddedtogether.

Jumpstrategy

Wecanfindthesumof34and47byusinganumberline,beginningat34andmaking ajumpof40totheright,thenanother7totheright.Wecanseethat34 + 47 = 81.

Theorderinwhichwedoadditiondoesnotmatter;theanswerwillbethesame whicheverorderweuse.Sowecouldbeginat47andjump30totheright,then another4totherighttogetthesameresult.Thenumberlineshowsthat47 + 34 = 81.

Partitioning

Usingthe partitioning method,wecansplitthenumbersintotheirplacevalueparts, thenaddthemtogetherstartingwiththelargestpartfirst.

Example1

a MikeandSallyare16and21.Whatistheircombinedage?

b Twooldcoinsare347and228yearsold.Whatistheircombinedage?

Solution

a 16 + 21 = 10 + 20 + 6 + 1

37years (Addtens,thenones.)

b 347 + 228 = 300 + 200 + 40 + 20 + 7 + 8 = 500 + 60 + 15 = 575years (Addhundreds,thentens, thenones.)

Compensation

Wecanaddmorethanisneeded,thensubtracttheextrathatwasaddedon.This strategyiscalled compensation

Example2

Add43and29. a

Solution

a 43 + 29 = 43 + 30 1 = 73 1 = 72

b 438 + 347 = 440 + 350 2 3 = 790 5 = 785

Add438and347. b

(Add1to29andimmediatelytake itaway.)

(Add2to438,add3to347,then subtract2andsubtract3.)

Thebestmentalstrategyistheonethatmakesthingseasyforyouandsavesyoutime. Youshouldpractisementalstrategiesandfindoutwhichstrategiesworkbestforyou. Alotdependsonthenumbersyouareworkingwith.

Teachers:readquestions 1–3 totheclassandaskthemtowritetheanswers.

1 Writethedoubleofeachnumber.

2 Doubleeachnumber,thenadd1.

3 Add9toeachnumberbyadding10,thentaking1away.

4 Workinpairs.Person1readsparts a–d toPerson2.Person2writestheaddition andthencalculatesthesummentallybeforewritingtheanswer.Thenswaproles forparts e–h.Checkeachother’sanswers.

2A Individual APPLYYOURLEARNING

1 Mentally calculate theseadditions. Recordthestrategyyouusetosolveeachone,forexample,‘jump’strategy.

2 Whichnumbermakesatotalof50whenaddedto:

3 Writethreedifferentsingledigitnumbersthataddto:

4 Usethe‘jump’strategytomentallycalculatethefollowingadditions.

5 Use‘partioning’tomentallycalculateeachaddition.Remember,theorderof additiondoesnotmatter.

6 a Therewere68Mazdasand45Toyotasinthecarpark.Howmanycarswere therealtogether?

b Therewere272boysand296girlsontherollofPalmerStreetSchool.How manychildrenwereontherollintotal?

7 Usethe‘compensation’strategytomentallycalculatetheseadditions.Thefirst onehasbeendoneforyou.

a 27 + 18 = 27 + 20 2 = 47 2 = 45

8 Scientistshavejustdiscoveredanewspeciesofbirdinthelargesourplumtreesofa remotepartofAfrica.Onescientistcountedthebirdshesawin10differenttrees: 18274162442933261119

a Mentallycalculatethenumberofbirdsthescientistsaw.

b Thescientistknowsthatatreewithanevennumberofbirdshasonlymaleand femalepairs.Treeswithoddnumbershaveallmales.Howmanymaleand femalepairsarethere?

9a Thisisa‘magicsquare’.Eachrow,eachcolumnandeachdiagonaladdsupto thesametotal.

Addthefirstcolumn.Ittotals18,soeveryrow,columnanddiagonalequals18. Thatmeansthemissingnumberinthebottomrowmustbe10.Themissing numberinthediagonalis6.Nowworkouttheothernumbers.

Copythesemagicsquaresandwritethemissingnumbers.

10 Makeamagicsquareofyourown.Seeifyourpartnercanfindthesolutionto yourmagicsquare.

11 Placeadditionsignsinthisstringofdigitssothatthesumofthenumbersis99. Youcangroupdigitstogether,forexample,98 + 7,buttheymustremaininthe ordergiven. 987654321

2B Thestandardaddition algorithm

An algorithm isasetofstepsusedtodocalculationsthatmaybetoodifficulttodo mentally.

Ifwewanttoadd39to45,wecanusethestandard additionalgorithm.

Setoutthenumbersoneundertheotheraccording totheirplacevalue.

Startwiththeonesdigits.Addthedigits.

Wesay,‘9onesplus5onesis14ones’. 14onesisthesameas1tenand4ones.

Write4intheonescolumnandcarry1tenintothe tenscolumn.

Nowlookatthetenscolumn.

Wesay,‘3tens + 4tens + 1ten(carriedfrom before) = 8tens’.

Writethe8inthetenscolumn.

39 + 45 = 84

Thestandardadditionalgorithmcanbeextendedtoaddnumbersofanysize.Allyou needtodoisaddthecolumnsfromrighttoleft,andcarrywheneveryouget10or higher.

Rounding isusefulforcheckingthereasonablenessofacalculationbyprovidingan estimation.Doestheanswermakesense?Ifwewereadding52and57,wewould expecttheanswertobecloseto50 + 60 = 110.Agood estimate usesnumbersclose totheoriginalnumbers.

Dependingonthesizeofthenumber,wecanroundtothenearest10,100,1000and soon.

Example3

Findthesumof315and568.

Theestimatetellsmemyanswerisreasonable.

Example4

Findthesumof3786and5949.

Theestimatetellsmemyanswerisreasonable.

Additionsthatinvolvemorethantwonumberscanalsobedonethisway.

Example5

Findthesumof2706,978and88.(Remembertoputthedigitsinthe correctplace-valuecolumns.)

(Addtheones,carrying2tensintothetenscolumn.Add thetens,includingthecarriedtensfrombefore.Addthe hundreds,carryingwherenecessary.Thenaddthe thousands.)

2B Wholeclass LEARNINGTOGETHER

1 Useplace-valueblockstomodeleachaddition.Thenrecordyourworking usingtheadditionalgorithm.

29 + 37 a

2 Checkyouranswersarereasonableusingrounding.

2B Individual APPLYYOURLEARNING

1 Usethestandardadditionalgorithmtocalculatetheseadditions. Thesehavenocarrying.

2 Usethestandardadditionalgorithmtocalculatetheseadditions.

Theseinvolvecarryingfromtheonestothetens.

3 Theseinvolvecarryingfromtheonestothetens,andfromthetenstothehundreds. 23

4 OnSaturday,JamesandTonidrove32kilometres.OnSunday,theydrove 326kilometres.Howmanykilometresdidtheydriveintotal?

5 Usethestandardadditionalgorithmtoworktheseout.

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+ 12848 + 176 i

6 Peterhas274marbles,Asafhas366marblesandKiahas185marbles.Howmany marblesdotheyhaveintotal?Useroundingtocheckyouranswerisreasonable.

7 Usethestandardadditionalgorithmtoworktheseout.

a Whatamountis $3525morethan $6778?

b Add2750litrestothesumof7750litresand5570litres.

c Whatamountis $1432morethanthesumof $2413and $3214?

d Addthesumof5128kilogramsand4736kilogramstothesumof 7394kilogramsand4328kilograms.

8 Jospent $187atthesupermarket, $288atthebutcherand $94atthebakery. Whatwasthetotalamountshespent?Useroundingtocheckyouransweris reasonable.

9 Laurenspilledchocolatemilkonherworksheetandcoveredsomeofthenumbers inthefollowingadditions.Writethemissingnumberforeachone.

10 Andrewadded2561 + 472andhisanswerwas7281.Ishecorrect?Ifnot,what mistakedidhemake?

2C

Mentalstrategiesfor subtraction

Whenweusesubtraction,weareeither‘takingaway’onenumberfromanotheror ‘buildingup’fromonenumbertoanother.

Youcanthinkaboutsubtractionastakingawayorasaddingon. Eitherway,subtractionisthedifferencebetweentwonumbers.

Takingaway

Forexample,27 18.Whenwetakeawayusinganumberlinewesay,‘27takeaway 18is…’.

Addingon

Wecanalsouseanumberlinetobuildupfromonenumbertothenextandwesay, ‘WhatdoIaddto18togetto27?’

Thementalstrategiesforsubtractionusetheideathatwecan‘breaknumbersapart’to makecalculationseasiertomanage.Sometimesthementalstrategiesweusearehard towritedown.Itisimportanttohaveconversationswithyourteacherandyour classmatesaboutthestrategiesyouusewhensubtracting‘inyourhead’. Therearemanymentalstrategiesforsubtraction.Herearesomeofthem.

Subtractabitatatime

Subtracttwoseparatepiecesinsteadofone.

Example6

Subtract17from43.

Builduptothelargernumber

Addontothesmallernumbertobuilduptothelargernumber.Keeptrackofwhat youhaveadded.

Example7

Subtract39from87.

Solution

Tocalculate87 39webuildupfrom39:

39 + 1 = 40 (Add1.)

40 + 47 = 87 (Add47.)

87 39 = 48 (Atotalof48hasbeenadded.)

Addthesametobothnumbers

Addingthesamenumbertobothnumbersdoesnotchangethedifferencebetween them.

Thinkofanadultandachildstandingtogether.Supposethedifferenceintheirheights is25centimetres.Iftheybothstandtogetheronabox,thedifferencebetweentheir heightsisstillthesame.

Example8

54 36:findtheresultbyaddingthesameamounttobothnumbers.

Solution

54 36 = 58 40 (Add4tobothnumbers.) = 18

Adding4tobothnumbersdoesnotchangethedifferencebetweenthem.The differencebetween58and40isthesameasthedifferencebetween54and36.

Part-part-whole

Additionandsubtractionare inverse operations.Thiscanhelpusfindan unknownnumberinaproblembyusing part-part-whole.

Subtractionistheinverseofaddition.

Additionistheinverseofsubtraction.

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Usinginverseoperationshelpsustochecktheaccuracyofourcalculation.

Wecanthinkofsubtractionintwoways:

• takingawayonenumberfromanother

• addingonfromonenumbertogettotheother.

Wecansubtractusingmentalstrategies–forexample:

• takingaway2piecesinsteadof1.

13 = 27 10 3 = 17 − 3 = 14

• buildinguptothelargernumber.36 17:add3,thenadd10,then add6.19hasbeenadded:36 17 = 19.

• addingthesametobothnumbers. 83 16 = 87 20 = 67

• Ifweknowthetotalandonepart,wecanfindthemissingpartby subtractingbecausesubtractionistheoppositeofaddition.

Thedifferencebetweentwonumbersistheresultwhenonenumberis subtractedfromanother.

1 Mentallysubtract13fromeachnumberbytakingaway10,thentaking away3.

2 Mentallysubtract127fromeachnumberbytakingaway100,thentaking away20andfinallytakingaway7.

3 Mentallysubtract99fromeachnumberbytakingaway100,thenadding1.

4 Addthenumberinbracketstobothnumbers,thencompletethesubtraction mentally.

5 Useinverseoperationstowriteadditionandsubtractionequationsbasedon thebarmodelbelow.

6 Tomhasworkedoutanaddition396 + 796 = 1182. Hewantstocheckheiscorrectbyusinginverseoperations.Which subtractionscouldheuse?

1 Dothesesubtractionsinyourheadby‘buildingup’tothelargernumber.

2 Usethe‘addthesametobothnumbers’methodtocalculatethesesubtractions.

3 Mentallysubtract24fromeachnumberbyfirstsubtracting20,andthen subtracting4.

4 Usethe‘addthesametobothnumbers’methodtosolvethesesubtractions mentally.

188 a

5 Solvetheseproblemsusingamentalstrategyofyourchoice.

a Therewere72childreninYear5.If43ofthechildrenweregirls,howmany boyswerethere?

b Afactoryemploys375menand288women.Howmanymorementhan womenareemployedinthefactory?

c Sumitrahas803stamps.HerfriendEnihas645stamps.Howmanymore stampsdoesSumitrahavethanEni?

d Thenurseryhas675petuniaplantsand397dahliaplants.Howmanymore petuniaplantsthandahliaplantsarethere?

e Mardi’sshoppingcametoatotalof $143.35.Shegavethecashiertwo $100 notes.HowmuchchangeshouldMardireceive?

f Thereare463childreninMountLeafySchool.78ofthechildrenareinYear5. IfallofYear5wentonanexcursion,howmanychildrenwouldbeleftat school?

6 Chooseyourownstrategytocalculatethesesubtractionsmentally.

371 a

7 Usementalstrategiestosolvetheseproblems.Insertadditionorsubtractionsigns tomakeeachstatementtrue.

a 8 4 6 7 = 13

b 27 13 8 3 = 3

c 49 121 642 777 = 35

d 264 391 227 443 = 871

2D Subtractionalgorithms

Sometimesyouneedtouseasubtractionalgorithmratherthanmentalstrategies.Here isastandardsubtractionalgorithm.

Calculate68 45.

Setoutthenumbersoneundertheother,andlinethemupinplace-valuecolumns. Writethenumbertobesubtractedfromtheothernumber.

Startwiththeonesdigits.Subtractthebottomdigitfromthetopdigit.

Wesay,‘8onestakeaway5onesis3ones’.

Write3intheonescolumn.

Nowworkwiththetensdigits.Subtractthebottomdigitfromthetopdigit.

Wesay,‘6tenstakeaway4tensis2tens’.

Write2inthetenscolumn.

68 45 = 23

However,notallsubtractionsareassimpleasthisexample.Sometimesthenumbers arenotaseasytodealwith.Therearetwodifferentalgorithmsyoucanuse,sochoose thealgorithmyoufeelmostcomfortablewith.

Tradingordecomposition

Findthedifferencebetween63and47.Thismethodisbasedontrading1tenfor 10onesand1hundredfor10tens,andsoon.

Tocalculate63 47,setoutthenumbersoneundertheotheraccordingtotheirplace value.

Startwiththeonesdigits.Therearenotenoughones.Weneedtotrade.

Trade10onesfor1teninthetopnumber.

Crossoutthe6andwritea5toshowthereare5tensleft. Writea1inthetopnumbernearthe3toshowthattherearenow13ones.

Nowwecansubtracttheonesdigits.

Wesay,‘13onestakeaway7onesis6ones’. Write6intheonescolumn.

Nowlookatthetenscolumn.

Wesay,‘5tenstakeaway4tensis1ten’. Write1inthetenscolumn.

63 47 = 16

Equaladdition

Thismethodisbasedonthementalstrategyofaddingthesametobothnumbers.Itis sometimescalledthe‘borrowandpayback’method.Whenthesameamountis addedtobothnumbers,thedifferencebetweenthemisthesame.

Tocalculate63 47,setoutthenumbersoneundertheotheraccordingtotheirplace value.

Startwiththeonesdigits.Therearenotenoughones.

Weadd10tobothnumbers.Thereisaspecialwaytodothis.

Because10onesisthesameas1ten,weadd10onestothetopnumberand1tento thebottomnumber.

Write1intheonescolumnofthetopnumber,sothe3becomes13.

Write1inthebottomnumbernearthe4.(Thisisaddedtothe4later.)

Wesay,‘13onestakeaway7onesis6ones’.

Write6intheonescolumn.

Nowlookatthetenscolumn.

Wesay,‘6tenstakeaway5tens(rememberthe1carriedfrombefore)is1ten’.

Write1inthetenscolumn.

63 47 = 16

2D Individual APPLYYOURLEARNING

Useoneofthesubtractionalgorithmstocompletetheseexercises.

1 Calculatethesesubtractions.

21 a

103 c

171 e

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65 b

56 d

523 f 1076 34 g

2 Calculatethesesubtractions.

68 a

3 OnMonday,Sofia’sbeanplantwas86centimetrestall.OnTuesday,itwas 92centimetrestall.HowmuchdidSofia’sbeanplantgrow?

4 Calculatethesesubtractions.

5 Theschoolbusseats64passengers.Thereare29peopleonthebus.How manyemptyseatsarethere?

6 AceCinemaseats865people.Thereare679peopleinthecinema.Howmany emptyseatsarethere?

7 Calculatethesesubtractions.

8 Thereare1423childreninHenleySchool.If846childrenareboys,howmany girlsarethere?

9 Trevor’sdadboughtacarfor $6375.Hesolditayearlaterfor $4990.How muchmoneydidTrevor’sdadlose?

10 Jimneeds $6325tobuyanewhome-theatresystem.Hehasalreadysaved $4897.Howmuchmoredoesheneedtosave?

11 a Take32847from56003.

b Findthedifferencebetween62497and43014.

c Howmuchmorethan42917is64164?

2E

Reviewquestions–Demonstrateyourmastery

1 Whatdoyouneedtoaddtoeachnumbertomakeatotalof250?

2 Mentallycalculatetheseadditions.

3 Minhhas57videosand28DVDsinhermoviecollection.Howmanymoviesdoes shehaveintotal?Explainwhichstrategyyouwouldusetosolvethismentally.

4 Usetheadditionalgorithmtocalculatethese.

5 Leannesold67T-shirtsatthemarket.Andrewsold188T-shirts.Howmany T-shirtsdidtheysellintotal?

6 Writethenumberthatis3489morethan2184.

7 Writethenumberthatis32904morethan3821.

8 Calculatetheseadditions.

9 Roundthesenumberstothenearest10000toestimatetheanswers,thenperform thecalculations. 32987 + 89678 a 9945 + 21432 b

c 58242 29201 d

10 Usetheinverseoperationtofindthemissingnumbers.

11 Findthedifferencebetween:

12 Completethesesubtractions.Explainthementalstrategyyouchose.

13 Writethenumberthatis: 239lessthan3857 a 483lessthan57239 b

14 ItisJedda’sfirstyearathighschool.Jedda’sparentsspent $2839onacomputer, $64onaschoolbag, $567onschooluniformsand $394onbooksforher.

a HowmuchdidJedda’sparentsspendintotal?

b Jedda’sparentshad $4000intheir‘gettingJeddareadyforhighschool’bank account.Howmuchmoneywasleftover?

15 Lachlanspilledstrawberrymilkonhisworksheetandcoveredsomeofthe numbersinthefollowingadditions.Writethemissingnumberforeachone.

16 MrGreenhas $1600tobuysomeskiequipment.Hespends $857onapairofskis and $385onaskijacketandtrousers.Doeshehaveenoughmoneyremainingto buyapairofskibootscosting $260?

2F Challenge–Ready,set,explore!

Romannumerals

Weusethedigits0, 1, 2, 3, 4, 5, 6, 7, 8and9 towritenumbersfrom0intothemillions andbeyond.ThesearecalledHindu-Arabic numbers.TheancientRomansuseda differentsystemforwritingnumbers,called Romannumerals.

Romannumeralsoriginatedinancient Rome.AlthoughRomannumeralshavenot beenusedfordoingmathematicsforalong time,westillusethemonclockfaces,to numberpagesatthebeginningofbooks andforfilm-releasedates.Wealsouse themtonumberkingsandqueenswho havethesamename(KingPhilipII,Queen ElizabethII)andtonumbersomesports eventsinaseries,suchastheXXVII OlympiadinSydney.

WritingRomannumerals

TheRomanswrotethenumbers1, 5, 10, 50, 100, 500and1000usingasingle upper-caseletter.ThesesinglelettersformthebasisoftheRomannumeralsystem. 1

TheRomansbuilttheirnumbersbycombiningnumerals.

I=1II=2III=3

IfanI,XorCwasplacedtotheleftofahigher-valuenumeral,itmeantthatthe smallernumeralwastobesubtracted.

Theywrotetheirnumeralfor4toshow‘onelessthan5’: IV=4

SinceV=5,thenumbersfrom5to8arebuiltupfromit.

VI=6VII=7VIII=8

TheRomanswrotetheirnumeralfor9toshow‘1lessthan10’: IX=9

XLmeans‘10lessthan50’=40.

XCmeans‘10lessthan100’=90.

Theydidnothaveaplace-valuesystemlikeours.Theywrotethelargervaluestothe leftofthesmallervalues,thenaddedthemtogether.

CLXV=100 + 50 + 10 + 5 = 165

TheRomansuseduptothreeofthesymbolsfor1, 10, 100or1000together,andonly onesymbolfor5, 50or500inanynumeral.

III=3CCC=300MMM=3000

Sometimesthenumeral4iswrittenasIIIIonaclockface,but theRomanswouldhavewrittenthisasIV.

Example10

WritethesenumbersinRomannumerals.

= 10 + 2=XII

= 40 + 7=XLVII

Numberswithninescanbetricky.

Example11

WritethesenumbersinRomannumerals.

= 30 + 8=XXXVIII

+ 4=LXXIV

a 49 = 40 + 9=XLIX

b 99 = 90 + 9=XCIX

c 94 = 90 + 4=XCIV

CDmeansonehundredlessthan500is400.CMmeansonehundredlessthan1000 is900.

Example12

WritethesenumbersinRomannumerals.

Challengequestions

1 WritetheseHindu-ArabicnumbersinRomannumerals. 29 a 165 b

c 57 d 444 e

2 WritetheseRomannumeralsinHindu-Arabicnumbers.

LXII a CXCVII b XLI c DCLXXII d CXLIX e

3 Measureeachoftheseitems,thenwritethemeasurementsinRomannumerals. Swapwithaclassmateandcompareyouranswers.

a Thelengthofyourmiddlefingerinmillimetres

b Thewidthofyourteacher’sdeskincentimetres

c Themassofyourschoolbaginkilograms

d Theheightoftheseatofyourchairincentimetres

e Thewidthoftheclassroominmetres,thenchangeittocentimetres

4 TryaddingusingonlyRomannumerals.Showyourworking. Forexample:CCXXXIV+CDXLVIII

=CCXXXIIII+CCCCXXXXVIII

=CCCCCCXXXXXXXVIIIIIII

=DCLXXVVII

=DCLXXXII

a LXX+XCVII

b CCXXIII+CCLXXXIX

c CDXLVII+CCXIX

d DidyoufinditdifficultusingRomannumeralstoadd?Discussthepossible reasonsfornotusingRomannumeralsformathematicsallthetime.

5 OnthePaulusfarm,therewereXLVIpigs,CCXXsheep,XCVIIIcows,XXXIVgoats andVIIIcats.

a Howmanyanimalswereonthefarmintotal?WriteyouranswerinRoman numerals.

b Howmanylegswerethereintotal?WriteyouranswerinRomannumerals.

6 Juliuscountedthenumberofpeopletopasshismarketstallinonehour. XXVIIpeoplehadblackhair,LXIXpeoplehadbrownhair,XXXIVpeoplehadfair hairandXIXpeoplehadredhair.

HowmanypeoplepassedJulius’sstallinthathour?DoyouradditionusingRoman numerals,thenuseHindu–Arabicnumberstocheckyouranswer.

7 TheancientRomansusedaqueductstogetwatertotownsandvillages.An aqueductisapipeorachannel.TheaqueductinBrutus’svillagedevelopedasmall leakandwaslosingwaterattherateofCVIIlitreseveryIIIhours.

a Howmuchwaterwouldbelosteachday?WriteyouranswerinRoman numerals.

b TheancientRomansusedonlyadditionandsubtraction,notmultiplication. Howdoyouthinktheywouldhavecalculatedtheanswertothisproblem?

8 Ascribewroteashortletterforasenator.IVlinesofthelettercontainedVIwords, IIIlineshadVIIwords,VlineshadIVwords,VIlineshadVwordsandIIlineshad VIIIwords.

a Howmanywordsdidthescribewriteintotal?

b TrydoingthecalculationusingonlyRomannumerals.

Usefulskillsforthistopic

• understandingthemultiplicationoftwo-digitnumbersbyasingle-digit

• understandingtheconnectionbetweenrepeatedadditionandmultiplication

• usinganarraytorepresentamultiplicationproblem

Vocabulary

Multiply

• Product • Estimation • Algorithm

Multiples

• Lowestcommonmultiple(LCM) • Array • Repeatedaddition • Commutativeproperty

• Distributiveproperty • Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage

1 Dirk’sBackyardBuildingCompanyhassupplied48squareconcretetilesforPam’s backyard.PamwantsDirktoarrangethetilestomakearectangle.

• HowmanydifferentwayscanDirkarrangethetiles?

• Drawallthepossibilities.

• Discussandcompareyourdrawingwithapartner,thencompareyourdrawings asaclass.

2 Thefarmerhas6basketsofapples;eachbaskethas24apples.Howmanyapples hashepicked?

• Howmanydifferentstrategiescanyouusetosolvetheproblem?

• Howdidyoucheckthereasonablenessofyouranswer?

Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication Multiplication

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Wecan multiply twonumberstogethertoworkoutatotal. Youcanthinkofmultiplicationasthetotalof‘lotsof’anumber. Forexample,Isaachasacollectionoftoycars,buthe’snotsurehowmanyhehas. Ratherthancountthemall,heassemblesthemintoanarray. Whenhedoesthis,Isaacfindshehasfourlots,eachoffivecars.

Thisiswrittenas4 × 5 = 20,orfourtimesfiveistwenty.SoIsaachasatotalof 20toycars.Hecancheckthisbycountingthemonebyoneifhewantsto. Multiplyingisafastandeasywayofcountinglargesetsofthings.

Uncorrected 4th sample

3A Arrays

Whenwecountbyfour,wesay: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52 …

Thefourscountingpatterncanbeshowninpicturescalled arrays,likethis:

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1 lot of 4

Weusethemultiplicationsymbol × asashortwayofsaying‘lotsof’.

Theanswertothemultiplicationisthenumberofobjectsinthearray. Arrayscanbedrawnsothattheystanduporliedown. Herearethetwoarraysforthemultiplication7 × 3 = 21.

7 × 3 3 × 7

Lying down: 3 lots of 7

Standing up: 7 lots of 3

Thenumberisthesameinboth.Itdoesnotmatterinwhichorderwedothe multiplication–wealwaysgetthesameanswer.

7 × 3 = 3 × 7 = 21

Whenwemultiplytwonumbers,theanswerwegetiscalledthe product ofthetwo numbers.

Uncorrected 4th

Theproductof6and4is24. Thereare6lotsof4in24. Wealsohave4lotsof6in24. 6 × 4 = 24isthesameas4 × 6 = 24. Thinkabouttheshapeofthearraysthat wehavebeenlookingat.Whichkindof shapeisalwaysused?Discusswhythisis thecase.

3A Wholeclass LEARNINGTOGETHER

1 Wecanusebothwordsandsymbolstodescribeanarraytable.Belowisan exampleofwordsandsymbolsforanarray.

• 7threes

• 7multipliedby3

• Thereare7columnswith3in eachcolumn.

• Thereare3rowswith7in eachrow.

Usethisexampletodescribethearraysin a and b a b

2 Draw6rowsof2apples.Writethetotalnumberofapplesunderyour drawing.Completethestatements:6 × 2 = 2 × =

3 Draw3rowsof4oranges.

Writethetotalnumberoforangesunderyourdrawing.

Completethestatements:3 × 4 = 4 × =

4 Draw4rowsof12watermelons.

Writethetotalnumberofwatermelonsunderyourdrawing.

Completethestatements:4 × 12 = 12 × =

5 rowsof stars × =

3A Individual APPLYYOURLEARNING

Copyandcompletethefollowing.

groupsof isequalto and × =

2 a Drawthearrayfortheproduct7 × 6 = 42.

b Whatdoyounoticeabouttheshapeofthearray?

c Isthereanotherwaytodrawthearray?

d Whatisdifferent?Whatisthesame?

3 Writetwomultiplicationstatementstorepresentthepicturebelow.

Whataretheproductsofyourmultiplicationstatements?

4 Howmanyeggsareinthecartonbelow?Howdoyouknow?Sharethestrategy youused.

5 Drawallthepossiblerectangulararraysthatgive24astheproduct.Writethe multiplicationstatementforeacharray.

6 Drawapictureofacaryardwith3rowsof6cars.

Howmanycarsarethereintotal?3 × 6 =

7 ThefloortilesinCara’sbathroomareinrowsof8. Thereare11rowsoftiles.DrawCara’sbathroomtiles.

Howmanytilesarethereintotal?8 × 11 =

8 Ian’skitchenhas48tilesin4rows.Howmanytilesareineachrow?

9 Afarmerplants72strawberryplantsinafield.Howmanydifferentcolumnsand rowscanthefarmercreatetoplantthestrawberries?Writeanequationforeach array.

10 Usethearraybelowtowriteawordedproblem.

3B Multiples

Weget multiples whenweskip-count.Thesearesomeofthemultiplesof7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91 …

Wegetmultiplesbybuildinguprectangulararrays. Thesearraysshowthefirst6multiplesof7:

Wheneveryoumultiplyawholenumberbyanotherwholenumber,yougetamultiple. Soweseethat7, 56, 609and700aremultiplesof7. 7 × 1 = 77 × 8 = 567 × 87 = 6097 × 100 = 700

Example1

Listthefirst10multiplesof8.

Solution

Themultiplesaretheanswertoeachmultiplicationfactupto10 × 8.

1 × 8 = 82 × 8 = 163 × 8 = 244 × 8 = 325 × 8 = 40 6 × 8 = 487 × 8 = 568 × 8 = 649 × 8 = 7210 × 8 = 80

Thisisthesameasskip-countingbyeight,stoppingatthetenthnumberinthe sequence.

Lowestcommonmultiple(LCM)

Herearethefirstfewmultiplesof3and4.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45 …

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

Thenumbers12, 24, 36 … areinbothlists.Wesaythattheyarecommonmultiplesof both3and4.

The lowestcommonmultiple(LCM) of3and4is12.Itisthesmallestcommon multiple.Findingthelowestcommonmultiplewillbeausefulskillforlateronwhen youstartaddingfractions.

Example2

FindtheLCMof3and5.

Solution

Listthemultiplesof3.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30 …

Listthemultiplesof5.

5, 10, 15, 20, 25, 30

Thefirsttwocommonmultiplesare15and30.Thelowestcommonmultiple(LCM) is15.

3B Wholeclass LEARNINGTOGETHER

1 Timedteamtask

Useaclockwithasecondhandtokeeptrackof1-minuteintervals.Seehow faryoucangetwithcorrectanswersin1minutewhenyouskip-countby: two a five b three c four d ten e six f

2 Whichnumbersbetween2and18: aremultiplesof3? a aremultiplesof3and4? b aremultiplesof4butnot3?

c aremultiplesof3butnot4? d

3 FindtheLCMofthefollowingpairsofnumbers. 2and3 a 3and4 b 3and7 c 4and5 d 10and12 e

4 TrueorFalse?

a TheLCMof4and8is16.

b TheLCMoftwonumbersistheproductofthetwonumbers.

c TheLCMof3, 4and5is60.

1 Listthefirst5multiplesof:

2 a Whichofthesenumbersaremultiplesof3? 16, 12, 6, 10, 23, 18, 21, 17, 33, 31, 43, 50, 22

b Whichofthesenumbersaremultiplesof7? 12, 14, 52, 49, 34, 84, 100, 28, 105

3 a Whatisthelargestmultipleof5between1and14?

b Whatisthelargestmultipleof7between60and68?

Drawfourbagsinyourbookandlabelthemasshownbelow.Sorteachnumber from1to36intothecorrectbag.Dosomenumbersfitintomorethanonebag?

5 Jamieisfindingthefirstcommonmultipleof3and6.Heworksout3 × 6 = 18,so 18isthefirstcommonmultipleof3and6.IsJamiecorrect?Explainyouranswer.

6 Janewalksherdogonthefootyovalevery2ndday.Emmawalksherdogonthe footyovalevery4thday.Theybothwalktheirdogsonthe1stMarchandmeeton theoval.HowmanymoretimeswillJaneandEmmameetontheovalbeforethe endofMarch?

3C

Mentalstrategiesfor multiplication

Thissectionisdesignedtohelpyoulearnyourmultiplicationtablesifyoudonotknow them‘offbyheart’.Youmayknowsomeoftheseideasalready.Itisagoodideatobe veryquickatrememberingyourmultiplicationfactsastheyareusedinotherareasof mathematics.

Youcanthinkaboutmultiplicationasa repeatedaddition.Forexample:

4 + 4 = 8

4 + 4 + 4 = 12

4 + 4 + 4 + 4 = 16

isthesameas1 × 4 = 4

isthesameas2 × 4 = 8

isthesameas3 × 4 = 12

isthesameas4 × 4 = 16

4 + 4 + 4 + 4 + 4 = 20isthesameas5 × 4 = 20

4 + 4 + 4 + 4 + 4 + 4 = 24isthesameas6 × 4 = 24

Whenwemultiplyanumberbyanyothernumber,wegetamultipleofthenumber.

So4, 8, 12, 16, 20and24aremultiplesof4.

Youcangetmoremultiplesof4byadding4atatime.Forexample:

4 + 4 + 4 + 4 + 4 + 4 + 4 = 7 × 4 = 28

4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 8 × 4 = 32, andsoon

4th sample

Ifyoucancountbyfive,youwouldfillinpartofthemultiplicationtablelikethis.

Wewilllookatsometipsandtricksforlearningmultiplicationtables.

Multiplyingby 6

Wecanuseashortcuttomultiplyanumberby6.Firstmultiplythenumberby2,then by3.Thisworksbecause6 = 2 × 3.

Example3

Calculate25 × 6.

Solution

25 × 6 = 25 × 2 × 3 = 50 × 3 = 150

Sometimesthereisamoreefficientstrategy.

Example4

Calculate19 × 6.

Solution

Youcanuseavariationofthemultiplying-by-20strategy,because19 = 20 1:

19 × 6 = 20 × 6 1 × 6 = 120 6 = 114

Multiplyingby 9

Ifwewanttoget9lotsofsomething,it’seasiertofind10lotsandtake1lotaway. Anexampleisbelow.

Example5

Calculate9 × 16.

Solution

9‘lotsof’16is10‘lotsof’16takeaway1‘lotof’16.

9 × 16 = 10 × 16 1 × 16 = 160 16 = 144

Multiplyingby 10

Numbersthataremultiplesof10endinzero.Forexample:

1 × 10 = 102 × 10 = 203 × 10 = 30

Tomultiplyawholenumberby10,placeazeroattheendofthenumber.

23 × 10 = 23099 × 10 = 990789302 × 10 = 7893020

Multiplyingby 11

Ifwewanttoget11lotsofsomething,wefind10lotsandadd1lot.

Example6

Calculate16 × 11.

Solution

16 × 11 = 16 × 10 + 16 × 1

= 160 + 16 = 176

Multiplyingby 20

Ifwewanttomultiplyby20,wedoublethenumber,thenmultiplyitby10.Thisworks because20 = 2 × 10.

Example7

Calculate12 × 20.

Solution

12 × 20 = 12 × 2 × 10 = 24 × 10 = 240

Multiplicationtablesarereallyjustlistsofmultiplesofanumber.Alwaystrytousethe mostefficientstrategy.

3C Wholeclass LEARNINGTOGETHER

1 Makeyourownmultiplicationbooklet

Inyourbookletexplainaruleformultiplyingeachofthedigits1–12.You candraworwriteamultiplicationandgiveanexampleofhowitcanbe solvedusingtheruleyouhavechosen.Istheremorethanoneruleforsome numbers?

3C Individual APPLYYOURLEARNING

1 Mentallymultiplyeachnumberby10.

2 Mentallymultiplyeachnumberby20.

3 Mentallymultiplyeachnumberby4.Remember:thequickwayistodouble,then doubleagain. 25 a

4 Mentallymultiplyeachnumberby9.

5 Mentallymultiplyeachnumberby11. 12 a

6 Mentallymultiplyeachnumberby6,bydoublingandthenmultiplyingby3.

9 a

7 HamptonHillsSchoolhas9classes,with26childrenineachclass.Howmany childrenarethereinHamptonHillsSchool?

8 Ifthereare11chocolatebiscuitsineachpacket,howmanybiscuitsin121packets?

9 a Writearuleformultiplyinganynumberby30.Testyourruleonfivenumbers. b Writearuleformultiplyinganynumberby300.Testyourruleonfivenumbers.

3D

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Multiplicationstrategies

The commutativeproperty ofmultiplicationmeansthatyoucanchangetheorderof thenumbersyoumultiply,andtheresultwillremainthesame.

Ifyouhave3bagswith4candiesineach,youcan multiplyitas3 × 4or4 × 3.Bothways,youwill have12candies.Thisshowsthattheorderin whichyoumultiplythenumbersdoesnotchange theanswer;itisalwaysthesame.

Thecommutativepropertyhelpsmakemultiplicationeasierandmoreflexible. However,itisimportanttorememberthatthispropertyonlyworksforadditionand multiplication,notforsubtractionordivision.

The distributiveproperty ofmultiplicationisawaytobreakdownamultiplication problemintosmaller,moremanageableparts.Itmeansthatwhenyoumultiplya numberbyasum,youcanmultiplyeachpartofthesumseparatelyandthenaddthe resultstogether.Thispropertyhelpsmakemultiplicationeasierandmoreflexible.

Wecanapplythedistributive propertytobreakapartanarray. Arrayscanbebrokenapartin manyways.Weneedtobreak apartanarraytomakeiteasierto findthetotal.Thearraysopposite demonstratetwodifferentways.

3D Wholeclass LEARNINGTOGETHER

1 Commutativepropertyofmultiplication

Theorderinwhichwewritetheproductoftwonumbersdoesnotmatter.Itwill alwaysgivethesameanswer.Forexample,ifyouknow3 × 4 = 12,youalsoknow 4 × 3 = 12.

Youwillneedtodownloadacopyofamultiplicationtable.Onthemultiplication table,everyproductabovethediagonalofperfectsquarescanalsobefound belowit.Colourintherepeatedfacts.

Nowyourmultiplicationchartshouldlooklikethis.

Goodnews!Younowonlyhavetorememberslightlymorethanhalfofthe multiplicationtable.Allyouneedtolearnnowarethemultiplicationsthatgivethe numbersinthewhiteboxes.

2 Youarearrangingdesksintheclassroom.Ifyouhave6rowswith8desksineach row,howmanydesksdoyouhaveintotal?Now,ifyouarrangethedesksin 8rowswith6desksineachrow,howmanydesksdoyouhave?Usethe commutativepropertytoexplainyouranswer.

3 Youareplanting6rowsofflowers,andeachrowhas14flowers.Showhowyou canusethedistributivepropertytofindthetotalnumberofflowers.

4 Youhave5shelves,andeachshelfholds23books.Showhowyoucanusethe distributivepropertytofindthetotalnumberofbooks.

3D Individual APPLYYOURLEARNING

1 a Ianiscoachingthetennisteam.Heneeds4tennisballsforeveryplayeronhis team.Howmanytennisballsdoesheneedforateamof9tennisplayers?

b Katehas7horsesinastable.Sheneedstogiveeachhorsenewshoes.How manyshoesdoessheneed?

2 Johnnoticedthatwhenhemultiplied4by3andthenadded3,itwasthesameas multiplying5by3.UseJohn’stricktosolvetheseproblems.Thefirstonehasbeen doneforyou.

a If4 × 3is12,whatis5 × 3?5 × 3 = 4 × 3 + 3 = 12 + 3 = 15

If3 × 3is9,whatis4 × 3? b

If5 × 8is40,whatis6 × 8? d

If5 × 7is35,whatis6 × 7? f

If8 × 8is64,whatis9 × 8? h

If8 × 6is48,whatis9 × 6? j

If11 × 11is121,whatis12 × 11? l

If11 × 6is66,whatis12 × 6? n

If6 × 5is30,whatis7 × 5? c

If11 × 4is44,whatis12 × 4? e

If7 × 7is49,whatis8 × 7? g

If6 × 8is48,whatis7 × 8? i

If11 × 8is88,whatis12 × 8? k

If11 × 9is99,whatis12 × 9? m

If12 × 8is96,whatis13 × 8? o

Chloenoticedthatwhenshemultiplied10by6andthensubtracted6,itwasthe sameasmultiplying9by6.UseChloe’sideatosolvetheseproblems.Thefirstone hasbeendoneforyou.

a 10 × 6 = 60, so9 × 6 = 60 6 = 54

b 10 × 8 = , so9 × 8 = .

c 10 × 9 = , so9 × 9 =

WhenImultiplyanumberby5,itisthesameasmultiplyingby10andthen dividingby2.Usethisruletosolvethefollowing.Thefirstonehasbeendone.

a 10 × 8 = 80, so5 × 8 = 80 ÷ 2 = 40

b 10 × 12 = , so5 × 12 =

c 10 × 7 = , so5 × 7 =

Dothemultiplicationsfirst,thenaddthechunkstofindtheproductof3and14.

3 × 14 = 3 × 10 + 3 × 4 = 30 + 12 = 42

Insteadofdrawingarrays,youcandrawmultiplicationdiagramstohelpyou‘see’ themultiplication.Thismultiplicationdiagramusesthechunks3 × 10and3 × 4to show3 × 14.

Useamultiplicationdiagramtocalculate8 × 17.

Withpractice,youcandothistypeofmultiplicationmentally.

Example9

Calculate17 × 4mentally.

Solution

Split17into10and7anddothesestepsinyourhead.

17 × 4 = 10 × 4 + 7 × 4 = 40 + 28 = 68 (Dothemultiplicationsfirst.)

Multiplyinglargernumbers

Multiplicationdiagramscanalsobeusedtobreakapartlargerproducts.

Hereisthemultiplicationdiagramfor13 × 17.Ithasbeenshadedtoshowthechunks.

10 × 1010 × 73 × 103 × 7

Ifwewanttodrawamultiplicationdiagramfor13 × 17,wefirstsplitthenumbers13 and17intotensandones.Thisworksbecause13 = 10 + 3and17 = 10 + 7.

Addtheproductsinthechunkstogettheproductof13and17.

13 × 17 = 10 × 10 + 10 × 7 + 3 × 10 + 3 × 7 = 100 + 70 + 30 + 21 = 221

Example10

a Drawamultiplicationdiagramtofindtheproductof16 × 19.

b Show16and19splitintotensandones.Showthechunksyougetwhenyou splitthenumbersintotensandones.

c Writetheproductsinsideeachchunk.

d Calculatetheproductsandfindtheirsum.Thisistheanswertothe multiplication16 × 19.

1 Drawmultiplicationdiagramsfortheproductsbelowandsolvethem.

3E Individual APPLYYOURLEARNING

1 Completeeachmultiplicationbywritingthemissingnumbers.

2 a Drawamultiplicationdiagramfor21 × 4.

b Shadeandlabelyourmultiplicationdiagramtoshowthenumberssplitintotens andones.Writetheproductforeachchunk.

c Addthetwoproductstocalculate21 × 4.

3 Drawamultiplicationdiagramtoshoweachproduct.Shadeandlabeltheproduct foreachchunk.Workouttheanswers.

32 × 12 a

21 × 43 b

38 × 41 c

23 × 92 d

4 Useamultiplicationdiagramtocalculate34 × 26.

3F

Multiplicationbya single-digitnumber

Themultiplication algorithm isaquickwaytoshowwhatwedidwiththe multiplicationdiagrams.Analgorithmislikearecipethatgivesyoustepstofollow.

Wecanuseamultiplicationdiagramtoseethat26 × 3is78.

Orwecanusethemultiplicationalgorithm.Tocalculate26 × 3,setoutthenumbers accordingtotheplacevalueoftheirdigits.

Tens Ones 21 6 × 3 8

Tens Ones 21 6 × 3 7 8

Wegetthisanswer: 26 × 3 = 78

First,weworkwiththeones.

Wesay3 × 6is18. Weknow18is1tenand8ones,sowewrite 8intheonescolumnandcarrythe1tothe tenscolumn.

Nowwegotothetenscolumn.

Wesay3times2is6.Addthecarried1tothe 6togive7.Write7inthetenscolumn.

Whatweareactuallydoingismultiplying3by 2tens.Thenweaddthe1tencarriedbefore. Thatiswhyweput7inthetenscolumn.

Wecandothesamekindofmultiplicationwitha3-digitnumber,likeinthefollowing example.

Example11

Multiply103by6usingthemultiplicationalgorithm. Solution 101 3 × 6 618

Startwiththeones. Say‘6times3is18’.18is8onesplus1ten. Write8intheonescolumnandcarrythe1ten.

Multiply6by0,whichis0,thenaddthecarried1. Write1inthetenscolumn.

Multiply6by1.Write6inthehundredscolumn. Theanswerto103 × 6is618.

Itisimportanttocheckthereasonablenessoftheanswerwhenmultiplying.Rounding isusefulforcheckingthereasonablenessofacalculationthroughprovidingan estimation oftheanswer.

Ifweweremultiplying253by3,wewouldexpecttheanswertobecloseto 250 × 3 = 750.Areasonableestimateusesnumbersclosetotheoriginalnumbers. Dependingonthesizeofthenumber,wecanroundtothenearest10, 100, 1000and soon.

21 53 × 3 759 roundtothenearestten250 × 3 750

Theestimateshowstheanswerisreasonable.

3F Wholeclass LEARNINGTOGETHER

1 Drawamultiplicationdiagramontheboardforeachofthese,thenshadeand labeltheproductineachchunk.Discusshoweachdiagramconnectstothe multiplicationalgorithm.

2 Theseproductscanbesolvedusingthemultiplicationalgorithm.Work througheachmultiplicationasaclass.Whichonesdidyoudowithoutany carrying?Whathappenedwhenyouhadazero?

3 Eachpersoninaclassof26studentseats4slicesofbreadeveryday.How manyslicesofbreaddoesthewholeclasseatinoneday?

a Calculatethisusingthemultiplicationalgorithm.

b Calculatethismentally.

c Discussthedifferentmentalstrategiesused.

4 Calculateeachoftheseusingthemultiplicationalgorithm.

a Howmanylegsdo3947chickenshave?

b Howmanyverticesdo49643triangleshave?

c Howmanylegsdo78994spidershave?

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d Howmanyverticesdo593574hexagonshave?

e Howmanylegsdo233995antshave?

3F Individual APPLYYOURLEARNING

1 Usethemultiplicationalgorithmtocalculateeachproduct.Thesehaveno carrying.

a 13 × 3

b 21 × 4

c 31 × 3

Thesecarryfromtheonesintothetens.

d 23 × 4

e 12 × 6

f 34 × 3

Thesecarryfromthetensintothehundreds.

g 41 × 9

h 72 × 4

i 83 × 3

Thesecarryintheones,tensandhundreds.

j 68 × 7

k 78 × 6

l 84 × 9

Thesemultiplybyamultipleof10.

× 6 m

× 4 n

× 8 o

Theseinvolvenumbersinthehundredsandthousands. 1201 × 4 p

× 3 q

× 7 r

2 Gobstopperscomeinpacketsof7.Howmanygobstoppersaretherein53 packets?

3 Agroupof8boyscountedtheirtoycars.Eachboyhad32cars.Howmany carswerethereintotal?

4 Duringtheholidays,17friendssaw9movieseach.Howmanymovietickets didtheybuyaltogether?

5 Helen’sfrontyardhas39rowsof8concretepavers.Howmanyconcrete paversarethereintotal?

6 Tomisbuyingpaintforamural.Eachtinofpaintcosts $22.Heneedstobuy 9tins?HowmuchdoesTomneedtospend?

3G Multiplicationbya2-digit number

Thestandardalgorithmfor multiplyingbyanumberwithtwo ormoredigitsisknownaslong multiplication.

Ontherightisamultiplication diagramfortheproduct14 × 27.

Thesumoftheproductsinthechunksis:

Longmultiplicationisaquickerwayofdoingthesamething.

First,setoutthenumberssothedigitslineup accordingtotheirplacevalue.

Startwiththeones.Multiplytheonesdigitin 27bytheonesdigitin14.

Say‘7times4is28’.Write8intheones columnandcarrythe2intothetenscolumn.

Nowworkwiththetens.Multiply7(theones digitin27)by1(thetensdigitin14).

Say‘7times1is7’,meaning7times1tenis7 tens.

Addthe2youcarriedfrombefore,making9 tens.

Write9inthetenscolumn.

Calculate123 × 45.

Nowmultiply2(thetensdigitin27)by4(the onesdigitin14).Thiswillgiveacertainnumber oftens.Sowrite0intheonescolumnnow.

Say‘2times4is8’,meaning2tenstimes4is 8tens.

Write8inthetenscolumn.

Next,multiply2(thetensdigitin27)by1(the tensdigitin14).

Say‘2times1is2’,meaning2tenstimes1ten is2hundreds.

Write2inthehundredscolumn.

Thefinalstepistoadd280to98. Theproductof14 × 27is378.

3G Wholeclass LEARNINGTOGETHER

1 Drawamultiplicationdiagramforeachproduct.Thendiscusshowthe diagramrelatestothelongmultiplicationalgorithmforeachproduct. 12 × 12 a 21 × 39 b

2 Theseproductscanbecalculatedusingthelongmultiplicationalgorithm. Workthrougheachoneasaclass.

3 Calculatethetotalnumberofeachitemifeachpersoninyourclasseats: 14biscuits a 28applepieces b 143sultanas c

3G Individual APPLYYOURLEARNING

1 Usethelongmultiplicationalgorithmtocalculatetheseproducts.

2 Twelvechildreneachown18T-shirts.HowmanyT-shirtsarethereintotal?

3 Sixteenstudentsearned $14eachovertheholidays.Howmuchdidtheyearn intotal?

4 Thereare27rowsofgumtreesinaplantation.Ifthereare16treesineachrow, howmanytreesarethereintotal?

5 Calculatetheseproducts. 235 × 17 a

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× 34 d

6 Peterrides118kilometreseachday.Howfardoesheridein: 1week? a 28days? b 13weeks? c

7 Rhiannonearns $36perhour.Howmuchdoessheearnifsheworks: 3hours? a 17hours? b 24hours? c 8shifts,each7hourslong? d 25shifts,each9hourslong? e

8 Liammultipliedthesumbelow.Findthe2mistakesthatLiammade.Whatisthe correctanswer? 85 3 91 3 × 65

9 Calculate:

10 Dannymeasuredhisheartrateas127beatsperminute.Ifhisheartbeatsatthe samerate,howmanytimeswillitbeatin: 4minutes? a 72minutes? b 923minutes? c 1hour? d 5hours? e 7hours23minutes? f

11 Pipistryingtomultiply48 × 21.Shethinksheransweris608.Useyourknowledge ofestimatingtocheckifsheisright.Whatisthecorrectanswer?

Writetheproductforeacharray.

2 Drawallthepossiblerectangulararraysfor36.Writetheproductforeacharray. 3 Mentallymultiplyeachnumberby6.

Mentallymultiplyeachnumberby20.

5 Thereare9chocolatebiscuitsineachpacketofCrunchyBiscuits.Mentally calculatehowmanybiscuitsareinthepacketsbelowandexplainyourstrategy. 14packets a 32packets b 144packets c

6 Rohannoticedthatwhenhemultiplied4 × 3andadded3,itwasthesameas multiplying5 × 3.UseRohan’sstrategytosolvethese.

a If5 × 3is15,whatis6 × 3?

b If8 × 4is32,whatis9 × 4?

c If6 × 8is48,whatis7 × 8?

d If8 × 8is64,whatis9 × 8?

e If12 × 8is96,whatis12 × 9?

f If11 × 11is121,whatis11 × 12?

7 Listthefirstfivemultiplesof:

a 6

b 4 c 7

d 25

e 50

8 Drawamultiplicationdiagramtoillustrateeachproduct.Then,shadeandlabelthe productsineachchunk.

a 19 × 5

b 15 × 11

9 Useamultiplicationdiagramtocalculate27 × 43.

10 Calculatetheseproductsusinganalgorithm.

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11 Lastyear,the23studentsinSteven’sclassread48bookseach.Howmanybooks didtheyreadintotal?

12 Kailani’sbackyardhas72rowsof198concretepavers.Whatisthetotalnumberof concretepavers?

13 Sarahhas $300tobuynotebooksforherclass.Eachnotebookcosts $9.Sheneeds tobuy32notebooks.

a WhatstrategycouldSarahusetoestimateifshehasenoughmoney?

b HowmuchwillSarahspendintotal?

14 Michaelisorganisingacharityrun.Hewantstoraise $1300.Eachparticipantpays aregistrationfeeof $25.Thereare48participants.

a Whatisthetotalheraised?Whydoyouthinkyouranswerisreasonable?

b HowmanymoreparticipantsdoesMichaelneedtoreach $1300?

3I Challenge–Ready,set,explore!

Findingmultiples

Itiseasytogenerateallthemultiplesofanumber.Youjustsayitsmultiplicationtable asfarasyouknowitandkeepaddingthenumberon,overandoveragain.

Forexample,nobodyreallyexpectsanyonetoknowtheir23timestable,butthefirst twomultiplesof23are23and46,andafterthatitisn’ttoodifficulttoget69and92as thenexttwo.Fromthenonweget115, 138, 161, 184, 207and230.Weknowthat 23 × 10 = 230,soalltheothersonthelistmustberighttoo.

Wecanseethat253mustalsobeamultipleof23–becauseitisjust23morethan 230–andifweadd23moreweget276.

Whataboutanumberlike11638?Wecouldworkitouteasilywithacalculator,of course.Justworkout11638 ÷ 23andtheanswerisexactly506.Couldwedoit withoutacalculator?

Weknowfromthelistabovethat115isamultipleof23.Itmakessensethat11500 mustalsobe.Now,11638 11500 = 138and138isinourlistofmultiplesof23.So 11638isalsoamultipleof23.

Wecanusethissortofthinkingtohelpanswerquestionsthatcalculatorscannot answerdirectly.Forexample,whatisthelargest4-digitmultipleof79?

Onewayofdoingthisistostartat7900.Thisis79 × 100.Keepadding79untilweget a5-digitnumber.Thesequencestarts7900, 7979, 8058, 8137, 8216, butwecansee thatthiswilltakealongtime.

Aquickerwayistofindoutif9999–whichisthelargestpossible4-digit number–isamultipleof79.Whenwedivide9999by79,theanswerwegetis 126 5696203.So9999isnotamultipleof79becausethebigstringofnumbersafter thedecimalpointtellsusthereisaremainderwhenwedivide.

Thecalculation9999 ÷ 79 = 126.5696203tellsusthat79goesinto9999atleast 126timesbutnotquite127times.Sothenumberwewantis126 × 79 = 9954(and not127 × 79 = 10033,whichhasfivedigits).

Wecouldusethistofindthelargestoddmultipleof79withfourdigits (9954 79 = 9875)orthesmallestevenmultipleof79withfivedigits (10033 + 79 = 10112).Wecandoallthiswithaminimumamountofguessingand checking.

Challengequestions

Useacalculator,buttrytobeefficient.Keeptrackoftheresultsofyoursearchsoyou don’tkeeptryingthesamenumbersoverandoveragain.

1 Findthesmallest5-digitmultipleof133.

2 Findthelargesteven5-digitmultipleof89.

3 Makealistofallthemultiplesof58between16500and16800.

4 Findthesmallestnumberyoucanthatisamultipleof73andstartsandends with8.

5 Findthesmallestmultipleof167whosedigitsadduptomorethan25.

Usefulskillsforthistopic

•

Vocabulary

Solvetheproblem

720studentsaregoingonanexcursion,andeachbuscanhold30students.

1 Howmanybusesareneededtotakeallthestudentsontheexcursion?

2 Ifthe720studentscouldfitevenlyon18buses,howmanystudentswouldeach bushold?

3 Whatstrategydidyouusetosolvethisproblem?Shareyourstrategieswitha partner.

Division Division Division Division DivisionDivision Division Division Division Division Division Division Division Division Division DivisionDivision Division Division Division Division Division Division DivisionDivision Division Division Division Division Division Division DivisionDivision Division Division Division Division Division Division DivisionDivision Division Division Division Division Division Division Division DivisionDivision Division Division Division Division Division Division Division DivisionDivision Division Division Division Division Division Division Division DivisionDivision Division Division Division Division Division Division Division DivisionDivision Division Division Division Division

Wheneverwesharesomethingequallywithotherpeople,weareusing division. Forexample,whenweshareapacketoflolliesequallyamongsomefriends,we usedivision.

Thereare24lolliesinapacket.Iffourfriendswanttosharethelollies,wecan givethem6each,sothateveryonehasthesameamountandtherearenomore toshareout.

Divisionisaboutsplittingorsharingquantitiesequally.

4A Factors

Factors

Sixstarscanbearrangedinrectangulararraysindifferentways.

threerows oftwo tworowsof three

sixrowsof one onerowof six

Wedescribetherectangulararraysusingmultiplication.Theproductofthesepairsof numbersis6.

Wesaythatthenumbers1,2,3and6are factors of6.Thenumber2isafactorof6 becausewecanfindanumbertomultiplyby2togive6.

Arrays

Everywholenumberhasatleasttwofactors. Let’slookathowarrayscanbeusedtofindthe factorsofanumber.

Hereare24eggs.

Canwearrange24eggsintoarectangulararray?In howmanydifferentwayscanthisbedone?When wefindallofthepossiblearraysfor24,wegetthe factorsof24.

Herearethepossiblearraysfor24.

Ofcourse,writing1 × 24 = 24isthesameaswriting24 × 1 = 24,and3 × 8isthe sameaswriting8 × 3,andsoon.

Fromnowonwewillwrite1 × 24whenwemeaneither1 × 24or24 × 1.

It’sagoodideatopairthefactorstoremindyouwhichotherfactoryoumultiplybyto gettheoriginalnumber.Thefactorpairsfor24canbeshowninthediagrambelow.

Thenumbers1,2,3,4,6,8,12and24arethefactorsof24. Thenumber24isa multiple ofeachofitsfactors.

Example1

a Drawthepossiblerectangulararraysfor20andlabelthemwiththe correspondingmultiplication.

b Listthepairsfactorsof20,pairingthemtomakesurethatyouhavethemall. Solution

b Thefactorpairsof20are 1, 2, 4, 5, 10 and 20.

Multiplicationtable

Wecanseefactorsonthemultiplicationtable.Tofindthefactorsof36,welookfor36 withinthemultiplicationtable.Wefollowthecolumnupandtherowtotheleftto findnumbersthathave36astheirproduct.

Weknowtherearesomerepeatedfacts.Wealsoknowthat1 × 36is36and2 × 18is 36,thoughthesefactorsarenotonthemultiplicationtable.

Factorsappearonthismultiplicationtableifbothfactorsarelessthan13. Thenumbersthatmultiplytogive36are: 3 × 12 = 364 × 9 = 366 × 6 = 36and1 × 36 = 362 × 18 = 36

Again,pairingthefactorshelpsusmakesurethatwehavethemall.Thefactorpairs of36are: 123469121836

Notethat6 × 6is36,so6ispairedwithitself.

4th

Primeandcompositenumbers

A primenumber isanumberwithonlytwofactors:itselfand1. Numbersthatarenotprimearecalled compositenumbers.Thenumber1isneither primenorcomposite.

Whenwewriteanumberasaproductoftwonumbers,thosetwo numbersarefactorsofthefirstnumber.

Anumberwithonlytwofactors–itselfand1–isaprimenumber. Acompositenumberhasmorethantwofactors. Thenumber1isneitherprimenorcomposite. 4A

Wholeclass LEARNINGTOGETHER

1 Drawrectangulararraysforthenumbers1to30.Eachpersoncouldhave responsibilityforadifferentnumber.Somenumberswillhavemorethan onepossiblearray.Usecountersorblocksorsimilartohelpyoufindthe possibilities.

Writestatementsaboutyourarraysusingmathematicallanguage,symbols andnumbers.Listthefactorsforeachnumber.

Discussthedifferentarrangementsasaclass.Whatdoyounotice?Which numbershaveonlyonearray?Whatistheconnectionbetweenthenumber ofarraysanumberhasanditsfactors?

2 Workwithapartnertofindallfactorsofeachnumber,pairingthefactorsto makesureyouhavethemall.

4A Individual APPLYYOURLEARNING

1 a Whichfactorispairedwith3togive12?

b Whichfactorispairedwith5togive75?

c Whichfactorispairedwith10togive900?

d Whichfactorispairedwith8togive96?

2 Completethefollowingstatements.

a 20 ÷ 5 = thereforefactorsof20are and b 32 ÷ 8 = thereforefactorsof32are and

c 45 ÷ 9 = thereforefactorsof45are and

3 a Listthefactorsof30.

b Listthefactorsof36.

c Drawfourbagsinyourbookasshown.Sorteachnumberfrom1to36 intothecorrectbag.

4 Statewhichnumberisnotafactorof72:12,8,36,5,24

5 1, 6and9arefactorsofanumberbetween50and60.Whatmightthe numberbe?

6 Howmanyrectangulararrayscanbedrawnforeachnumber?

7 a 17 b 23 c 29 d

Writethefactorsofeachnumber. e Whatdoyounotice? f

Whatgroupdothesenumbersbelongto?

7 Asnumbersgetlarger,theyalwayshavemorefactors.TrueorFalse?Explain youranswer.

4B Makingaconnectionbetween multiplicationanddivision

Ifwehaveanumberofballoonstoshareequally,wecandoitintwoways.

1Howmanygroups?

Ifwehave24balloonsandwegive8balloonstoeachchild,howmanychildren arethere?

Ifwesplit24balloonsintogroupsof8,threechildrenget8balloonseach. Wecandrawanarraytoshowthis.

3lotsof8make24,orwecanwrite3 × 8 = 24.

Wesaythat‘24dividedby8is3’becausewecanbreakup24into3equalgroups of8.Wewritethedivisionlikethis:

24 ÷ 8 = 3

Thesymbol ÷ isusedfordivisionandmeans‘dividedby’or‘howmanygroups’.

2Howmanyineachgroup?

Ifweshare24balloonsamong8children,howmanydoeseachchildreceive?

Wewanttomake8equalgroups.Wedothisbyhandingoutoneballoontoeach child.Thisuses8balloons.Thenwedothesameagain.Wecandothis3times,so eachchildgets3balloons.

Wecanseethisfromthisarray.

Wecanwritethisindifferentways.

8lotsof3make248 × 3 = 2424 ÷ 8 = 3

Sodividing24by8isthesameasasking:

‘WhichnumberdoImultiply8bytoget24?’

Somespecialnames

Inthisdivision,24isthe dividend,8isthe divisor and3isthe quotient. 24 ÷ 8 = 3

Thedivisoristhenumberyoudivideby.

Inverseoperations

Wesaythatdivisionisthe inverseoperation tomultiplication.Forexample,takingoff yourshoesistheinverseofputtingthemon.Mathematically,wemeandivisionundoes multiplication.

Assoonasyouknowamultiplication,youimmediatelygettwodivisionfacts.

Let’slookatthemultiplicationtable.

Wecanseetwowaysofmultiplyingtoget54: 9 × 6 = 54and6 × 9 = 54

Wecanreversethemultiplicationstofind: 54 ÷ 6 = 9and54 ÷ 9 = 6

Wecanusethemultiplicationtableinreversetodocalculationsinvolvingdivision.

Example2

Completethesentencebyfillinginthegaps.

If2 × 3 = 6,then6 ÷ 3 = and6 ÷ 2 = Solution

If2 × 3 = 6,then6 ÷ 3 = 2and6 ÷ 2 = 3. Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember

Divisioncanbeconsideredassplittingintoequalgroupsorassharing.

Divisionistheinverseoperationtomultiplication.

Wholeclass LEARNINGTOGETHER

1 Makeonearrayforeachnumberbelowusingcountersorbottletops.Then,write thecorrespondingmultiplicationanddivisionstatements.Thefirstonehasbeen doneforyou.

2 Halveeachnumberbymentallydividingitby2.(Checkyouranswerbydoubling theresult.)

3 Mentallydivideeachnumberby4byhalvingandhalvingagain.(Checkyouranswer bydoublingtheresultanddoublingagain.)

4 Numbersthataremultiplesof10haveazeroattheend.Divideeachnumberby 10bymentally‘choppingoff’theendzero.Sayyouranswertothepersonnext toyou.

a 50 b

5 Herearethefactorsof18. 1236918

Nowcompleteeachstatement.

6 Writethefactorsof36,thencompleteeachstatement.

= 36 a

4B

UNCORRECTED

Individual APPLYYOURLEARNING

1 Completeeachstatement.Thefirstonehasbeendoneforyou.

a If3 × 5 = 15,then15 ÷ 3 = 5and15 ÷ 5 = 3.

b If2 × 4 = 8,then8 ÷ 4 = and8 ÷ 2 = .

c If6 × 3 = 18,then18 ÷ 6 = and18 ÷ 3 = .

d If7 × 8 = 56,then56 ÷ 8 = and56 ÷ 7 =

e If313 × 279 = 87327,then87327 ÷ 279 = and87327 ÷ 313 =

2 Herearethefactorsof12. 1234612

Nowcompleteeachstatement.

÷ = 6 j

Uncorrected 4th

Checkwhethereachdivisioncalculationiscorrectbyusingyourknowledgeofthe multiplicationtable.Thefirsttwohavebeendoneforyou.

a Does121 ÷ 11 = 11?Yes,because11 × 11 = 121.

b Does98 ÷ 8 = 12?No,because12 × 8 = 96.

24 ÷ 8 = 3 c

Checkwhethereachdivisioncalculationiscorrectbydoingthecorresponding multiplication.Thefirsttwohavebeendoneforyou.

a Does504 ÷ 63 = 7? No,504 ÷ 63doesnotequal7because63 × 7 = 441.

62 3 × 7 441

b Does243 ÷ 9 = 27? Yes,243 ÷ 9 = 27because27 × 9 = 243.

26 7 × 9 243

Does72 ÷ 3 = 24? c

Does161 ÷ 7 = 23? e

Does181 ÷ 16 = 12? g

Does343 ÷ 49 = 7? i

Does478 ÷ 239 = 2? d

Does612 ÷ 9 = 68? f

Does200 ÷ 23 = 9? h

Does1836 ÷ 68 = 27? j

Divisibilityby10and5

Wehavealreadyseenthatanumberthatendsin0isamultipleof10.

Forexample:

2 × 10 = 20and20 ÷ 10 = 2

127 × 10 = 1270and1270 ÷ 10 = 127

Numbersthatendin0canbedividedexactlyby10.Thenumber10dividesthese numberswithzeroremainder.

Anumberis divisible by10ifitendsin0.

Uncorrected 4th

Anumberthatendsin5or0isamultipleof5.

2 × 5 = 10and10 ÷ 5 = 2

125 × 5 = 625and625 ÷ 5 = 125

Allnumbersthatendin0or5aredivisibleby5.Thenumber5dividesthesenumbers with0remainder.

Anumberisdivisibleby5ifitendsin5or0.

Divisibilityby2,4and8

Evennumbersaremultiplesof2.Evennumbersendin0,2,4,6or8.Becausetheyare multiplesof2,evennumbersarealsodivisibleby2.

13 × 2 = 26and26 ÷ 2 = 13

148 × 2 = 296and296 ÷ 2 = 148

Anumberisdivisibleby2ifitendsin0,2,4,6or8.

Ifyoudrawanarraytoshow100,itshowsthat100isdivisibleby4.Ifyouthendraw anotherarraytoshow24,youcanseethat124isdivisibleby4. 100 124 24

Anynumberofhundredsisdivisibleby4,because100isdivisibleby4.

So,youonlyneedtothinkaboutthelasttwodigitstofindifanumberisdivisibleby4.

Anumberisdivisibleby4ifitslasttwodigitsmakeanumberthatisdivisibleby4.

Divide1000by8

8 125 ) 10 20 40

Anynumberofthousandsisdivisibleby8because1000isdivisibleby8.So,youonly needtothinkaboutthelastthreedigitstofindifanumberisdivisibleby8.Usethe divisionalgorithmexplainedinthenextsectiontodothis,ifnecessary.

Anumberisdivisibleby8ifthelastthreedigitsmakeanumberdivisibleby8.

Divisibilityby3,6and9

Lookatthefirstfewmultiplesof3after9.Whathappensifyouaddthedigitsineach ofthesenumbers?

Number Sumofitsdigits

Theanswersarealldivisibleby3.

Ifthesumofitsdigitsisdivisibleby3,thenumberisdivisibleby3. Whyisthisruletrue?Thekeytotheruleisthat:

100 = 99 + 1and10 = 9 + 1

Anymultipleof3isdivisibleby3,so9and99aredivisibleby3.

Lookatthenumber132.

Weknowthat99isdivisibleby3and9isdivisibleby3. If1 + 3 + 2isdivisibleby3then132isdivisibleby3 1 + 3 + 2 = 6,and6isdivisible by3,so132isdivisibleby3.

Anumberisdivisibleby3ifthesumofitsdigitsisdivisibleby3.

Example3

Is189divisibleby3?Usethesetwomethodstofindout.

a Divide189by3.

b Usethedivisibilitytest.

Solution

a 3 63 ) 189 r0

b Addthedigits.

So189isdivisibleby3.

1 + 8 + 9 = 18 18isdivisibleby3. So189isdivisibleby3.

Ifanumberisdivisibleby2and3thenitmustbedivisibleby6,since2 × 3 = 6.

Anumberisdivisibleby6ifitisevenanddivisibleby3.

Example4

Testthesenumbersfordivisibilityby6.

Solution

a 324isanevennumber,soitisdivisibleby2.Thesumofitsdigitsis 3 + 2 + 4 = 9,and9isdivisibleby3.So324isdivisibleby2and3,andthis tellsusitisdivisibleby6.

b 106isanevennumber,soitisdivisibleby2.Thesumofitsdigitsis 1 + 0 + 6 = 7,so106isnotdivisibleby3.106isnotdivisibleby6.

c 163isanoddnumberandnotdivisibleby2.So163isnotdivisibleby6.

Thetestfordivisibilityby9issimilartothetestfordivisibilityby3.Weaddthedigits togetherandcheckifthesumisdivisibleby9.

Number

Sumofitsdigits

573 5 + 7 + 3 = 15

201006 2 + 0 + 1 + 0 + 0 + 6 = 9

Thesumofthedigitsfor573isnotdivisibleby9,so573isnotdivisibleby9.

Thesumofthedigitsfor201006isdivisibleby9,so201006isdivisibleby9.

Anumberisdivisibleby9ifthesumofitsdigitsisdivisibleby9.

Divisibilityby7

Thereisnoeasytestfordivisibilityby7.Soifwewanttochecktoseewhethera numberisdivisibleby7,wedoashortdivision,whichisexplainedinthenextsection.

Divisibilitytofindprimenumbers

Wecanusedivisibilityteststohelpusfindprimenumbers.

Example5

Findtheprimefactorisationof999.

Solution

9isafactor,as999 = 9 × 111

9 = 3 × 3

111isdivisibleby3because1 + 1 + 1 = 3

Divide111by3: 3 37 ) 11 21

111 = 3 × 37and37isaprimenumber.

Sotheprimefactorisationis:999 = 3 × 3 × 3 × 37

Divisibilitybyacompositenumber

Ifweknowthatanumberisdivisiblebyacompositenumber,thenitfollowsthatthe numberisalsodivisiblebytheprimefactorsofthatcompositenumber.

Wecantestthisideausingthenumber234andoneofitsfactors,6.

Weknowthat6isacompositenumber,becauseithasfactors1,2,3and6.

First,weestablishthat6isafactorof234,usingdivision.

6 39 ) 23 54

Nowwetesttoseeif2and3arefactorsof234usingdivision.

2 117 ) 23 143 78 ) 23 24

Thenumber234isdivisibleby6andby2and3.Theconnectionisthat2 × 3 = 6.

So,anumberthatisdivisiblebyacompositenumberisalsodivisiblebythefactorsof thatcompositenumber.Sometimesthesefactorsareprimefactorsasinthecaseof2 and3above.

Itfollowsthatifanumberisdivisiblebytwodifferentprimenumbers,itisalsodivisible bytheproductofthosetwonumbers.Forexample,100isdivisibleby2and5.Itis alsodivisiblebytheproductof2and5,whichis10.

Wholeclass LEARNINGTOGETHER

1 Asaclass,writestatementsdemonstratingthefollowing.

Anumberthatisdivisiblebytwoormoreprimefactorsisalsodivisibleby theirproduct.

Forexample,‘60isdivisibleby2and5,soitisdivisibleby10’and‘3and13 areprimefactorsof156,so39isalsoafactorof156’.

2 Copythe4-digitnumber4 2 andfillintheblankstomakeanumber thatisdivisibleby:

4C Individual APPLYYOURLEARNING

1 Usetheshortdivisionalgorithmtocalculatethese.Usemultiplicationtocheck youranswers.

÷ 3 a

2 Usethedivisibilitytesttoworkoutwhetherthesenumbersaredivisibleby3. 4989111313891

3 Testthesenumbersfordivisibilityby5. 25405561820025387

4 Howmanydifferentdigitscanyouputafter46tomakea3-digitnumber divisibleby4?

5 Foreachofthenumbersbelow,copyandcompletethetwostatementsabout factorsanddivisibility.Thefirstnumberhasbeendoneforyou. Thenumber isdivisiblebytheprimenumbers and ,soitisalso divisibleby . Also, and areprimefactorsof ,so isalsoafactorof

63Thenumber 63 isdivisiblebytheprimenumbers 7 and 3,soitisalso divisibleby 21.Also, 7 and 3 are primefactorsof 63,so 21 isalsoa factorof 63.

Theshortdivisionalgorithm

Thedivision algorithm isunusualbecauseitstartsontheleftofthenumberandshares outthebiggestpiecesfirst.

Let’sdivide84by4.Youprobablyknowtheanswer,orcanworkitoutusingother methods,butitiseasiertouseasimpleexamplethefirsttimeweusethedivision algorithm.

Todivide84by4wetrytomake4equalgroups.Webeginwiththetens.Thereare 8tens.Ifweshare8tensamong4people,eachperson’sshareis2tens.so,8tens dividedby4is2tens.

Nextwesharetheones.Thereare4ones.Ifweshare4onesamong4people,each person’sshareis1one.

Ifweshare84among4people,eachpersongets2tensand1one,whichisthe sameas21.

84 ÷ 4 = 2tens + 1one = 21

Werecordthisusingthenotation4) 84 = 21.

Ifwewanttodivide336by7,wecanuseshortdivision.

7 48 ) 33 56

7into3hundredswillnotgo.Therearenot enoughhundreds.

7goesinto33tens4timeswith5tens leftover.

Write4tensintheanswerlineandcarrythe5. Nowdivide7into56:56 = 7 × 8.

Write8onesintheanswerline.

So336 ÷ 7 = 48.

Example6

Calculate708 ÷ 7andcheckyourworkbydoingthemultiplication.

Solution

1 Usetheshortdivisionalgorithmtocalculate:

2 Usetheshortdivisionalgorithmtocheckwhichofthesenumbershas3asafactor. 122336817399910031008

3 Usetheshortdivisionalgorithmtocheckwhichofthesenumbershas4asafactor. 24342849147100167396

4 Doyouneedtousetheshortdivisionalgorithmtocheckwhichofthesenumbers has5asafactor?Explainashortcutyoucoulduse,thenwritedownthelistof numberswith5asafactor.

243449147100168395672

5 Mannyworksinafruitshopmakingupbagsof9bananas.Writethenumberof bagsthatcanbemadeupandtheremainderoutof:

99bananas a 127bananas b

593bananas c 1026bananas d

6 Footballsaresoldinboxesof6.Writethenumberofboxesandtheremainderfor:

66footballs a 302footballs b

888footballs c 1000footballs d

7 Hannahstoresherbeadsinsmallpacketsof9.Howmanysmallpacketswill Hannahhaveifshehas:

2007beads? a 585beads? b

5283beads? c 18324beads? d

Dothemultiplicationforeachoftheabovetomakesureyouarecorrect. e

Reviewquestions–Demonstrateyourmastery

1 Findthefactorsofeachnumber.Pairthefactorstomakesureyouhavethemall.

2 Listthefactorsof50. a Listthefactorsof54. b

3a Whichofthesenumbersarenotfactorsof24? 234567

b Whichofthesenumbersarenotfactorsof20? 123456

c Whichofthesenumbersarenotfactorsof48? 67891011

4 Completeeachstatement.

a If9 × 8 = 72,then72 ÷ 9 = and72 ÷ 8 = .

b If12 × 8 = 96,then96 ÷ 12 = and96 ÷ 8 =

c If87 × 93 = 8091,then8091 ÷ 87 = and8091 ÷ 93 = .

4th

5 Completethesestatements.

78 ÷ 3 = 26,so3 × 26 = a

208 ÷ 16 = 13,so13 × 16 = b

472 ÷ 8 = 59,so59 × 8 = c 15 × 32 = 480,so480 ÷ 15 = d

84 × 31 = 2604,so2604 ÷ 84 = e

52 × 23 = 1196,so1196 ÷ 23 = . f

6 Flowerscomeincontainersthatholdexactlythefollowingamounts.

Whichcontainerscouldbeusedfor: bunchesof6? a bunchesof4? b bunchesof8? c

7 Checkthateachdivisioncalculationiscorrectbydoingthecorresponding multiplication.

÷ 9 = 13 a

8 Divide1890byeachnumber.

9 Divide2304byeachnumber.

10 Useadivisionalgorithmtocheckwhichofthesenumbershas7asafactor. 1057431771196026174

11 Copythisnumbergridonto1-centimetregridpaper.

Colourthegridusingthiscode.

• Colourthenumbersdivisible by6green.

• Colourthenumbersdivisible by7red.

• Leavethenumbersdivisible by5blue.

Whichwordcanyousee?

12 Calculatethesedivisionsusingadivisionalgorithm,thencheckyourworkby multiplying.

13 Thereare330schoolbagstobeplacedonhooks.Howmanyrowsarethereif thereare:

5hooksperrow? a 3hooksperrow? b 11hooksperrow? c 2hooksperrow? d

14 Thereare6048lightglobesinacarton.

a Howmanyboxesof4lightglobesisthis?

b Howmanyboxesof8lightglobesisthis?

c Howmanyboxesof7lightglobesisthis?

d Howmanyboxesof9lightglobesisthis?

4F Challenge–Ready,set,explore!

Testfordivisibilityby7

1 Totestanumbertoseeifitisdivisibleby7,doublethelastdigitandsubtractit fromtheremaining‘choppedoff’number.Iftheresultisdivisibleby7,thensois theoriginalnumber.Applythisrulerepeatedlyuntilyougetanumberyouknowis divisibleby7.Forexample:826.Double6is12.Take12fromthe‘choppedoff’82. Now82 12 = 70.Weknow70isdivisibleby7,so826isalsodivisibleby7.

a Whichofthesenumbersisdivisibleby7?102919641813682

b Fillintheblankstomakeanumberthatisdivisibleby7.34 6

c Fillintheblankstomakeanumberthatisdivisibleby7and2.34 6

d Fillintheblankstomakeanumberthatisdivisibleby7,2and3butnot4. 34 6

2 Finda10-digitnumberthatusesallthedigits0–9andisdivisibleby1,2,3,4,5,6, 7,8,9and10.

Usefulskillsforthistopic

• understandingofwholenumbersandnumberlines

• experienceinpartitioningnumbersandbreakingobjectsintoequalparts

• abilitytoworkcomfortablywithmultiplesandfactors

Vocabulary

Numerator

• Denominator • Vinculum

• Unitfraction • Properfraction • Improperfraction • Mixednumbers • Equivalentfractions • Likefractions

• Unlikefractions •

Thewordfractioncomesfromthelatinword’frango’whichmeans‘Ibreak’. Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage

Whatfractionhasbeenshaded?

Thisentiretangramisonewhole.

1 Whatfractionofthistangramhasbeenshadedgreen?

2 Whatfractionofthistangramhasbeenshadedred?

3 Whatfractionofthistangramhasbeenshadedorange?

4 Howmanycolourscanyoufindthatrepresent 1 8 ofthe entireshape?

Explainyourthinkingtoapartneranddiscusswiththeclass.

Fractions Fractions Fractions Fractions FractionsFractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions FractionsFractions Fractions Fractions Fractions Fractions Fractions Fractions FractionsFractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions FractionsFractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions FractionsFractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions FractionsFractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions FractionsFractions Fractions Fractions Fractions Fractions Fractions Fractions Fractions FractionsFractions Fractions Fractions Fractions Fractions

Wecanusefractionstoexplainarelationshipbetweenpartof anobjectandthewholeobject.Forinstance,wecanhavea wholecake,butitistoobigforonepersontoeat.

Instead,wecutoutasingleslicetoeat,whichisasmallerpiece ofthecake.

Thesliceofcakeisafractionofthewholecake.Afractionisa smallerpieceofalargerwhole.

Fractionscanalsodescribetherelationshipbetweenapartofacollectionandthe wholenumberofobjectsinthatcollection.

Hereare5puppies.

Thereare3whitepuppiesinthewholecollection, so3outofthe5puppiesarewhite.Wewritethis asafraction.

The3whitepuppiesare 3 5 ofthewhole collection.

The number of white puppies The number of puppies in

5A Namingandrepresenting fractions

Afractionmaybepartofawhole:

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Example1

Hudson’sblanketisdividedinto4equalpieces.Eachpieceisadifferentcolour.

WhatfractionofHudson’sblanketisyellow?

Solution

1partoutof4equalpartsoftheblanketisyellow.So, 1 4 ofHudson’sblanket isyellow.

Or,afractionmaybepartofacollection:

Example2

Sabinahasacollectionof19stamps.7ofherstampsarefromMalta. WhatfractionofherstampcollectionisfromMalta?

Solution

Drawacollectionof19stamps.Circle7ofthestampsandlabelthispartofthe collection‘Maltese’.

Writeafractionwith19onthebottomtorepresentthetotalnumberofstampsin Sabina’scollection,and7onthetoptorepresentthenumberofstampsthatare fromMalta.

Write:‘ 7 19 ofSabina’stotalcollectionofstampsarefromMalta’.

Somespecialnames

Afractionhasanumberontopofthelineandanumber belowtheline.Thesenumbershavespecialnames.

Thetopnumberiscalledthe numerator.Thenumberbelow iscalledthe denominator.Theeasywaytoremember whereitgoesistosay‘Dfordenominator,Dfordown’.

Wecandraw11blocksandshowthefraction 5 11 .The numeratoristhenumberofblockscircled.Thedenominatoris thetotalnumberofblocks.

Thecircledblocksare 5 11 ofthetotalcollectionofblocks.

Thelinebetweenthenumeratorandthedenominatoris calledthe vinculum

Numberlines

Fractionscanbemarkedonanumberline.

Wereadtheseas‘one-quarter’,‘two-quarters’,‘three-quarters’,andsoon.

4th sample

Example3

Whatfractionsareshownbythestarsonthenumberline?

Thefirststaris 3 5 .Thesecondstaris 9 5 . A unitfraction isafractionwherethenumeratoris1andthedenominatorisany wholenumber.So,thefractions

and 1 4819302 areunitfractions.

Example4

Showtheunitfractions

Withunitfractions,thelargerthedenominator,thesmallerthefraction.

Usingshapes

Sofarinthischapterwehavedividedpartofanumberlineintoequalpartsand markedfractionsonit.Wecanalsouseshapestoshowfractionsaspartofawhole.

Squares

Drawasquare.Nowdrawdiagonallinestojoinopposite corners.Shadeoneofthetriangles.

Whatfractionofthesquareisshaded?

1partoutofatotalof4equalpartsisshaded.

Wewrite:‘1 4 ofthesquareisshaded’.

Whatfractionisshadednow?

2partsoutofatotalof4equalpartsareshaded.

Wewrite:‘2 4 ofthesquareisshaded’.

Drawasquarewithtwodiagonallines,averticallinethrough thecentreandahorizontallinethroughthecentre.

Eachpartofthesquarehasthesamearea.Ifwecutoutthe partsofthesquareandputthemontopofeachother,they wouldallbethesamesize.

Thereare8equalparts.

Wecanshadeany3ofthe8equalpartstoshowthe fraction 3 8.Herearesomedifferentwaystodothis.

Example5

Whatfractionofeachsquareisshaded? a b

Solution

a 7partsoutofatotalof8partsareshaded,sowewrite: ‘7 8 ofthesquareisshaded’.

b 4partsoutofatotalof8partsareshaded,sowewrite: ‘4 8 ofthesquareisshaded’.

Example6

Shade 5 8 oftheareaofthissquare.

Solution

Dividetheshapeinto8partsthatarethesamesizeandshape. Shadeany5ofthe8parts.Herearesomedifferentsolutions.

Circles

Whenwebuyapizza,itisusuallyintheshapeofacircle,andcanbecutinto4,6 or8pieces.

Ifapizzaiscutinto4 equalpieces,eachpiece is 1 4

Example7

Ifapizzaiscutinto6 equalpieces,eachpiece is 1 6

Thispizzahasbeencutinto4equalpieces. 2ofthe4pieceshavenotbeeneaten.

Writetwodifferentfractionsforthepartofthepizza thatremains.

Ifapizzaiscutinto8 equalpieces,eachpiece is 1 8

Solution

2piecesoutofatotalof4piecesremain.

Wecanwrite: ‘2 4 ofthepizzahasnotbeeneaten’.

Wecanalsoseethat2outof4piecesisthesameasonehalf. Sowecanalsowrite‘1 2 ofthepizzahasnotbeeneaten’.

5A Wholeclass LEARNINGTOGETHER

1 Writethesefractionsinwords.

2 Copyandcompletethesesentences.Drawanumberlinetohelpyou.

a 5 2 ishalfwaybetween_______and_______.

b 10 5 isthesameas_______.

c 6 4 ishalfwaybetween_______and_______.

3 Writingmultiplesof 1 3 onanumberline Startwithanumberlinewiththewholenumbersmarkedonit.

Step1

Dividethepieceoflinebetween0and1into3equalpieces.Markthedividing pointsas 1 3 and 2 3 .

Step2

Continuetomarkthirdsacrossthenumberline.

Markthepointsyoucometoas 3 3, 4 3, 5 3, 6 3, 7 3,… Copythesestatements.Useyournumberlinetohelpyoufillintheblanks.

a Thenumbers 1 3, 2 3, 3 3, 4 3 …arecalledthe______________of 1 3 .

b Thenumber 3 3 isthesameas______________.

c Thenumber □ 3 isthesameas2.

d Thenumber 9 3 isthesameas________.

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4 Discussthebestwaytodrawandshadeacircletorepresentthirds.

5 a Cutanequilateraltrianglefromapieceofpaper.

b Howcouldthistrianglebecutorfoldedsothatitrepresentsquarters?

c Howcouldyoucheckthateachpieceisthesamesizeastheothers?

d Show 3 4 usingyourequilateraltriangle.

1 Writethefractionsthatmatchthesedescriptions.

Numerator2,denominator5 a Numerator4,denominator8 b

2 Foreachofthesechocolatebars,thedottedoutlinesshowwhichpieceshavebeen eaten.Theremainingchocolateisshowninbrown.Writethefractionofeach chocolatebarthatremains.

3 Foreachofthese,writeafractiontorepresentthecircledmarblesasafractionof thewholecollection.

4 Writeafractiontorepresentthecircleddiscsasafractionofthewholecollection.

5 Writetheseasfractions.

Three-quarters a One-third b

Two-thirds c Seven-quarters d

Nine-halves e 72thirds f

6 a Drawanumberlinefrom0to1.Mark 1 4, 1 2 and 3 4 onit.

b Drawanumberlinefrom0to3.Mark 1 3, 4 3, 5 3 and 6 3 onit.

c Drawanumberlinefrom0to5.Mark 3 2, 3 4 and 3 3 onit.

d Drawanumberlinefrom0to4.Mark 1 5, 10 5 , 15 5 and 9 5 onit.

7 Drawfourrectangles.Dividethemintosixthsanduseshadingtorepresenteachof thesefractions. 5 6 a 3

8 Writethefractionthatdescribeshowmuchofeachsquareisshaded.

9 Writethefractionthatdescribeshowmuchofeachcircleisshaded.

5B Equivalentfractions

Inthissection,welookat equivalentfractions:fractionsthathavethesamevalue.

Halves,quartersandeighths

Takeastripofpaper.Folditovertomake2equalpieces,thenshade1piece. 1 has been shaded. 2

Leaveyourpaperstripclosedonthehalffold.Folditagaintomake4equalparts. Unfoldyourpaperstrip.Itshouldlooksomethinglikethis:

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Youcanseethat 2 4 isshaded.Thismeansthat 2 4 isequalto 1 2 .

Foldyourpaperstripshutonthequarterfoldandfolditagain. Unfoldyourpaperstrip.Itshouldnowlooklikethis:

Youcanseethat 4 8 isshaded.Thismeansthat 4 8 isequalto 2 4 andto 1 2 .

Thefractions 1 2, 2 4 and 4 8 areequal.Wecallthemequivalentfractions.

Wecanshowequivalentfractionsonthenumberline. Startwithanumberlinewith0and1markedonit.

Dividethedistancebetween0and1into2equalpieces.Eachpieceis 1 2

Divideeachofthosepiecesinto2equalpieces.Eachpieceis 1 4 .

Nowdivideeachofthosepiecesinto2equalpieces.Eachpieceis

Wecangetanequivalentfractionifwemultiplythenumeratorandthe denominatorbythesamewholenumber.

Halves,sixthsandtwelfths

Takealongstripofpaper.Folditinhalftomake2equalparts.

Openthepaperstripandshadeonepart. 1 2 isshaded.

Closeyourpaperstriponthehalffoldagain,thenfolditinto3equalparts.Openitup. Itshouldlooksomethinglikethis.

3 6 ofthepaperstripisshaded.

Closeyourpaperstriponthesixthfoldagain,thenfoldthepaperinto2equalpieces. Openitup.Itshouldlooksomethinglikethis.

6 12 ofyourpaperstripisnowshaded.

Thefractions 1 2 , 3 6 and 6 12 areequal.Theyareequivalentfractions.

Wecangetanequivalentfractionifwedividethenumeratorandthedenominator bythesamewholenumber.

Thirdsandsixths

Wecanshowthirdsandsixthsonthenumberline. Startwithanumberlinemarked0,1and2.

Dividethedistancebetween0and1into3equalpieces,andthendividethedistance between1and2into3equalpieces.

Yournumberlineisnowmarkedinthirds.

Divideeachthirdinto2equalpieces.Eachpieceis 1 6 .

Lookingatthenumberline,wecanseethat:

Whathappensifwecut 1 6 intotwoequalpieces?Wegettwelfths.

Example9

Showthat 3 4 and 6 8 areequivalentbydrawingthemonanumberline.

Simplestform

Herearetwoequivalentfractions: 6 15 and 2 5

Checkmentallythattheyareequivalent.Whatdidyoudividethenumeratorand denominatorby?Doyouthinkthat 2 5 iseasiertoworkwiththan 6 15?Mostpeople thinkso,becausethenumbers2and5aresmallerthan6and15.

Wedividethenumeratoranddenominatorby3.

Wecannotgetafractionequivalentto 2 5 withanevensmallernumeratorand denominator,becausewecannotfindawholenumberthatdividesboth2and5 (exceptfor1,ofcourse).

Sowesay 2 5 isthesimplestformofthefraction 6 15 . Example10

Writethefraction 8 20 insimplestform.

Solution

Thelargestwholenumberwecandivide8and20byis4.

8 20 = 8 ÷ 4 20 ÷ 4 = 2 5

So 2 5 isthesimplestformofthefraction 8 20 .

1 Copyandcompletethesestatements.

2 Drawcirclesorrectanglesdivideduptorepresenttheseequivalentfractions.

3 Drawatablewiththreecolumnsontothewhiteboardasfollows:

Sorteachfractionintothecorrectcolumn.

4 a Foldasquareinto3equalsections.Openitout.Nowfolditinhalfinthe otherdirection.Usecolouranddotstoshowthat

b Isitpossibletoshowthat

5B Individual APPLYYOURLEARNING

1 Completethemissingdenominatorstomakeequivalentfractions.

2 Writethenumberthenumeratoranddenominatorweremultipliedbyto arriveateachequivalentfraction.Thefirstonehasbeendoneforyou.

3 Drawatablewiththreecolumnsintoyourworkbookandlabelasfollows:

Sorteachfractionintothecorrectcolumn.

4 Matcheachfractioningroup1withanequivalentfractioningroup2.

5 Workoutthesimplestformofthesefractions.

5C Comparingandordering fractions

Comparingfractionswiththesamedenominator

It’seasytocomparefractionsthathavethe samedenominator.Thisrectanglehasbeen dividedintoquarterstocompare 1 4 and 3 4

1 4 3 4 is

Iftwofractionshavethesamedenominator,theonewiththelarger numeratoristhelargerfraction.

Theserectangleshave beendividedinto seventhstocompare 3 7 and 5 7

Comparingunitfractions

Fractionsthathave1asthenumeratorarecalled unitfractions.Forexample, 1 2 , 1 3 , 1 4 , 1 10 , 1 100 , 1 837 , 1 1000000 , andsoon.

Aunitfractionhasanumeratorof1.

Let’scomparetwounitfractions.

2equalpiecesmake1whole.

3equalpiecesmake1whole.

1 3 issmallerthan 1 2 becauseittakes3lotsof 1 3 tomakeawhole.

Itonlytakes2lotsof 1 2 tomakeawhole.

Whenwehavetwounitfractions,theonewiththelargerdenominatoristhe smallerfraction.

Comparingfractionswithdifferentdenominators

Tocomparetwoormorefractionswithdifferentdenominators,wecanmarkthemon thenumberline.Thefractiontotherightonthenumberlineisthelargerfraction.

Example11

Orderthesefractionsfromsmallesttolargest.

5 8 , 1 2 , 4 6

Solution

Drawanumberlinemarkedwithhalves,sixthsandeighths.

Theorder,fromsmallesttolargest,is

Usingequivalentfractionstocomparefractionswithdifferent denominators

Thebestwaytocomparefractionswithdifferentdenominatorsistochangethemso theyarethesametypeoffraction,meaningtheymusthavethesamedenominator. Wecanuseourknowledgeofequivalentfractionstodothis.

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Iftwofractionshavethesamedenominator,theonewiththelarger numeratoristhelargerfraction.

Whenwehavetwounitfractions,thefractionwiththelarger denominatorissmaller.

Anumberlinecanbeusedtocomparefractionswithdifferent denominators.

Iffractionshavedifferentdenominators,wecanchangethemsothey havethesamedenominator.

Whichfractionineachpairislarger:

Canyoufindtwofractionsthataregreaterthan 3 8 andlessthan 3 4 ?

4 Writethesefractionsinorder,smallesttolargest.

3 8 , 1 8 , 3 4 , 5 8 , 4 4 a

5 AnnieandherbrotherAndrewweregivenasamesizeblockofchocolate eachasatreat.Anniehaseaten 2 3 ofherchocolatebar.Andrewhaseaten 8outofhis15pieces.Whohaseatenthemost?

6 Whichfractionislarger: 1 2 or 48 100 ? Explainyouranswer.

7 MrsTee’sclassandMrCher’sclassbothhavethesamenumberofstudents inthem. 5 6 ofthestudentsarepresentinMrsTee’sclass. 2 3 ofthestudents arepresentinMrCher’sclass.Whichclasshasmorestudentsatschool?

8 Ben,MaxandTomcompletedamathsquizatschool.Bengot 3 4 ofhisanswers correct.Maxgot 3 5 ofhisanswerscorrect.Tomgot16correctoutofthe 20questions.Whogotthehighestscoreandwhatwasit?Whogotthe lowestscoreandwhatwasit?

Properandimproperfractions

Ifthenumeratorofafractionislessthanthedenominator,wecallita properfraction Forexample, 1 3 and 4 5 areproperfractions.

Ifthenumeratorisgreaterthanorequaltothedenominator,thefractioniscalledan improperfraction.Forexample, 4 3 and 3 3 areimproperfractions.

4 isaproperfraction. a 6 5 isanimproperfraction. b 2 2 isanimproperfraction. c 89 99 isaproperfraction. d

Wholenumbersasfractions

Allwholenumberscanbewrittenasfractions.Forexample,1 = 2 2 and2 = 4 2

Ifthenumeratorandthedenominatorarethesamenumber,wegetafractionthatis equivalentto1.Forexample, 3 3 = 1and 10 10 = 1.

Ifthenumeratorisamultipleofthedenominator,wegetawholenumber.

Forexample, 6 3 = 2and 9 3 = 3.

Mixednumbers

Amixednumberisawholenumberplusafractionsmallerthan1. 1 1 2 isamixednumber.Itmeans1plus 1 2 more.

Let’sbuildupthirdstoseewhathappenswhenwegetmorethanonewhole.

Ifwedividearectangleintothreeequalpieces,eachpieceis 1 3 ofthewhole.

Five-thirdsisthesameas1 + 2 3 = 12 3 .

Six-thirdsisthesameas2.

Seven-thirdsisthesameas2 1 3

Wecanalsoshowthisonthenumberline.

Hereisanumberlinemarked0,1,2and3.

Ifwemarkthenumberlineinthirdsandlabelacrossthenumberline

Wecanre-labelthenumberlineusingwholenumbersandmixednumbers.

Writetheseimproperfractionsasmixednumbers.

1 Drawatablewiththreecolumnsontothewhiteboardandlabelasfollows:

Sortthesefractionsintooneofthreecolumns.

2 Wecanturndominoesontheirsidetogetafraction.Thisdominocanberead aseither 6 5 or 5 6

Writethefractionforeachdominobelow.Aretheyproperorimproperfractions?

3 Youwillneedaclasssetofdominoes.

a Take2dominoesfromtheclasssetandstandthemontheirends.Writethe fractionsdown.Converteachtoamixednumberifyoucan.Whichisthelarger fraction?

b Take5dominoesfromtheclasssetandstandthemontheirends.Sortthem fromsmallestfractiontolargest.

c Take10dominoesfromtheclasssetandusethemtomakefractions.Sortthem into2groups:‘largerthan 1 2 ’or‘smallerthan 1 2’.Turnthedominoupsidedown ifyougetafractionequivalentto 1 2 .

1 Writethesefractionsaswholenumbers.

2 Drawapictureusingshadedpartsofrectanglestoshoweachmixednumber.

3 Convertthesemixednumberstoimproperfractions.

4 Converttheseimproperfractionstomixednumbers.

5 Giuseppe’sschoolhadaPizzaDay.Eachclasshadsomeleftoverpizza.Write thefractionfortheamountremainingineachclassasafractionandasa mixednumber.

5E Addingandsubtractingfractions withthesamedenominator

Additionoffractionswiththesamedenominatorislikeotheradditions.

3balloons + 4balloons = 7balloons + =

3eighths + 4eighths = 7eighths

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Likewise,subtractionoffractionswiththesamedenominatorislikeothersubtractions.

7eighths 4eighths = 3eighths

Whenweaddfractionswiththesamedenominator,weaddthenumerators.

Whenwesubtractfractionswiththesamedenominator,wesubtractthe numerators.

Thisishowweshowtheadditionoftwofractionsonthenumberline. Toworkout 3 8 + 4 8

• wedividethenumberlineintoeighths

• thenweshowthetwofractionsassegmentsonthenumberline

• toaddthefractions,wemovethesecondsegmentnexttothefirstone.

Whenaddingfractionswiththesamedenominators,addthe numerators.

Whensubtractingfractionswiththesamedenominator,subtractthe numerators.

5E Wholeclass LEARNINGTOGETHER

1 Drawrectanglepicturestoshoweachadditionorsubtraction.

2 Drawnumberlinestoshowtheseadditionsorsubtractions.Writetheanswer foreach.

5E Individual APPLYYOURLEARNING

1 Drawanumberlinetocalculateeachadditionorsubtraction.

2 Calculate:

3 Arkyhas3watertanksofthesamesize.Hecheckedthewatertankseachday duringaveryrainyweekandwrotedownhowmuchwaterwascollectedeach dayasafractionofonewholetank.

a Onwhichday(s)wasmorethanonetankofwatercollected?

b Howmuchwaterwascollectedintotalfortheweek?

c HowmuchmorewaterwascollectedonThursdaythanMonday?

d Whatwasthedifferenceinwatercollectedontheweekendcomparedto duringtheweek?

4 Mikeaddsthreefractionswiththesamedenominatortoget 16 20 .Whatcouldthe threefractionsbe?

Addingandsubtractingfractions withdifferentdenominators

Additionandsubtractionoffractions withdifferentdenominatorsrequires renamingsotheybecomelike fractionsandhavethesame denominator.Thentheycaneasily beaddedorsubtracted.

Weuseourknowledgeofequivalent fractionstohelpus,forexample, weknow 2 5 isthesameas 4 10 or 6 15 or 8 20

Afractionwallisanothertoolthat canbehelpfulinfindingan equivalentfraction.

Ifweweretoadd 2 5 + 7 10,wecouldchangethe 2 5 toitsequivalentfraction 4 10 .

4th sample

5F Wholeclass LEARNINGTOGETHER

1 Usethefractionwalltohelpsolveeachadditionorsubtraction.Remember torenameoneofthefractionssobothfractionshavethesamedenominator first.

2 Drawdiagramstoprovethefollowing.

5F

Individual

APPLYYOURLEARNING

1 Useafractionwalltocalculateeachadditionorsubtraction.Rememberto renameoneofthefractionssobothfractionshavethesamedenominatorfirst.

2 TheSmithfamilybought2dozeneggsatthemarket.Theyused 2 6 oftheeggsin anomelette.Howmanyeggswereleft?

3 Thesumof3fractionsis1 7 8 .Noneofthefractionshavethesame denominator.Allofthefractionsareproperfractions.Allofthedenominatorsare factorsof8.Whatcouldthefractionsbe?

cupbutter

1tbspgoldensyrup

1 2 tspbakingsoda

2tbspboilingwater

Method

a Preheattheovento180degreesCelsius.Lineabakingtraywithbakingpaper.

b Mixtogetherflour,sugar,coconutandrolledoats.

c Meltbutterandgoldensyrup.Dissolvebakingsodaintheboilingwaterand addtobutterandgoldensyrup.Stirbuttermixtureintothedryingredients.

d Placeleveltablespoonfulsofmixtureontocoldgreasedtraysandflattenwith afork.

e Bakeforabout15minutesoruntilgolden.Leaveonthetrayfor5minutes, thenplaceonawireracktocool.

Thisrecipewillmakeapproximately6cookies.Usingthisrecipe,howcouldyou adjusttheingredientlisttomake24cookies?

5G Reviewquestions–

1 Eachchocolatebarhasbeenbrokenintoanumberofequalpieces.Writethe fractionshownbytheshadedpartofeachbar.

2 Hereare10frogs.Someofthefrogshavespots.Writeafractionforthenumberof frogsthathavespotsaspartofthewholegroupoffrogs.

3 Writeafractiontorepresentthecircledcollectionofdiscsaspartofthetotal numberofdiscs.

4 Writethefractionsforthese. seven-eighths a two-fifths b eight-tenths c five-quarters d twoandfive-sixths e sixandtwo-thirds f

Uncorrected 4th sample pages

5a Drawanumberlinefrom0to1.Mark 3 4 , 1 2 and 1 6 onit.

b Drawanumberlinefrom0to2.Mark 2 3 , 4 3 , 1 2 and 5 6 onit.

6 Thisbirdhousehasthreefloors. Therearethreekindsofbirds:green, blackandyellow.

a Whatfractionofthebirdsonthe secondfloorareblack?

b Whatfractionofthebirdsonthe firstfloorareyellow?

c Whatfractionofthebirdsonthe firstfloorareblack?

d Whatfractionofthebirdsonthe groundfloorarenotgreen?

e Whatfractionofthebirdsintheentirebirdhouseareyellow?

f Whatfractionofthebirdsintheentirebirdhousearegreen?

g Whatfractionofthebirdsintheentirebirdhouseareblack?

7 Writethefractionthatdescribeshowmuchofeachsquareisshaded.

8 Fillinthenumeratoranddenominatortomakeequivalentfractions. 1 □ = □ 6 = 12 24 a 2 □ = 5 □ b 6 □ = □ 4 = 12 16 c

9 Drawatablewiththreecolumnsinyourbookandlabelasfollows: Smallerthan 2 3 Equivalentto 2 3 Largerthan 2 3

Sorteachfractionintothecorrectcolumn.

10 Writethesefractionsinorder,smallesttolargest.

8 , 1 4 , 1 3 , 1 6 , 1 10

11 Whichfractionineachpairislarger:

2 or 6 8 ? a

3 or 3 12 ? b 1 4 or 3 16 ? c

8 or 3 4 ? d 4 5 or 9 10 ? e

12 Billate 4 6 ofhischocolatebar.Stefanate 10 12 ofhis.Whoatemoreofhis chocolatebar?

5H Challenge–Ready,set,explore!

Tiger-stripefractions

Fractionscanberepresentedbydividingarectangleintoequalpartsandshadingsome ofthepartsinadarkercolour.Ifthepartsareshadedintwoalternatingcolours,one darkandtheotherlight,wegeta‘tiger-stripepattern’.

Thepartoftherectangleshadedinthedarkercolourrepresentsafractionwecalla ‘tiger-stripefraction’.

Thesediagramsshowtiger-stripefractionsfor 3 7, 5 9, 2 4 and 3 6 .

1 Whichofthesearetiger-stripefractions? 4 9 ,

2 Listallthetiger-stripefractionsthathave9asanumerator.

3 Therearetwoequivalentfractionsinthelistinparta,butonlyoneofthemisa tiger-stripefraction.Whatarethosefractions?Makeupanotherpairofequivalent fractions,onlyoneofwhichisatiger-stripefraction.

4 Whatisthelargesttiger-stripefraction?Whatisthesmallesttiger-stripefraction? Explainyouranswer.

5 Explainwhyeverytiger-stripefractionwithanevendenominatorisequivalent to 1 2 .

Usefulskillsforthistopic

• understandingthebase-10system(hundreds,tensandones)

• understandingthebase-10systemextendstotenthsandhundredthsandbeyond

• understandingthatfractionsarepartofawholeorpartofacollection

• usingnumberlinestorepresentnumbers,includingfractions

Vocabulary

Decimalpoint

• Decimalnotation • Tenths • Hundredths • Thousandths • Comparing • Rounding

• Theword‘decimal’comesfromtheLatinword‘decem’,meaning‘ten’. Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage

Sameordifferent?

Is3.5thesameas 3 5 ?

Explainyourreasonsforagreeingordisagreeing.

Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals Decimals Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals Decimals Decimals Decimals DecimalsDecimals Decimals Decimals Decimals Decimals

Decimalsarecommonlyusedineverydaylife,forexample,inmeasurementand money.Decimalsbreakdownawholenumberintosmallerparts.

$2 75isadecimalnumbermeasuringmoney.

3.5kgisadecimalmeasuringweight.

1.75misadecimalmeasuringheightorlength.

1.5Lisadecimalmeasuringliquid.

Inthischapterweseehowdecimalnumbersarebuiltupfromwholenumbersand fractions,suchas 1 10 , 1 100 , 1 1000,andsoon.

Whenweusedecimalnumbers,weextendtheplace-valueworkwehavedoneto includetenths,hundredthsandthousandths.

6A Tenths

Tenthsbetween0and1

Thisnumberlineshows0and1.

Ifwecutthelengthbetween0and1into10equalpieces,eachpiecehasalengthof 1 10 .

Welabelthefirstpoint 1 10,thencontinuetolabelacrossthenumberline.

Wewrite 1 10 as0 1whenweusethedecimalwayofwritingnumbers.

Whenthenumberlineis cutinto tenths,welabelthefirstmarkertotherightofthe zeroas0.1.Wereadthisas‘zeropointone’.

0.10

0.1isthesameas 1 10 .

The decimalpoint sitstotherightoftheonesplaceandtellsusthattheplacesafterit arepartsofthewhole.Thefirstplacetotherightofthedecimalpointisfortenths.We write0.1inaplace-valuechartlikethis:

Onthenumberline,welabelthesecondpointtotherightofthezeroas0.2.Weread thisas‘zeropoint two’.

0.2isthesameas 2 10 .

Wewrite0.2inaplace-valuechartlikethis:

Wecontinuetolabelacrossthenumberlineintenths.Thisnumberlinestopsat1,but wecouldkeepgoingbeyond1forevermarkingintenths.

1isthesameas 10 10

Inthedecimalsystem,1isthesameas1 0.

Tenthsarepartofourbase-10numbersystem,sotheyfollowthesamerulesas wholenumbers:

10hundredsmake1thousand 10tensmake1hundred 10onesmake1ten 10tenthsmake1one

Example1

Mark0.7onanumberline.

Solution

Drawanumberlinemarkedwith0and1.Cutthelengthbetween0and1into10 equalpieces.Thelengthofeachpieceisone-tenth.Labelthenumberlinebetween 0and1intenths,goingfromlefttoright.Theseventhmarkerafter0is0.7or seven-tenths.

Numberslargerthan1

Nowwearegoingtoseewhathappenswhenwegopastthenumber1onthe numberline.

Thisnumberlineismarkedwiththewholenumbers0,1and2andthetenthsbetween 0and1.

Ifwestartat0andstepbytenths,wesay:

Whenwewritethisindecimals,itbecomes: 0.10.20.30.40.50.60.70.80.91

Noticethat 10 10 isthesameas1.

Ifwekeeptakingstepsofone-tenthonthenumberline,whatdoyouthinkwillcome after1?

Wecutthelengthbetween1and2into10equalpiecessothateachpiecehasa lengthofone-tenth.Welabelacrossthenumberline.

Ifwetakeonestepofone-tenthfrom, 10 10 wearriveat 11 10,whichisthesameas 1 + 10 10 or1 1 10 .

Thedecimalwayofwriting1 1 10 is.1.1.Thisiscalled decimalnotation.

4th

Thenumbersthatfollow1willbe: 1 1 10 1and1tenth1.1onepointone 1 2 10 1and2tenths1.2onepointtwo 1 3 10 1and3tenths1 3onepointthree 1 4 10 1and4tenths1 4onepointfour 1 5 10 1and5tenths1 5onepointfive 1 6 10 1and6tenths1.6onepointsix 1 7 10 1and7tenths1.7onepointseven 1 8 10 1and8tenths1.8onepointeight 1 9 10 1and9tenths1.9onepointnine 2 2and0tenths2.0twopointzero

Onaplace-valuechart,we write1as1.0toshowthattherearezerotenths.

Ifwetakeastepofone-tenthtotheright1ofonthenumberline,thenumberof tenthshasincreasedby1.

Wereadthenumber45.7as‘forty-fivepointseven’.

Thenumber45totheleftofthedecimalpointtellsusthatwehave4tensand5ones. The7totherighttellsusthatwehave7tenthsmore.

45.7isthesameas45 7 10 .

Thisnumberlineshowsthat45.7isbetween45and46.

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Thedecimalpointseparatesawholenumberandadecimal.

Thedigitstotheleftofthedecimalpointtellusthewholenumber part.

Thedigitinthefirstplacetotherightofthedecimalpointtellsusthe numberoftenths.

Sortthesenumbersintotwogroups:

Weplaceeachnumberonthenumberline,thendividethemintotwogroups.

Example3

Write3metresand75cmasadecimal.

Solution

3metresand75cmhasawholepart,whichis3metres,andapartofametre, whichis75centimetres.

Indecimalnotation,thedecimalpointseparatesthemetresandcentimetresand wewouldrecordthisas3.75m.

Example4

Write5dollarsand35centsasadecimal.

Solution

5dollarsand35centshasawholepart,whichis5dollars,andapartofadollar, whichis35cents.

Withmoney,thedecimalpointseparatesthedollarsandcentsandwewouldrecord thisas $5 35.

Wholeclass LEARNINGTOGETHER

1 Youwillneedahalf-metrepieceofstreamerorpaper.Foldyourstreamerinto 10equalpiecesandlabelthefoldmarksintenthsusingdecimalnotation. Completethesesentences.

a Eachofthe10piecesis 1 □ or0.□

b 3piecesisequalto 3 □ or0.□

c 7piecesisequaltothefraction______andthedecimal______.

2 Readthesenumbersaloud.Howwouldyourecordthemasadecimal?Thefirst onehasbeendoneforyou.

a 5onesand8tenths Answer:5.8

b 3tens,2onesand 1tenth

c 7hundreds,6onesand7tenths

d 4tens,9onesand 6 10

e 5ones,7tenthsand2tens

f 3tenths,5tensand7hundreds

6A Individual APPLYYOURLEARNING

1 Drawanumberlinemarkedintenthsfrom0to1,thenshowthesedecimals.

2 Copyandcompletethisplace-valuechart.

a 45.6

b 8 2

c 60.8

d 2.3

e 32 f 500 4

3 Writethesedecimalsasfractions.

4 Writethesefractionsasdecimals.

5 Writethenumbershownoneachabacusasadecimal,theninwords.

One-tenthcanberepresentedbytakingasquareandthinkingofitas1.Then dividethesquareinto10equalparts,asshown.Theshadedpartrepresents 1 10 or 0 1ofthesquare. 1

Writeadecimalfortheshadedpartofeachsquare.

7 Writethesefractionsasdecimals.

1and 1 10 a 4 10 b 7and 9 10 c

8 Writethesedecimalsasfractions. 3 2 a 0 3 b 1 5 c

9 Orderthefollowingfromsmallesttolargest.

a 0.4, 3 10,0.8, 1 2

b 1 10,1 3,0 9, 17 10

10 Threestudentsrecordedtheiranswerto 5 10 + 8 10 inthreedifferentways.

Thefirststudentrecordedtheiransweras 13 10

Thesecondstudentrecordedtheiransweras1.3

Thethirdstudentrecordedtheiransweras1and 3 10

Whowascorrect?

11 Writethefollowingasadecimal.

a 2dollarsand80cents

b 23 4

c 12metresand95centimetres

d 89dollarsand50cents

e 1 20

f 16centimetresand4millimetres

g 72 10

6B Hundredths

Justaswecut1intotenequalpiecestogettenths,wecancutonetenthintoten equalpiecestoget hundredths.Whenwecuteachtenthintotenequalpieces,it makes100piecesbetween0and1.Eachpieceisequalto 1 100

Wewrite 1 100 as0.01whenusingthedecimalwayofwritingnumbers.

Thisgivesnewmarkerson thenumberline.Thefirstoneis 1 100 or0.01,asshown below.

0.01isthesameas 1 100 .

Imaginecuttingalargeballofplaydoughinto10equalpiecesandcuttingeachof those10piecesinto10smallerpieces.Eachsmallpiecewouldbeonehundredthofthe originallargeball.

Onaplace-valuechart,thesecondplacetotherightofthedecimalpointhasthe valueofhundredths.

0 01iswrittenlikethis.

Wecancountinhundredthsonthenumberline.Welabelthefirstpointonthe numberlinetotherightofzeroas0.01andkeepgoing.Weneedto‘zoom’inonthe numberlinetoseehundredthsbetween0and0.1.

Whenwegetto10hundredths,thisisthesamemarkerasone-tenthor0.1.Wecan alsowrite0.1 as0.10.

Ifwekeepgoingup byone-hundredth,weget:

0.11,whichisthesameas 1 10 + 1 100,or 11 100,orelevenhundredths

0.12,whichisthesameas 1 10 + 2 100,or 12 100,ortwelvehundredths

0.13,whichisthesameas 1 10 + 3 100,or 13 100,orthirteenhundredths

0 14,whichisthesameas 1 10 + 4 100,or 14 100,orfourteenhundredths

0 15,whichisthesameas 1 10 + 5 100,or 15 100,orfifteenhundredths

0.16,whichisthesameas 1 10 + 6 100,or 16 100,orsixteenhundredths

0.17,whichisthesameas 1 10 + 7 100,or 17 100,orseventeenhundredths

0.18,whichisthesameas 1 10 + 8 100,or 18 100,oreighteenhundredths

0.19,whichisthesameas 1 10 + 9 100,or 19 100,ornineteenhundredths

0 2,whichisthesameas 2 10,or 20 100,ortwentyhundredths,ortwotenths.

Wecankeeplabellinginhundredthsacrossthenumberlineuntilwegetto 100hundredths,whichisthesameas1.

1isthesameas 100 100 .

4th

Example5

Mark0.68onanumberline. Solution

Drawanumberlinemarkedwith0and1.Cutthelengthbetween0and1into10 equalpieces.Eachpieceisone-tenth.Labelthenumberlineintenths,fromleftto right.Thesixthmarkerafter0is0.6.Nowcutthelengthbetween0.6and0.7into 10equalpieces.Eachpiece isone-hundredth.Theeighthmarkeris0.68.

Convert 7 100 toadecimal.

Wehave0ones,0tenthsand7hundredths,sowewritea7inthehundredths place.

Convert 87 100 toadecimal.

Solution

100 = 80 100 + 7 100 = 8 10 + 7 100

Wehave0ones,8tenthsand7hundredths,sowewritean8inthetenthsplace anda7inthehundredthsplace.

87 100 = 0.87

Wholeclass LEARNINGTOGETHER

1 Drawnumberlinesfordecimalnumbersthathavehundredthsinthem. Youcanuse1cmgridpaperandyourrulertousemmformarkinghundedths.

Recordthesedecimalnumbersonthenumberline.

2 Drawanumberlinestartingat2andendingat4. Recordthesedecimalnumbersonthenumberline.

6B Individual APPLYYOURLEARNING

1 Copyandcompletethisplace-valuechart.

a 2.33

b 10.82

c 153 18

d 49 02

2 Writethenumbershownoneachabacusinnumbers,theninwords.

3 Onehundredthcanberepresentedbytakingasquareandthinkingofitas1. Thendividethesquareinto100equalparts,asshown.Theshadedpartrepresents 1 100 or0 01ofthesquare.

2 100 a 7

b 91

c 12

d 137

5 Writethesedecimalsasfractions.

0 01 a 2.08 b 9.22 c 0 66 d

7.99 e

6 Writethesenumbersasdecimals.Thefirstonehasbeendoneforyou.

a 3tens,4ones,0tenthsand5hundredths Answer:34 05

b 2ones,8tens, 6tenthsand3hundredths

c 7hundredths,9hundreds,8tens,1tenthand4ones

d 5hundredths,2ones,2tenthsand5tens

e 7tenths,6hundredths,1hundredand7tens

7 Anewtake-awaystorewasopeninginthemainstreet.Intherushtogetthe menuprinted,theownermadesomeerrorswithpricingonthemenuandforgot torecordthepricesaswewouldnormallyrecordmoney.

Canyouhelpwiththecorrections?Thefirstonehasbeendoneforyou.

a Burger12.5 Answer: $12.50

b Chips7.5

c Toastedsandwich10.75

d DimSim3 4

e Smallsoftdrink2 8

f Largesoftdrink5.4

6C Thousandths

Wecancuthundredthsintotenequalpiecestoget thousandths. Toshowthousandthsonthenumberline,imaginethatyouarelookingthrougha magnifyingglass.

Whenwecuteachhundredthintotenpieces,thereare1000piecesbetween0and1, soeachpieceiscalled 1 1000 .

Wewrite 1 1000 as0 001whenweareusingdecimals.

0.001isthesameas 1 1000 .

Onaplace-valuechart,thethirdplacetotherightofthedecimalpointhasthevalue ofthousandths.One-thousandthiswrittenonaplace-valuechartlikethis:

Wecancountinthousandthsonthenumberline.Welabelthefirstmarkeronthe numberlinetotherightofzeroas0.001andthenkeepgoing:

0.0020.0030.0040.0050.0060.0070.0080.009

Whenwegetto10thousandths,weseethatthisisthesamemarkerasone hundredthor0.01.

0 01isthesameas 10 1000 and0 010

Ifwekeepgoingupbyone-thousandths,weget:

0.011,whichisthesameas

0 012,whichisthesameas

0.013,whichisthesameas

0.014,whichisthesameas

0.017,whichisthesameas

0.018,whichisthesameas

0 019,whichisthesameas

0.02,whichisthesameas

Wecankeeplabellinginthousandthsacrossthenumberlineuntilwegetto 100thousandths,whichisthesameas0.1.

0.1isthesameas 100 1000 .

Wecancontinuelabellinginthousandthsacrossthenumberlineuntilwegetto 1000thousandths,whichisthesameas1.

1isthesameas 1000 1000

Theplace-valuesystemkeepsgoingforever.Youmighthavealsoseen ten-thousandths,hundred-thousandthsandmillionths.

SAMPLEPAGES

Drawanabacuswiththedecimal1.919onit.

Howtoreaddecimals

Wereadthedigitsafteradecimalpointbysayingthedigitsinorder.Forexample, 3.215is‘threepointtwoonefive’. Example9

Convert 23

toadecimal.

Wehave0ones,0tenths,2hundredthsand3thousandths,sowewritea0inthe tenthsplace,a2inthehundredthsplaceanda3inthethousandthsplace.

.023

Example10

Convert0 002toafraction. a Convert0 104to afraction. b

Solution

a Wehave0ones,0tenths,0hundredthsand2thousandths.Sowewrite:

0.002 = 2 1000 = 1 500 (insimplestform)

b 0.104 = 1tenth + 0hundredths + 4thousandths = 1 10 + 0

(insimplestform)

1 Isthereanumberinbetween?(Activity)

Totheteacher:Thinkoftwowholenumbers.Writethematoppositeendsofthe board,withthelargernumberontheright.Askforanumberinbetweenthese two.Writethenewnumberbetweenthetwonumbersalreadywrittenonthe board.Ruboutoneofthefirstnumbersandaskforanumberbetweenthetwo numbersnowontheboard.Eventuallyyouwillgettotwoconsecutivewhole numbers.Studentswillthenhavetouseadecimalnumberforthenumberthey makeupbetweenthetwonumbersontheboard.Keepgoinguntilthenumbers ontheboardareonlyone-thousandthapart.Repeatseveraltimesuntilstudents realisethatthereisalwaysanumberinbetween.Youcouldgobeyond thousandthsandkeepgoingonforever.

2 1kilogramisequalto1000grams.Thedecimalpointisusedtoseparatethe kilogramsfromthegrams.4kgand500grams=4 500 1000 kg,or4.5kg.

Convertthefollowingintodecimals forkilograms:

12kgand250g a 3kgand300g b 750g c 50kgand560g d

3 1litreisequalto1000millilitres.Thedecimalpointisusedtoseparatethelitres fromthemillilitres.8litresand650millilitres=8 650 1000 L,or8.65L.

Convertthefollowingintodecimals forlitres:

10litresand700mL a 35litresand450mL b 5litresand990mL c 350mL d

1 Writethenumbershownoneachabacusasadecimalnumber,andtheninwords.

2 Copyandcompletethisplace-valuechart.

a 0 236

b 1.732

c 456.007

d 121.893

e 909 674

3 Writethesefractionsasdecimals.

4 Writethesedecimalsasfractions.

Comparingdecimals

Whichnumberislarger:2.1or1.9?

Wecanseethis onanumberline.

Weknowthatnumbersincreaseinsizeaswegototherightonanumberline.So2.1 islargerthan1.9.

Thereisoftena shortcutto comparing decimalnumbers.

Thewholenumberpartof1.9is1.Thewholenumberpartof2.1is2.Thenumber withthelargerwhole numberpartisthelargernumber,so2islarger.

Whathappensifthewholenumberpartsoftwonumbersbeingcomparedarethe same?

Thisnumberlineshows2.3and2.5.

Wecanseethat2 5islargerthan2 3becauseitliestotherightof2 3onthenumber line.

Or,we canworkitoutthisway:

2 3isthesameas2 3 10

2.5isthesame2 5 10 .

Thewholenumberpartsarethesame,but:

2.5is5tenthsmorethan2 and 2.3isonly3tenthsmorethan2. So2.5islargerthan2.3.

Example11

Putthesenumbersinorder,smallesttolargest: 2.8, 3.7, 1.5, 2.6

Solution

Lineupthenumbersundereachother.

Orderthenumbersbycomparingthehighestvaluedigits,inthiscasethatmeans startingwiththeones. 1 52 83 7 2.6

Therearetwonumberswiththedigit2intheonesplace,soweneedtocompare thetenths.Since8tenthsislargerthan6tenths,2.8islargerthan2.6.

Ourorderingisnow done:1.5, 2.6, 2.8, 3.7.

Itisbesttocomparedecimalnumbersstartingfromtheleft.Wecancompareanytwo decimalnumbersinthisway.Lineupthenumberssothattheirplace-value componentsareoneundertheother.Makesurethedecimalpointsarealigned. Startwiththewholenumberparts.

Ifthewholenumberpartsarethesame,comparethetenths.

Ifthetenthsarethesame,comparethehundredths.

Ifthehundredthsarethesame,comparethethousandths,andsoon. Thefirsttimeyoufindthatonedigitislargerthananotherinthesameplace,thenthe numberwiththelargerdigitisthelargerofthetwonumbers.

Example12

Whichislarger:1.37or1.214?

Solution

Thiscanbedoneinanumberofways. Alignthedigits.

Comparetheones:thesearethesame.

Comparethetenths:3tenthsislargerthan2tenths,so1.37islargerthan1.214

Alternatively,onanumber linewecanseethat1 214islessthan1 3and1 37is greaterthan1 3

6D Wholeclass LEARNINGTOGETHER

1 Drawanumberlineforeachsetofnumbers,thenorderthemfromsmallest tolargest.

3.25, 4.5, 2.9 a 1 3, 1 87, 1 45 b

2.09, 1.6, 2, 1.72, 1.9 c

2 Stringnumberline:

Everybodywritesadecimalnumberonacard.Theythentaketurnstopeg theirdecimalnumberonastringnumberline.Discussthecorrectplacement ofnumbers.

3 Makethelargestnumber: Workinpairs.

Drawaplace-valuechartlikethisone.

Rolla10-sideddie(marked0–9)andcalloutthenumber.Writethenumber inoneoftheboxesonyourplace-valuechart.Onceyouhavewrittenthe number,youcannotchangeitsplace.

Repeatthisstepthreemoretimes,thencomparethenumbersyouhavemade. Thestudentwiththelargestnumberisthewinner.

6D Individual APPLYYOURLEARNING

1 Drawanumberlineforthesepairsofdecimalnumbers,thendecidewhich oneofeachpairislarger.

2 Fivefriendsmeasuredtheirheights.Thesearetheresults: Hui1.55m,Ben1.64m,Sally1.6m,Lin1.49m,Jack1.72m

a Whoisthetallest?

b Whois theshortest?

c Whoisclosesttobeing2metrestall?

3 Whichdecimalineachpairislarger?

4 Writethesenumbersinorder,fromlargesttosmallest.

.4021.4991.

0040 110 1660 01 c 0.68930.380.30.099 d

5 Whichofthesedecimalsisclosestto1?

6 Showeachpairofnumbersonthenumberline.Thensaywhichoneissmaller.

7 Whichislarger?

8 Putthesefractionsanddecimalsinorder,fromsmallesttolargest.

Justas rounding isusedforwholenumbers, roundingcanbeusefulfordecimalsaswell.This isparticularlytruewhenweareworkingwith moneyandmeasurements.

Wemightnotneedtoknowtheexactamountof moneyneededforanupcomingspend,butit maybeusefultohaveanestimate.

ShouldIbudgetfor $100or $50foratriptothe supermarket?

Whataboutplanningatriptoasportsgame?

Willthefamilycostbecloserto $70or $120? Estimatingwithmeasurementscanalsobeuseful.

ApproximatelyhowmanymetresoffencingwillI needforavegetablegarden?Howmanylitresof waterwillbeneededtofillawatertank?

Wedon’talwaysneedprecisedecimalnumbers;awholenumberestimateis sufficient.Considerthewholenumberssurroundingadecimalandidentifywhich oneisnearest.That’sthenumbertoroundtoforyourestimate.

Anumberlinecanbeusefulindeterminingwhichwholenumberadecimalisclosestto. Wecanseewhat15.45isclosesttoonthenumberlinebelow.15.45isbetween15 and16.

15 45iscloserto15than16soweround15 45to15.

Whenthenumberisexactlyhalfwaybetweentwonumbers,theconventionisto roundtothegreaterofthetwoaswedowithwholenumbers.7.5,forexample, roundsto8.

1 Whatwholenumberiseachofthefollowingdecimalsclosestto?Drawa numberlinetohelp.

$12.95 a

.3m b

$65.15 d 36.83kg e

.7cm c

.4mL f

2 Thinkofanumberwithonedecimalplacethatcanberoundedtoup20. Whataboutanumberwith1decimalplacethatcanberoundeddownup 20?Showyourthinkingonanumberline.

1 Drawanumberlinetoshoweachdecimalandthewholenumbersthatare eithersideofit.

2 Roundeachdecimaltothenearestwholenumber.Useanumberlinetohelp:

.5km a

.15 d

3 I’mthinkingofadecimalnumberwithonedecimalplace.

Thetenthisanevennumber.

Whenroundedtothenearestwholenumber,mynumberis18. Whatcouldmynumberbe?

4 I’mthinkingofadecimalnumberwithtwodecimalplaces. Thetenthisanoddnumber.

Thehundredthisanoddnumber.

Whenroundedtothenearestwholenumber,mynumberis56. Whatcouldmynumberbe?

5 Usingthedigits2,8,3,6and9,makedecimalnumberswithtwodecimal placesthatcouldberoundedto:

1 Copyandcompletethisplace-valuechart.

a 23 803

b 999 876

c 20 07

d 402 024

2 Writethedecimalnumberforeachletter.

3 Drawanumberlinefrom0to1.Markitintenths,thenshowthesedecimals.

4 Whatdecimalpartofeachsquareisshaded?

5 Writethenumbershownoneachabacusinnumbers,theninwords.

6 Writethesefractionsasdecimals.

7 Writethesedecimalsasfractions. 0 003 a

.43 d

8 Joanne,MarthaandIngridmeasuredtheirheightsas1 24m, 1 99mand1 362m respectively.Whoisthetallestandwhoistheshortest?

9 ClaudeandRebeccaweredoingsomecooking.Claudeweighedtwolotsofflour, eachone0 3kg.Rebeccaweighedout0 58kgofflour.Whohadmoreflour?

10 Whichislarger?

a 6 7or7 6?

b 4.009or4.9?

c 1.4or1.392?

d 1.5or1.932?

11 Writeeachgroup ofnumbersinorder,fromlargesttosmallest.

a 1.23.40.8 5.6

b 0 020 0080 8020 228

c 0.3470.5930.0090.051

d 100 2100 02100 222100 022

12 Roundeachdecimaltothenearestwholenumber: 164 65m a $8 50 b 29.39cm c $999.99 d

6G Challenge–Ready,set,explore!

Maths crossword

UNCORRECTEDSAMPLEPAGES

Across

1 24 is the of 2 and 12

5 100 m × 100 m

6 little line in a fraction

9 top number in a fraction

11 divides a number exactly

12 one hundredth of a metre

14 lowest common multiple of 8 and 12

15 one thousandth of a kilometre

17 0.375 as a fraction

20 852 741 more than 147 259

22 larger of 0.8 and 0.634

23 number with two factors, 1 and itself

24 quarters in three and a half

25 out of a hundred

Down

2 bottom number in a fraction

3 one-third of 38.16 (4 words)

4 seven elevens

7 unit of measurement for volume

8 larger of 43.2 and 43.1999867

10 one-half as a decimal

13 ten cubed

16 one cubic centimetre of water

18 add two numbers together

19 subtract

21 one hundred

Usefulskillsforthistopic

• understandingofdecimalsonthenumberline

• convertingdecimalstofractionsandvice-versa

• comparingdecimalnumbersofdifferentlengths

Vocabulary

• Percentage

• Equivalent

‘Percent’comesfromtheLatinwordspercentum,meaning‘outofonehundred’. Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage

Numberline

Placethesefractions,decimalsandpercentagesonanumberlinestartingat0and endingat2.Couldyouaddinsomebenchmarkstostart?

Haveagoonyourown,thenshareyourthinkingwithapartnerbeforediscussingasa wholeclasswhyyouhaveplacedthenumberswheretheyare.Youarealwaysfreeto changeyourmindatanytime!

Fractions,decimals andpercentages Fractions,decimals andpercentages andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals Fractions,decimals Fractions,decimals andpercentages andpercentages Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages Fractions,decimals andpercentages

Inthischapter,weextendourunderstandingoffractionsanddecimalstoinclude percentages.

Percentagesareanotherwayofwritingfractionordecimalquantities.

Thereare100beadsonFreya’snecklace.Fiveofthe beadsarered.

Wecanwritethisasafraction.

5outof100beadsareredor 5 100 beadsarered.

Wecanalsowriteitasadecimal.

5outof100beadsareredor0.05ofthebeads arered.

5 out of 100 beads

Orwecansaythat‘5percent’ofthebeadsarered.‘5percent’isaquickwayof saying‘5outofonehundred’.

Percentisrepresentedbythissymbol %.Thetotalamountis100%.’Percent’ means’outofonehundred’,so’5percent’isanotherwayofsaying’5out of100’.

Weusepercentagestodescribeanamount,forexample,5% of100is5.

7A Fractionsandpercentages

A percentage isanotherwayofwritingafractionwithadenominatorof100.

Wecanunderstandpercentagebydrawingapictureofasquaredividedinto 100equalparts,thenshadingsomeofthem.

2outof100partsare shaded.

2% ofthesquareis shaded.

Example1

50outof100partsare shaded.

50% ofthesquareis shaded.

100outof100parts areshaded.

100% ofthesquareis shaded.

Thissquareisdividedinto100equalparts.Whatpercentageisshaded?

Solution

23outofthe100partsareshaded. So23% ofthesquareisshaded.

Wecanalsowritefractionsthatdonothave100asadenominatoraspercentages. Firstconvertthefractiontoan equivalent fractionwithadenominatorof100,then writethefractionasapercentage.Forexample:

Youcanalsomultiplythefractionby100%

Example2

Convertthesefractionsintopercentagesbymultiplyingby100%.

Wholenumberscanbe convertedtopercentages.

Toconvertapercentagetoanequivalentfractionormixednumber,firstwriteitasa fractionwithadenominatorof100,thensimplifyit.Forexample:

Example3

Writethesepercentagesasfractionsormixednumbers.

Solution

a 80%= 80 100 = 4 5

b 40%= 40 100 = 2 5

c 75%= 75 100 = 3 4

d 12%= 12 100 = 3 25

e 140%= 140 100 = 1 40 100 = 12 5

Weoftenuse100% todescribe‘all’ofsomething.Forexample,‘100% oftheaudience enjoyedthemovie’meansthatalloftheaudiencelikedthemovie.

Percentageslessthan100% describesomethinglessthanthewhole.Forexample, ‘only86% ofpeoplevoted’meansnoteveryonevoted.Also,‘Teresaisnotfeeling 100% today’meansthatTeresadoesnotfeelcompletelywell.

Wealsousepercentagesgreaterthan100% toindicatethatsomethingwasmorethan thewhole.Forexample,threeglassesoforangejuicegive120% oftherecommended dailyintakeofvitaminC,whichismorethanrequired.

UNCORRECTEDSAMPLEPAGES

1 Createeachsituationinyourclassroom,thensaywhatthepercentageis.

a Thereare10childrenstanding;8ofthemhaveonehandintheair.

b Thereare5childrenstanding;3ofthemaresmiling.

c Thereare2peoplestanding;1ofthemisolderthan10.

7A Individual APPLYYOURLEARNING

1 Writeeachoftheseasapercentage.

10outof100 a 50outof100 b 78outof100 c

2 Writethepercentageforeachsituation.

a 87outof100childrenatQueenstownSchoollikecricket.

b 62outof100plantsinGraeme’sgardenarenativeplants.

c InVictoria,8litresoutofevery100litresofwaterconsumedareusedby privatehomes.

3 Writethesefractionsaspercentages.

4 Writethesepercentagesasfractionsintheirsimplestform.

5 Writethesemixednumbersaspercentages.

6 100peopleraninaCitytoSurffunrun.56ofthemwereadults.Whatpercentage oftherunnerswerechildren?

7 AtKangarooFlatSchool,34outof50childrencatchthebuseachday.What percentageisthis?

8 3 5 ofthehousesinHelen’sstreetaredouble-storeyhouses.Whatpercentageof thehousesaresinglestorey?

9 75% ofthechildreninMei’sfamilyareboys.Whatfractionaregirls?

10 15outofthe45studentswhoattendThredboRegionalSchoolareover10years ofage.Whatpercentageisthis?

11 InJanuary,LakeEildonwasat42% ofcapacity.ByDecember,itwasonly33% full. Bywhatpercentageofcapacityhadthewaterleveldropped?

12 Liambought40lollies.10ofthe40lolliesweregreen.Whatpercentageofthe lollieswerenotgreen?

13 StudentsatBendigoPrimarySchoolweresurveyedabouttheirfavouriteice-cream flavour.25% ofstudentslikedstrawberry,40% likedchocolate,andtherestliked vanilla.Whatpercentageofstudentslikedvanillaice-creamthebest?

14 100childrenenrolledforLittleAthletics.13% enrolledonMonday,29% on Tuesday,8% onWednesdayand16% onThursday.Ifenrolmentswerenottakenat theweekend,whatfractionofthechildrenenrolledonFriday?

15 Inastreetthereare25houses.16haveadoublegarage,6haveasinglegarage and3haveacarport.Workoutthepercentageforeachtypeofvehicleshelter.

7B

Decimalsandpercentages

Adecimalcanalsobewrittenasapercentage.Toconvertadecimaltoapercentage, firstwriteitasafractionwithadenominatorof100.Adecimalthathashundredthsas thelastplaceconvertseasilytoapercentage.

Example4

Writethesedecimalsaspercentages.

Apercentagecanbeconvertedtoadecimalbyfirstwritingitasafractionwitha denominatorof100,thenconvertingittoadecimal.Forexample: 34%= 34

= 0.34

Example5

Writethesepercentagesasdecimals.

Solution

a 24%= 24

b 50%= 50

c 3%= 3

1 Selectdiceoftwodifferentcoloursandrollthemtogiveadecimalnumber.One dieisthetenthsdigit,andtheotherdieisthehundredthsdigit. Forexample,usingbluefortenthsandredforhundredths,therollshownhere

givesthenumber0.46

Rollthedice10times andconverteachdecimalnumberintoapercentage.

2 Selectdiceofthreedifferentcoloursandrollthemtogiveadecimalnumber.One dieistheonesdigit,anotherdieisthetenthsdigitandthethirddieisthe hundredthsdigit.

Forexample,therollshownhere

wouldgivethenumber2 46

Rollthedice10timesandconverteachdecimalnumberintoapercentage.

7B Individual APPLYYOURLEARNING

1 Writethesedecimalsaspercentages.

2 Writethesepercentagesasdecimals.

3 Kwameworked0.5ofthetotalhoursneededforapart-timejobthisweek.What percentageofthetotalhoursdidhework?

4 Asavingsaccounthasanannualinterestrateof0.04.Whatistheinterestratein percentageterms?

5 Inabasketballgame,aplayerscored0.6ofthetotalpointsfortheirteam.What percentageofthepoints didtheplayerscore?

6 Thetemperatureincreasedby25%.Whatistheincreaseindecimalform?

7 Yoursavingsaccounthasgrownby12%.Whatisthegrowthrateindecimalform?

8 Explaininyourownwordshowtochangeadecimaltoapercentage.Usean exampletodemonstrateyourexplanation.

9 Explaininyourownwordshowtochangeapercentagetoadecimal.Usean exampletodemonstrateyourexplanation.

7C

Reviewquestions–Demonstrateyourmastery

1 Writeeachoftheseasapercentage.

2 Writeeachnumberasapercentage.

3 Writethemissingfractions(ormixednumbers),decimalsandpercentages.

4 Duringaproject,65% oftheworkwascompletedonthefirstday.Whatisthe completionpercentageindecimalform?

5 Asavingsaccountoffersanannualinterestrateof7%.Whatisthisinterestrateas adecimal?

6 Oliviascored 17 25 onhermathtest.Whatisherscoreasapercentage?

7 Astoreisofferinga15% discountonasmartphone.Whatisthefractionformof thediscountpercentage?

8 Atournamenthad50participants.Only0 8ofthemshowedupfortheevent. Whatpercentageoftheparticipantsattended?Howmanyparticipantsattended?

Yourclassisorganisingasustainablemarkettoraisefundsforanenvironmental charity.Youwillselleco-friendlyproductssuchasreusablewaterbottles,recycled clothing,andplantseeds.

1 Pricingtheitems: Youhave50reusablewaterbottles,100piecesofrecycledclothing,and200 packetsofplantseedstosell.

Decideonarealisticpriceforeachitem.Makesuretousedecimals(e.g. $3.50for areusablewaterbottle).

2 Calculatingsales:

Ifyousell70% ofthereusablewaterbottles,60% oftherecycledclothing and80% oftheplantseeds,howmanyofeachitemdidyousell?

Converteachofthesepercentagestofractionsanddecimals.

3 Totalsales:

Calculatethetotalsalesforeachtypeofproduct. Addthetotalsalestofindouthowmuchmoneyyoumadeintotal.

4 Profitcalculation:

Ifthecosttomakeeachreusablewaterbottleis $1.50,eachpieceofrecycled clothingis $2.00andeachpacketofplantseedsis $0.50,calculatethetotalcost. Subtractthetotalcostfromthetotalsalestofindouttheprofit.

5 Bonustask:

Ifyouwanttooffera15% discountonallitemsforthelasthourofthemarket, howmuchwilleachitemcostafterthediscount?

Calculatethenewtotalsalesandprofitwiththediscountapplied.

Usefulskillsforthistopic

• measuringlengthsanddistancesincentimetresandmetresusingrulersandtape measures

• recordinglengthsanddistancesincentimetresandmetres

• recallingquicklythemultiplicationfactsto12 × 12

Vocabulary

Centimetre

Squarecentimetre

1 Ifarectanglehassidelengthswhicharealloddnumbers,theperimeterwillalso beodd.

2 1800metresismorethan8kilometres.

3 ThereisonlyonerectangleIcandrawwithanareaof24squarecentimetres.

4 Ionlyneedtoknowonesidelengthtocalculatetheperimeterofasquare.

UNCORRECTEDSAMPLEPAGES

Length,perimeter andarea andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea andarea andarea andarea andarea andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea andarea Length,perimeter andarea andarea Length,perimeter andarea andarea Length,perimeter andarea Length,perimeter andarea andarea Length,perimeter andarea andarea Length,perimeter andarea andarea Length,perimeter andarea andarea andarea Length,perimeter andarea Length,perimeter andarea andarea Length,perimeter andarea Length,perimeter andarea andarea Length,perimeter andarea Length,perimeter andarea andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea Length,perimeter andarea

InAustraliaweusethe metricsystem ofmeasurement. Theunitsoflengthinthemetricsystemarebasedonthemetre(m).

Length isthewidth,height,depthordistancearoundanobject.

Thedistancearoundanobjectisalsocalledthe perimeter

The area isthesizeofthesurfaceoramountofspaceinsideanobject.

Thissoccerpitchhassidelengthsshowninmetres.

Theperimeteristheentiredistancearoundthecourt.

Theareaistheentirespaceinsidethecourt.

8A Length

Choosingunits

Therearetwoimportantthingstorememberwhenmeasuringlength.First,youneed toselectthemostsuitableunitofmeasurementfortheobjectyouwanttomeasure. Second,youneedtoselecttherightmeasuringinstrument.

Weusemetrestomeasureitemsthatcouldbesteppedoutand countedinpaces.Ametreisaboutthelengthofoneadultpace.

Theword metre isoftenabbreviatedto‘m’whenitiswritten.

Thebedis1metrewideand 2metreslong.

Theswimmingpoolis 50metreslong.

About1metre

Aswellasthemetre,wealsousethecentimetre,millimetreandkilometretomeasure length.

Theprefixbeforetheword‘metre’tellsusaboutthesizeoftheunit.

Theprefix‘centi’tellsusthattheunitisone-hundredthofthebaseunit.

So 1centimetre = 1 100 metre and 1metre = 100centimetres

Weabbreviatetheword centimetre to‘cm’asashortwayofwritingit.

SAMPLEPAGES

Youwouldprobablyusecentimetrestomeasure itemsthatareaboutthesizeofyourhand.

Thecoverofthismathsbookis28cmlongand 21cmwide.

Theprefix‘milli’tellsusthattheunitisone thousandthofthebaseunit.

So 1millimetre = 1 1000 metre and 1metre = 1000millimetres.

Weabbreviate millimetres to‘mm’.Wewould usemillimetrestomeasureitemsthatareabout thesizeofyourfingernail.

UNCORRECTEDSAMPLEPAGES

Builders,furnituremakers,architectsandelectriciansnearlyalwaysusemillimetres, evenforverylargemeasurements.

Thestampis24mmhigh and29mmwide.

Thepaperclipis32mmlong and8mmwide.

Theprefix‘kilo’tellsusthattheunitisone thousandtimesthebaseunit.

So, 1kilometre = 1000metres and 1metre = 1 1000 kilometre

Weabbreviatetheword kilometre tokm.

Weusekilometrestomeasurelarge distances.Forexample,itisabout4500km fromPerthtoBrisbanebyroad.

Weneedtomakesensiblechoiceswhenusingmeasurementunits.

Example1

a JanewantstomeasureWang’sshoe.Whichunitofmeasurementshould sheuse?

b Aliwantstomeasurethewidthofthefootballoval.Whichunitof measurementshouldheuse?

Solution

a JanecanuseherrulertofindthelengthofWang’sshoe,socentimetresor millimetresarethebestunitstouse.

b Alicanpaceoutthedistance,sohecoulduseatapemeasureoratrundle wheelandrecordthedistanceinmetres.

Example2

Measurethelengthofthispenciltothenearestmillimetre.

Solution

Thepencilis73millimetreslong.

Thestandardunitofmeasurementisthemetre. Weabbreviatethistom.

Thereare100centimetresin1metre. Thereare1000millimetresin1metre. Thereare1000metresin1kilometre.

1a Measureeachstudent’sheightincentimetres,thencutalengthof streamertomatchtheheight.Writeeachstudent’snameandheighton theirstreamer.Orderfromshortesttotallest.

b Measureeachstudent’sarmspanincentimetres,fromfingertiptofingertip witharmsstretchedout.Cutalengthofstreamerofadifferentcolour tomatchthislength.Writeeachstudent’snameandarmspanontheir streamer.

c Compareeachstudent’sarmspanwiththeirheight,thendiscussthe results.

2 Onepaceisapproximatelyequalto1metre.Usepacestofindameasurement inyourschoolthatis: lessthan1metre a between3metresand5metres b morethan10metres. c Measureeachitemtocheck.

8A Individual APPLYYOURLEARNING

1 Writethelengthofeachpencilincentimetres.

2 Writethelengthofeachpencilinmillimetres.

3 Writethelengthofeachpencil.

4a Howmuchisleftwhen50cmiscutfroma1mpieceofribbon?

b Howmuchisleftwhen5cmiscutfroma1mpieceofribbon?

c Howmuchisleftwhen62cmiscutfroma1mpieceofribbon?

5 Howmany5cmpiecesofribboncanEshacutfroma35cmlength?

6a Albertmeasuredthewidthofhisbedroomdoor.Itwasaswideas5ofhisshoe lengths.Albert’sshoeis15centimetreslong.HowwideisAlbert’sdoor?

b Albert’sbrotherEdwardmeasuredthesamedoor.Hefoundittobeaswideas 3ofhisshoelengths.IsEdward’sshoesmallerorlargerthanAlbert’sshoe?

7 Carlacuta60cmpieceofropeinto4piecesofequallength.Whatwasthelength ofeachpiece?

8 Astorageroomcanfitexactly8boxesalongitswidth,11boxesalongitslength and6boxesfromfloortoceiling.Ifalltheboxesareidenticalcubeswitheachside equalto60cm,whatarethedimensionsoftheroom?Youmightneedtodrawa pictureormakeamodelusingcubestohelpyou.

9 Joseph’sclassweremeasuringthedistancetheycouldthrowashot-put.Thetape measuretheywereusingwasbrokenoffatthe20-centimetremark.Itlooked likethis.

a Josephthrew6metres40centimetres.Whatdidthetapemeasureshow?

b WhenJoshuameasuredanotherthrow,thetapemeasureshowed6metres 15centimetres.Whatwasthetruemeasurement?

c Tobeabletousethistapemeasuretoaccuratelymeasurethedistanceofa shot-putthrow,theusermustadd/subtract20centimetres.(Choosethe rightword.)

8B Convertingmeasurements

Weusedifferentunitstomeasuredifferentlengths.Sometimeswewanttochange fromoneunittoanother.

Metresandcentimetres

Weknowthat1metreisequalto100centimetres,sotoconvertfromcentimetresto metreswemake‘lotsof100centimetres’.Forexample:

382centimetres = 3‘lotsof100centimetres’ + 82centimetres = 3metres82centimetres.

Example3

Convert412centimetrestometres.

Solution

412centimetres = 4‘lotsof100centimetres’ + 12centimetres = 4metres12centimetres.

Weconvertmetrestocentimetresbychangingeachmetreinto100centimetres.

Example4

Convert6metres43centimetrestocentimetres.

Solution

6metres43centimetres = 6‘lotsof100centimetres’ + 43centimetres = 643centimetres

Centimetresandmillimetres

Sincethereare100centimetresin1metreand1000millimetresin1metre,weknow thatthereare10millimetresinonecentimetre.Youcanalsoseethisonarulerortape measure.

Toconvertfrommillimetrestocentimetreswemake‘lotsof10millimetres’.

Uncorrected

Example5

Convert30millimetrestocentimetres.

Solution

30millimetres = 3‘lotsof10millimetres’ = 3centimetres

Kilometres

Thereare1000metresin1kilometre.Toconvertmetrestokilometreswemake‘lotsof 1000metres’.

Example6

Convert4213metrestokilometres.

Solution

4213metres = 4‘lotsof1000metres’ + 213metres = 4kilometres213metres

Weconvertkilometrestometresbychangingeachkilometreinto1000metres.

Example7

Convert7kilometres802metrestometres.

Solution

7km802m = 7‘lotsof1000m’ + 802m = 7000 + 802m = 7802m

Toconvertfrommetrestocentimetres,changeeachmetreinto 100centimetres.

Toconvertfromcentimetrestometres,changeeach‘lotof 100centimetres’into1metre.

Toconvertfromcentimetrestomillimetres,changeeachcentimetre into10millimetres.

Toconvertfrommillimetrestocentimetres,changeeach‘lotof 10millimetres’into1centimetre.

Toconvertfromkilometrestometres,changeeachkilometreinto 1000metres.

Toconvertfrommetrestokilometres,changeeach‘lotof 1000metres’into1kilometre.

8B

Wholeclass LEARNINGTOGETHER

1 Converteachmeasurementbelowtotheunitwritteninbrackets.Aruleror tapemeasuremayhelpyouchecksomeofyouranswers.

130cm(metresandcentimetres) a 12000m(kilometres) b 1m32cm(centimetres) c 43mm(centimetresandmillimetres) d

62cm(millimetres) e 4km825m(metres) f

2 Imaginethatyouandyourthreefriendswereplacedonthefloorinastraight line,foottohead.Whatwouldbethetotallengthincentimetres?Convert thismeasurementtomillimetres.

3 Ifyouweremeasuringeachofthefollowing,whatwouldbethemost appropriateunittomeasuretouse?

a Thelengthofyourclassroomforsomenewcarpet

b Thedistancearoundyourheadforanewhat

c Thedistancearoundtheovalforarunningrace

d Thewidthofyourfingerforanewring

e ThedistancefromMelbournetoPhillipIslandforaschoolexcursion

8B Individual APPLYYOURLEARNING

1 Writeeachmeasurementinmetres.

2 Writeeachmeasurementinmetresandcentimetres. 125cm a 387cm b 514cm c 644cm d

3 Converteachmeasurementtocentimetres. 6m a 1m85cm b 3m10cm c 7m8cm d

4 Converteachmeasurementtocentimetres.

5 Converteachmeasurementtocentimetresandmillimetres.

6 Converteachmeasurementtomillimetres. 12cm a 130cm b 15cm2mm c 135cm9mm d

7 Writeeachmeasurementinkilometresandmetres. 6270m a 10000m b 2680m c 23780m d

8 Converteachmeasurementtometres. 11km a 5km123m b 6km90m c 7km3m d

9 Malikandhismothermeasuredawindowsothattheywouldknowwhatsize curtainstobuy.Thewidthofthewindowwas340cmandtheheightwas180cm. Thecurtainshopneededthemeasurementsinmillimetres.ConvertMark’s measurementstomillimetres.

10 a Howmuchisleftwhen188cmiscutfroma3mpieceofribbon? b Howmuchisleftwhen1m74cmiscutfroma3mpieceofribbon?

11 Benjamincut5piecesoftimberfroma5-metrelength.Eachpiecemeasured 700millimetres.Howlongwastheremainingpieceoftimber?

12 Kathswims10lapsofa25-metreswimmingpooleachweekday.Sheswims 20lapsonSaturdayandSunday.Howfardoessheswiminkilometresandmetres overthewholeweek?

13 Ifyouweremeasuringeachofthefollowing,whatwouldbethemostappropriate unittomeasuretouse?

a Theheightofyourbedroomforsomewallpaper

b Thedistancearoundyourdog’storsoforawintercoat

c thedistancefromyourhometoschool

d theheightofabakingtinforbakingaspongecake

14 Jamesismakinglabels.Hemakes10thatare45mmlong,10thatare75mmlong and10thatare95mmlong.Whatisthetotallengthofalllabelsinmetresand centimetres?

15 Thesearethetraveldistancesfromhometoschoolbyschoolbusforfivestudents travelingonthesameroute.

Zhi:12kilometres400metres

Jason:13kilometres200metres

Brock:12kilometres800metres

Gordon:5kilometres200metres

Sue:4kilometres500metres

a HowmuchfurtherisitforBrockandGordontotravelaltogetherthanforSue andJasontogether?

b Jasonmissedthebusandneedshisparentstodrivehimtothenextstop. Whosestopwillheneedtogettoandhowfarisit?

8C

Perimeter

Theword‘perimeter’comesfromtwoGreekwords: peri,meaning‘around’and metron,meaning‘measure’.Soperimetermeansthemeasureordistancearound something.Itisthelengtharoundtheedge.

Imaginewalkingaroundtheedgeofabasketballcourtandcountingeachmetreas youpaceitout.Youwouldwalk15metres,28metres,15metres,then28metres again,asyouwalkedallfoursidesoftherectangle.

Theperimeterofthebasketballcourtisthesumoftheselengths.

Perimeter = 15 + 28 + 15 + 28 = 86metres

Amoreefficientwaytocalculatetheperimeterofarectangle,wouldbetodoublethe lengthsofthetwoadjacentsidesandaddthem.

Perimeter =(2 × 15)+(2 × 28) = 30 + 56 = 86metres

Example8

ThisisMax’sgarden. Calculatetheperimeterof Max’sgarden.

Solution

IfMaxwalksaroundtheedgeofhisgarden,hewalks:

perimeter = 8 + 5 + 9 + 4 = 26m

TheperimeterofMax’sgardenis26metres.

Example9

Calculatetheperimeterofthistriangle.

Theperimeterofthetriangleisthesumofthelengthsofitssides.

= 13 + 12 + 5 = 30cm

Theperimeterofthetriangleis30centimetres.

Example10

Calculatetheperimeterofthis irregularhexagon.

Theperimeterofthehexagonisthesumofthelengthsofitssides.

Theperimeterofthehexagonis35centimetres.

8C Wholeclass LEARNINGTOGETHER

1 Workinpairs.Chooseasuitableunitformeasuringeachofthefollowing. Estimate,thenmeasure,theperimeterof: thecoverofthisbook a thetopofyourdesk b theclassroomdoor c thedooroftheclassroomcupboard. d Askyourpartnertocheckyourmeasurements.

8C Individual APPLYYOURLEARNING

1 Theseshapesaredrawnon1-centimetregridpaper.Calculatetheperimeterof eachshape.(Notdrawntoscale.)

2 Calculatetheperimeterofeachshapebelow.Allmeasurementsareincentimetres, soremembertoput‘cm’aftereachanswer.(Theshapesarenotdrawntoscale.)

3 Usewholesquareson1-centimetregridpaper.Drawshapesthathavea perimeterof:

Canyoudrawmorethanoneshapeforeachperimeter? e

Whichperimeterallowsyoutodrawthelargestnumberofdifferentshapes? Discussyouranswerwithafriend. f

4 Calculatetheperimeterofeachrectanglebyaddingthemeasurements,thendoubling them.The and = marksontherectanglesshowwhichsidesareequalinlength.

5 Themeasurementofonesideofeachsquareisgivenbelow.Sidesmarkedwitha dashareequalinlength.

Calculatetheperimeterofeachsquarebymultiplyingthesidemeasurementby4.

6 Usethemeasurementsforeachshapetoworkoutthemeasurementsthatarenot given.Thencalculatetheperimeter.

Theregioninsidethisrectanglehasbeenshaded.

Howcanwemeasurehowmuchofthepageiscoveredbytheshadedregion?We startwithasquarethathasasidelengthof1unit.Wecallthisa unitsquare.

Therectanglewewanttomeasureisincentimetres,soweuseaunitsquarethat measures1cm × 1cmandhasanareaof1 squarecentimetre.Theshortwayof writing1squarecentimetreis1cm2 .

Tofindtheareaoftherectangle,wecounthowmanysquarecentimetresfitinsideit.

Wecanseethat8unitsquaresfitinside,sotherectanglehasanareaof8cm2 .

Example11

Theserectangleshavebeendrawnon1-centimetregridpaper.Findtheareaof eachrectanglebycountingthesquarecentimetres.

Solution

a Therectangleismadeupof20unitsquares.Eachunitsquarecovers1cm2,so theareaoftherectangleis20cm2 .

b Therectangleismadeupof20unitsquares.Eachunitsquarecovers1cm2,so theareaoftherectangleis20cm2 .

Squarecentimetresareusefulformeasuringandcalculatingsmallareas,butalarger unitisneededformeasuringlargerareas,suchasabasketballcourtorthefloorofa classroom.Weuse squaremetres Remember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember RememberRemember Remember Remember Remember RememberRemember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember

Theareaofarectangleisthe‘size’ofthesurfaceinsideit.

Wemeasureareabycountingthenumberofunitsquaresthatfit insidetherectanglewithoutanyoverlap.

Thereisaquickerwayoffindingtheareaofarectanglethancountinglittlesquares. Wecanfindtheareaofarectanglebyfindingtheproductofitslengthandwidth.

Wecanseethisbydrawingeachsquarecentimetreinsidetherectangle.

Wehave3rows,eachcontaining5unitsquares.Sowehave3 × 5squaresintotal. Thesidelengthsfortherectangleaboveare3cmand5cm. Wecanfindtheareaoftherectanglebymultiplyingitslengthbyitswidth.

Area = length × width = 5 × 3 = 15cm2

Thisistheformulaforcalculatingtheareaofarectangle.Itworksforallrectangles.

Area = length × width

Thelengthandwidthmustusethesameunitofmeasurementandtheareawillthen measurethosesquareunits.

Asquareisaspecialtypeofrectangle.Itswidthanditslengthareequal.

Area = length × length = length2 (Wereadthisas‘lengthsquared’.)

Rememberthebasketballcourtwithasidelengthof28metresandwidth15metres.

Wecancalculatetheareaofthebasketballcourtbymultiplyingthelengthby thewidth.

Area = 28 × 15 = 420metres2

• Theformulaforcalculatingtheareaofarectangleis:

area = length × width

• Theformulaforcalculatingtheareaofasquareis:

area = length2

8D Wholeclass LEARNINGTOGETHER

1 Copytheserectanglesontothewhiteboardand,asaclass,discusshowto findtheareaofeach.

2 Samanthadrewarectanglewithanareaof24cm2.ShetoldherbrotherTom thatallrectangleswithanareaof24cm2 haveaperimeterof20cm.Doyou agreewithSamantha?Drawthreedifferentrectangles,eachwithanareaof 24cm2,thendiscusswhetherSamantha’sstatementistrue.

3 Tapetogetherorcutupsheetsofnewspapertomake: asquarewithsidelengthsof1metre i arectanglewithsidelengthsof50centimetresand2metres ii arectanglewithsidelengthsof25centimetresand4metres. iii

Calculatetheperimeterandareaofeachnewspapershape.Whatdoyou notice? a

Couldyoumakeanotherrectanglewiththesamearea?Whatmighttheside lengthsbe? b

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1 Theserectanglesaredrawnon1-centimetregridpaper.Whatistheareaofeach rectangle?

2 Drawthefollowingsquares,thenmarkinthe1cmgridlines.Calculatetheareaof eachsquare.

a Asquarewithasidelengthof4cm

b Asquarewithasidelengthof5cm

3 Drawthreedifferentrectangles,eachwithanareaof20cm2

4 Copythistable,thencalculatetheareaandperimeterofeachrectangle.

5 ThisisaplanofKim’shome.Calculatetheareaofeachroomandthehall.

6 Jacquisellsplasticgrassatapriceof $100persquaremetre.

a Adrianwantstocoveranareaof27m2.HowmuchwillJacquichargehim?

b Lucaswantstocoverasquarewithsidelengthsof6metres.Howmuchwill Jacquichargehim?

8E Reviewquestions–Demonstrateyourmastery

1 Writethesemeasurementsinmetres.

2 Writethesemeasurementsinmetresandcentimetres.

3 Convertthesemeasurementstocentimetres.

4 Convertthesemeasurementstocentimetres.

5 Convertthesemeasurementstocentimetresandmillimetres.

6 Convertthesemeasurementstomillimetres.

7 Convertthesemeasurementstokilometresandmetresortoawholenumberof kilometres.

8 Convertthesemeasurementstometres.

4km a

6km38m b 98km103m c

9 Catherinerodeherbicyclearoundthebiketrack8times.Eachlapwas1km450m. HowmanykilometresandmetresdidCatherineride?

10 Harrycutthreepiecesoftimberfroma3m20cmlength.EachpieceHarrycut measured220mm.Howlongwastheremainingpieceoftimber?

11a Howmuchofa7kmjourneyisleftwhenRitahastravelled2km800m?

b Howmuchofa7kmjourneyisleftwhenAidanhastravelled4227m?

12a Completethefollowingtable.

b Writetheorderforareafromsmallesttolargest.

c Writetheorderforperimeterfromsmallesttolargest.

d Whichrectangleschangedorderinparts b and c?

13 Therearesixrectanglesofthesesizes.

4mby2m A

5mby2m B

6mby2m C

9mby2m D

10mby2m E

15mby2m F

a Calculatetheareasofalloftherectangles.

b Whichtworectangleshaveatotalareaof50m2?

8F Challenge–Ready,set,explore!

Backyardritz

Youwillneedtodosomehomeworkfirstforthisactivity!

1 Drawanaccurateplanofyourbackyard.Drawittoscale.Forexample,youmight useascalewhere1metreequals2centimetres.

Ifyoudonothaveabackyard,drawaplanforabackyardthatcouldfitintoa space10metreswideand9metreslong.Includeawashingline,cubbyhouseand gardenbeds.

2 Calculatetheperimeterandareaofyourbackyard.

Challengequestions

1 Plan-Usehardwarestorecataloguestohelpyouplanyourdreambackyard. Youhaveabudgetof $10000.Staywithinyourpricelimit.Youmustincludean areafor: animals • playingsportandgames • entertaining • storage • flowers,plantsandgrass. •

2 Design-Produceadesignofyourdreambackyardtoscale.Yourdesignmustfit ontoanA3pieceofpaper.

3 Costs-Provideanaccuratecostbreakdownforallitems.Youwillbesupplyingall ofthelabouryourself,sotherewillbenocostsfortradesotherthanelectricians andplumbersiftheyareneeded.

4 Time-Estimatehowlongeachjobwilltaketocomplete.Drawupatimelineof wheneachpartoftheconstructionwillstartandend.Includeastartdateandan enddate.

5 Construct-Makea3Dmodelofthedesign.

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Usefulskillsforthistopic

• readingscalesontapemeasuresandmeasuringcontainers

• measuringlengthusingarulerortapemeasure

Vocabulary

Grams • Kilograms • Cubicmillimetres • Cubiccentimetres • Cubicmetres

• Litres • Millilitres • Scale • Calibrations • Volume • Capacity • Tonnes • Milligrams • Mass • Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage

Discusswithapartner

Imagineyouarehelpingtosetupascienceexperimentinvolvingsomecooking.You willhaveaccesstoarangeofresourcesbutneedtomakesomedecisionsbeforethe restoftheclassarrive.

Atableisinthecentreoftheroomforworkingaroundandplacingingredientson. Whatmeasurementsmightyouneedtoknowaboutthistableandwhy?Howwould yourecordthemeasurements?

Abagofflourisstoredonanearbybench.Whatmeasurementsmightyouneedto knowabouttheflourandwhy?Howwouldyourecordthesemeasurements?

Anemptycleanwaterbottleisalsoavailable.Whatmeasurementsmightyouneedto knowabouttheflourandwhy?Howwouldyourecordthesemeasurements?

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Volume isthespaceanobjecttakesup.Itismeasuredin cubes.Weneedtoknowaboutvolumewhenpackingboxes, fillingawatertank,orcalculatingtheamountofsandneeded forasandpit.

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Mass istheamountofmatterinanobject, whichweexperienceasweight.Itismeasured inmilligrams,gramsandkilograms.Weneedto knowaboutmasswhenmakingsureour backpackisn’ttooheavy,measuringingredients forbakingacake,orensuringourpetgetsthe rightamountoffood.

Capacity istheamountofliquidanobjector containercanhold.Itismeasuredinmillilitres, litresandkilolitres.Weneedtoknowabout capacitywhencheckingifawaterbottlecan holdenoughdrinkforatripormakingsureour fishtankhasenoughwaterforourpetfish.

9A Volume

Thisisanopenrectangularbox.Themeasurementsof thisboxare:

length = 3cm, width = 5cm, height = 2cm

Itdoesnotmatterwhichmeasurementswecallthelength,widthorheight.Ifweturn theboxaround,itsdimensionsarethesame.

Howcanwemeasurethevolumeofthisbox?Todothis,westartwithacubeofside length1cmandcallita‘unitcube’.

Wesayitsvolumeis1cubiccentimetrebecauseitis1cm × 1cm × 1cm.Theshortway ofwriting1cubiccentimetreis1cm3

Agoodexampleofa cubiccentimetre (1cm3) isabase-10one.Acenticubeisalsoa cubiccentimetre.Wecanusebase-10onesorcenticubestomeasurehowlargethe rectangularboxis.

Thisdiagramshowstheboxmadeupofunitcubes.Ithastwolayers.Eachlayeris showninadifferentcolour.

Eachlayercontains3cubesinitswidthand5cubesinitslength,making3 × 5 = 15. Eachunitcubehasavolumeof1cm3,sothismeansonelayerhasavolumeof15cm3

Thereare2layersofcubes.Sothevolumeoftherectangularboxis2 × 15,or 2 × 3 × 5cm3,making30cm3

Thevolumeofarectangularboxincubiccentimetresisthenumberofcentimetre cubesrequiredtomakeit.

Asolidintheshapeofarectangularboxiscalledarectangularprism.Ithas6faces. Eachfaceisarectangle.

Acubeisaspecialkindofrectangularprism.Eachofits6facesisasquare.

Example1

Findthevolumeofarectangularboxmeasuring5cmlong,8cmwideand 3cmhigh.

Solution

Usebase-10onestoconstructthe rectangularbox.

Eachlayerhas5 × 8cubes,or40cm3 . Thereare3layers.

Sothevolumeoftherectangularbox: = 5 × 8 × 3 = 120cm3

Example2

Findthevolumeofacubemeasuring3cmlong,3cmwideand3cmhigh.

Solution

Eachlayerhas3 × 3cubes,or9cm3 . Thereare3layers.Thevolume: = 3 × 3 × 3

27cm3

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Volumeisameasurementoftheamountofspacesomethingtakesup. Thevolumeofarectangularboxincubiccentimetresisthenumberof 1cm × 1cm × 1cmcubesthatfitinsideit.

1 Usebase-10onesorcenticubestobuildtheserectangularprisms.Thenfindthe volumeofeachrectangularprismbycountingthenumberofcubes.

2 a Usebase-10onesorcenticubestobuild therectangularprismontheright.Then finditsvolumebycountingthenumberof cubesused.

b Usethesamenumberofcubesto constructthreeotherrectangularprisms, eachwiththesamevolume.Sketchyour threeprisms.

c Lookattheprisminpart a.Workoutthenumberofcubesitcontainsby multiplyingitslength,widthandheight.

9A Individual APPLYYOURLEARNING

1 Writethesemeasurementsincm3.Thefirstonehasbeendoneforyou. 10cubiccentimetres = 10cm3

a 25cubiccentimetres=______

b 60cubiccentimetres=______

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c 48cubiccentimetres=______

2 a Usebase-10onesorcenticubestobuildtheserectangularprisms.

A B C

b Countthenumberofblocksyouuseineachlayer,thenaddthemtogetherto findthevolumeofeachprism.

c Listtheprismsinorderofvolume,fromsmallesttolargest.

d Workoutthenumberofcubesineachprismbymultiplyinginsteadofcounting.

3 Usebase-10onesorcenticubestobuildrectangularprismswiththesedimensions. Countthenumberofblocksyouuseandfindthevolumeofeachprism.

a 4cubes 2cubes 2cubes ____cm3

b 4cubes 3cubes 2cubes ____cm3

c 5cubes 2cubes 2cubes ____cm3

d 3cubes 2cubes 2cubes ____cm3

9B

Theformulaforcalculating volume

Wecanfindthevolumeofarectangularprismby findingtheproductofitslength,widthandheight. Thisisquickerthancountinglotsoflittlecubes.

Thesidelengthsofthisrectangularprismare 3cm, 5cmand4cm.

Eachlayeroftheprismhas3 × 5 = 15cubic centimetres.

Thereare4layers,sowehave3 × 5 × 4cm3 = 60cm3 intotal.

So,thevolumeoftheprismistheproductofitslength,itswidthanditsheight.

Volume = length × width × height

= 3 × 5 × 4 = 60cm3

Thisistheformulaforcalculatingthevolumeofarectangularprism.

Itworksforallrectangularprisms.Makesureyouhavethesameunitforthelength, widthandheight.

Volume = length × width × height

Example3

Calculatethevolumeofarectangularprismwithlength6cm,width3cmand height2cm.

Solution

Volume = length × width × height

= 6cm × 3cm × 2cm = 36cm3

Acubeisaspecialrectangularprismbecauseitslength,widthandheightareequal.

Theformulaforfindingthevolumeofacubeis:

Volume = length × length × length = length3

Calculatethevolumeofacubewithsidelength4cm.

Solution

Volume = length3 = 4cm × 4cm × 4cm = 64cm3

9B

Wholeclass LEARNINGTOGETHER

1 a Collectsomesmallboxesandmeasuretheirlength,widthandheightto thenearestcentimetre.

b Usetheformulaforcalculatingthevolumeofarectangularprismto estimatethevolumeofeachbox.(Itisnotacompletelyaccurate measurementbecauseyouhaveroundedthemeasurementsforlength, widthandheight.)

c Writealabelforeachboxexplaininghowyoucalculateditsvolume.

2 Buildamodelofacubiccentimetreusingclay,plasticine,paperor cardboard.

9B Individual APPLYYOURLEARNING

1 Calculatethevolumeofeachrectangularprism.(Theyarenotdrawntoscale.)

2 Calculatethevolumeofeachrectangularprism.

3 Calculatethevolumeofacubethathassidelength:

4 a Calculatethevolumeofaboxwithlength14cm,width13cmand height15cm.

b Whatisthevolumeofriceintheboxifitisfilledtoaheightof10cm?

9C Cubicmetres

Itisnotalwayspracticaltouseasmallmeasuringunitlikeacubiccentimetrefor measuringvolume.Theobjectmightbesolargethatyougethugenumbers.

Weusealargerunitofmeasurementforlargerobjects:weusethemetre.The cubic metre isusedwhenmeasurementsaremadeinmetres.

Weusethemetretomeasurethelengthof largeobjects.Thebasicunittomeasuretheir volumeisacubeof1m × 1m × 1m.Itsvolume iscalled‘onecubicmetre’.

Wethenusetheformulaforcalculating volumetocalculatethevolumeincubic metres.

Example5

Thisshippingcontainermeasures6m × 2m × 2m. Calculateitsvolume.

Example6

Calculatethevolumeofarectangularprismwithdimensionslength = 50cm, width = 2mandheight = 3m.

Solution

Firstconvert50cmtometressothatallunitsarethesame:

50cm = 0 5m

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LEARNINGTOGETHER

1 Namethreeobjectsthathaveavolumeof:

a about1m3

b morethan1m3

c lessthan1m3

2 Youaregoingtomakeamodelofacubicmetre.

a Predictwhetheryouwillbeabletofititthroughthedooroftheclassroom orthroughawindow.

b Userolled-upnewspaperandtapetomakeahollowstructurewith dimensions:length = 1m,width = 1mandheight = 1m.Therolled newspaperwillbecometheedgesofaboxwithavolumeof1m3 .

c Howmanypeoplecancomfortablyfitinsideyourcubicmetre?

d Howmanycubiccentimetresequalonecubicmetre?

3 Writethemeasurementsoffourdifferentrectangularprismsthathavea volumeof36m3

4 a Makeanestimate(orapproximation)ofthevolumeofyourclassroomby measuringitslength,widthandheighttothenearestmetre.

b Airhasamassofapproximately1 2kilogramsforeverycubicmetreata temperatureof20° C.Assume theclassroomtemperatureisaconstant20° C.Calculatetheapproximatemassoftheairinyourclassroom.

1 Classifytheobjectsbelowintothesethreegroups:

Volumeisless than1m3

Volumeismorethan1m3 butlessthan10m3

Volumeismore than10m3

Atelephonebox a Aswimmingpool b

Aschoolbag c Yourclassroom d

Yourbathroomathome e Ashoebox f

2 Calculatethevolumeofeachrectangularprism.(Theyarenotdrawntoscale.)

3 Calculatethevolumeoftherectangularprismsinthistable.

a 5m 4m 3m ____m3

b 2m 10m 4m ____m3

c 12m 5m 2m ____m3

d 6m 3m 10m ____m3

e 7m 10m 4m ____m3

4 Calculatethevolumeofacubewithsidelength:

a 3m

b 7m

c 10m

5 Calculatethevolumeoftheserectangularprisms.Length = L, Width = Wand Height = H.Givethevolumeincubicmetres.

a L = 20cm, W = 3m, H = 2m

b H = 4m, W = 50cm, L = 6m

c H = 4m50cm, L = 1m, W = 0.05m

d W = 0 02m, L = 0 12m, H = 10m

6 Thisisarectangularprism.

Height = 4 m

Width = 5 m

Length = 10 m

a Calculatethevolumeoftherectangularprism.

b Whathappenstothevolumeifyoudoubleonlythelength?

c Whathappenstothevolumeifyoudoublethelength and thewidth?

d Whathappenstothevolumeifyoudoubleallofthedimensions?

9D Mass

Theunitsofmeasurementweuseformeasuringmassaremilligrams,grams,kilograms andtonnes.Wecanconvertfromoneunittoanotherbymultiplyingordividingby 1000.

Thebasicunitformeasuringmassisthe kilogram (kg).

Theprefix‘kilo’meansonethousand.Thereare1000 grams (g)in1kilogram.

1000grams = 1kilogram

Ifwehave2kgandwewanttoknowhow manygramsthatis,wemultiplyby1000.

1kgisthesameas1000g

So2kg = 2 × 1000g = 2000g

Ifwehave3000gandwewanttoknowhow manykilogramsthatis,wedivideby1000.

1000g = 1kg

So3000 ÷ 1000 = 3kg

Apaperclipweighsabout1gram,soa kilogramofpaperclipswouldbeabout 1000paperclips!

Example7

WhenJosephwasbornheweighed5kilograms. Howmanygramsisthat?

Solution

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Weneedtomultiplythekilogramsby1000tofindthenumberofgrams.

5kg = 5 × 1000g = 5000g

Thereare5000gramsin5kilograms.

Example8

WhenLachlanwasbornheweighed3585grams. Howmanykilogramsisthat?

Solution

Weneedtodividethegramsby1000tofindthenumberofkilograms.

3585g = 3585 ÷ 1000kg = 3 585kg

Thereare3 585kgin3585grams.

Forverysmallamountsweuse milligrams (mg).Apinchofsaltweighs about1milligram.

Theprefix‘milli’meansone-thousandth.

Onemilligramis 1 1000 ofagram(g).

So1000milligrams = 1gram

Example9

Ritadrinkstwocupsofmilkperday.Onecupofmilkcontains300milligramsof calcium.Therecommendeddailyintakeforachildof11is0 9grams.IsRitagetting enoughcalciumfromthe milkshedrinks?

Solution

1000mg = 1g.Toconvertmilligramstograms,divideby1000.

300mg = 300 ÷ 1000g = 0.3g

Twocupsofmilkis0.6gofcalcium.Ritaneedsatleastthreecupsofmilktoget herrecommendeddailyintake ofcalcium.Sotwocupsisnotenough.

Forveryheavyobjectsweusetonnes(t).Thereare1000kilogramsin1tonne(t).

So 1000kilograms(kg) = 1tonne(t).

Example10

Dale’scarweighs1587kilograms.Howmanytonnesisthat?

Solution

1t = 1000kg.Toconvertkilogramstotonnes,divideby1000.

1587kg = (1587 ÷ 1000) t = 1.587t

Dale’scarweighs1.587tonnes.

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Thestandardunitofmeasurementformassisthekilogram.

Toconverttonnestokilograms,kilogramstogramsorgramsto milligramsmultiplyby1000.

Toconvertmilligramstograms,gramstokilogramsorkilogramsto tonnes,divideby1000.

9D Wholeclass LEARNINGTOGETHER

1 Youwillneedakilogramweightandasetofscales.

a Drawachartwith3columnswiththeseheadings: ‘Lessthan1kg’;‘About1kg’and‘Morethan1kg’.

b Comparedifferentobjectsintheclassroomtothe1kgmassbyholding eachinyourhand.

c Drawapictureofeachobjectintheappropriatecolumn.

d Finally,measurethemassofeachobjectusingthescalesandwritethe measurementunderneaththepictureoftheobject.

2 Workingroupsofthree.Compareakilogrammasswithtwo500gmasses. Dotheyfeelthesame?

a Howmanykilogramsaretherein500g?

b Next,comparefour250gmasseswithasinglekilogrammass.Dothey feelthesame? Howmanykilogramsaretherein250g?

3 Estimatehowmanyoftheseitemsyouwouldneedtohaveatotalmassof 1 2 kg.Checkyourestimateusingscales.

Mathsbooks a Dictionaries b Pencils c

4 Converttograms. 1250mg a 3400mg b 3500mg c 275mg d 3kg e 12kg f 0.03kg g 124kg h

5 a Estimatethemassofa1-centimetre-cubeblock.Weighit.

b Itmightbehardtoaccuratelymeasurethemassof1block.

Sothistimeweighahundred1-centimetreblocksanddividetheresult by100.Wasthemassof1blockdifferenttowhatyoufoundinpart a? Discusswhichisthemoreaccuratemeasurement.

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9D Individual APPLYYOURLEARNING

1 Write‘morethan 1 2 kg’or‘lessthan 1 2 kg’foreach.

a 0.6kg b

c 0.2kg d 510g e 0.3kg f

2 Write‘morethan 1 4 kg’or‘lessthan 1 4 kg’foreach.

.5kg a

3 Converttograms.

4 Writethesemassesinkilograms.

5 Convertthesemeasurementstotonnes.

6 a Howmanykilogramsaretherein3560g?

b Howmanygramsaretherein0.7kg?

c Howmanygramsare therein450mg?

d Howmanykilogramsaretherein11 4 t?

e Howmanytonnesandkilogramsaretherein2500kg?

f Howmanymilligramsaretherein41 4 g?

7 a Joannabought750gofpeas.Howmanykilogramsdidshebuy?

b Joelcarried13.68kgofbricksinhiswheelbarrow.Howmanygramsdid hecarry?

c Chrisput0.125kgofbutterinthecakemixture.Howmanygramsof butterdidChrisuse?

8 TobuildhisdrivewayLesneeds8250kgofsand,4850kgofgraveland 10500kgofcrushedrock.Leswantstogetallofhismaterialsdeliveredin containersbyasingletruck.Isthispossibleifthetruckcancarryamaximum loadof25tonnes?

9E Capacity

Thevolumeofacontainerorprismcanbemeasuredincubiccentimetresorcubic metres.Thereisaspecialwayofmeasuringvolumewhenliquidsorgasesareinvolved. Weuse millilitres (mL)or litres (L).

1000millilitres = 1litre

Weuseavarietyofdifferentmeasuringcontainerstomeasureliquidvolume.

Thewordcapacityisusedtodescribehowmuchliquidacontainercanhold.A1litre jughasacapacityof1litre,evenifitdoesnotactuallyhaveanyliquidinit.

Measuringjugsandcontainershave scales ontheirsidesthataremarkedwithlines. Theselinesarecalled calibrations orgraduatedscales,andtheyenableyoutomeasure liquidsaccurately.

Whenmeasuring,itisimportanttohaveyoureyelevelwiththetopoftheliquidinthe container.Thisenablesyoutoreadthescaleaccurately.

Didyouknow? 1millilitreofwaterweighs1gramandhasavolumeof1cm3

Example11

a Alidrank1litre250millilitresofsoftdrink.Howmanymillilitresisthat?

b Nadiadrank1340millilitresofjuice.Whatisthatinlitresandmillilitres?

Solution

a 1litre250millilitres = 1000mL + 250mL = 1250mL

b 1340millilitres = 1000mL + 340mL = 1L340mL

Litres(L)andmillilitres(mL)areusedtomeasurethevolumeofliquids andthecapacityofcontainers.

9E Wholeclass LEARNINGTOGETHER

1 Estimate,thenmeasure,thecapacityofthesecontainersinmillilitres.

2 Findtwocontainersthatyouestimatewillholdlessthan1litreofwater. Estimatehowmanymillilitreseachcontainerholds.Checkyourestimatesby pouringwaterintoameasuringjugmarkedinmillilitres.

3 Findtwocontainersthatyouestimatewillholdmorethan1litreofwater. Estimatehowmanylitreseachcontainerholds.Checkyourestimatesby pouringwaterintoameasuringjugmarkedinlitres.

9E Individual APPLYYOURLEARNING

1 Wouldyouusemillilitres(mL)orlitres(L)tomeasuretheamountofliquidin: thepetroltankofacar? a ateacup? b amedicinebottle? c abucketofwater? d asoupbowl? e alargefireextinguisher? f

2 Readthescaleforeachmeasurement.

3 Convertthesemeasurementstomillilitres. 1litre a 2litres b 5litres200millilitres c 27litres d 7litres100millilitres e 13litres100millilitres f

4 Convertthesemeasurementstolitres,orlitresandmillilitres.

5 Annabellahasa2-litre,a3-litreanda1.5-litrecontainer.Shehasabucketwith 4750mLofwaterinit.Howmuchmorewaterwillsheneedtofillthethree containers?

6 Acontainerfullofoilis15cmlong,10cmdeepand30cmhigh.Whatvolumeof oildoesitcontain?

7 Whichrectangularcontainerholdsmoreliquid?

L = 10cmH = 4cmW = 5cm a

L = 12cmH = 3cmW = 6cm b

L = 9cmH = 7cmW = 3cm c

8 Victoriafilledher130cmlongrectangularbathwithwater.Thebathis60cmwide and40cmdeep.Whatvolumeofwaterisinthebathifshefillsthebathto: 5cm? a 10cm? b 27cm? c 40cm? d

9F Reviewquestions–

1 Calculatethevolumeofeachrectangularprism.(Theprismsarenotdrawnto scale.)

2 Calculatethevolumeofrectangularprismswiththefollowingdimensions.

3 Thisfigurewasmadefromcenticubes.

a Whatisitsvolumeincm3?

b Howmanymoreunitcubesareneededtomakea7 × 3 × 4cm3 rectangular prism?

4th

4 Calculatethevolumeofacubewithsidelength: 4cm a 7cm b 3m c 22mm d

5 Lookatthedimensionsofthisrectangularprism.

Height = 3 m

Width = 4 m

Length = 8 m

a Calculatethevolumeoftheprism.

b Whathappenstothevolumeifyoumultiplythelengthby3?

c Whathappenstothevolumeifyoumultiplythelength and thewidthby3?

d Whathappensifyoumultiply all ofthedimensionsby3?

6 Classifytheobjectsbelowintooneofthesethreegroups.

Volumelessthan1litre Volumebetween1litreand3litres Volumemorethan3litres

Abowlofsoup a Aswimmingpool b Acupofcoffee c Alargebottleofsoftdrink d Alaundrysink e Abucket f

7 Calculatethevolumeoftheserectangularprisms.Givethevolumeincubicmetres.

8 Readthescaleforeachmeasurement.

9 Patpoured4800mLinto4containers,fillingthemtothetop.Eachcontaineris 20cmhighand12cmwide.Howdeepiseachcontainer?

10 Agreengrocerknowsthateachlargeappleheissellingweighsabout150g.

a Acustomeraskshimfor11 2 kgofapples.Howmanyapplesshouldhepickout?

b Anothercustomerwantsabout1kgofapples.Howmanyshouldhepickout?

c Discusshowyouworkedthisout.

9F Challenge–Ready,set,explore!

Buildingblockchallenge

Inthischallenge,wewillexplorehowyoucanalsofindthevolumeofobjectsthatare notrectangularprisms.Readthroughtheexamplesbelowandcompletethequestions.

Example12

Thisobjectismadeupof1cm × 1cm × 1cmcubes.

a Countthecubestofindthevolumeof theobject.

b Howmanymorecubeswouldyouneedto makea5cm × 1cm × 3cmrectangularprism?

Solution

a Countingcubesgivesatotalvolumeof11cm3

b A5cm × 1cm × 3cmrectangularprismhasa volumeof15cm3

Theobjectneedsfourmoreunitcubesto becomea5cm × 1cm × 3cmrectangularprism.

Someobjectshavecubesthatarehidden.Youneedtogetusedtopicturingor‘seeing’ thehiddencubes.

Example13

Thisobjectismadefrom2cm3 cubes.

a Countthecubestofindthevolumeoftheobject.

b Howmanymorecubesareneededtomakea 2cm × 2cm × 2cmcube?

Solution

a Countingcubes,includingthecubehiddeninthecorner,givesatotalvolume of4cm3 .

b A2cm × 2cm × 2cmcubehasavolumeof8cm3.Theobjectaboveneeds fourmore1cm3 cubestobecomea2cm × 2cm × 2cmcube.

Challengequestions

1 Thisobjectwasmadefrom1cm × 1cm × 1cmcubes.

a Howmany1cm × 1cm × 1cmcubeshavebeenusedtomakeit?

b Howmanymorecubesareneededtomakea9cm × 1cm × 3cmprism?

2 Thisobjectwasbuiltfrom1cmcubes.

a Howmany1cm × 1cm × 1cmcubeswereusedtobuildthisobject?

b Howmanymore1cm3 cubeswouldyouneedtomakea3cm × 3cm × 3cm cube?

3 Thisstaircasewasbuiltfrom1cmcubes.

a Howmany1cm × 1cm × 1cmcubeswereusedtobuildthesestairs?

b Howmanymore1cm × 1cm × 1cmcubeswouldyouneedtomakea 4cm × 4cm × 4cmcube?

4 Thisobjectwasmadefrom1cm × 1cm × 1cmcubes.

a Howmany1cm × 1cm × 1cmcubeswereused?

b Howmanymore1cm × 1cm × 1cmcubeswouldyouneedtomakea 4cm × 3cm × 2cmrectangularprism?

5 Thisobjectwasmadefrom1cm × 1cm × 1cmcubes.

a Howmany1cm × 1cm × 1cmcubeswereused?

b Howmanymore1cm × 1cm × 1cmcubeswouldyouneedtomakea 5cm × 5cm × 2cmprism?

c Howmanydifferentwayscanyouworkouttheanswertopart b?Showat leastoneothersolution.

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Usefulskillsforthistopic

• understandingoftherelationshipbetweenunitsoftime

Vocabulary

Antemeridiem

•

• Postmeridiem

Duration

• Elapsedtime

• Second

•

• Time

Lookattheclocksbelow.Oneismissingtheminutehandandtheotherismissingthe hourhand.

Discussthesequestions:

1 Whichofthehandsistheminutehandandwhichisthehourhand?

2 Ontheclockwiththeminutehandonly,istheminutehandpastortothehour? Explainyourthinking.

3 Howmanyminutespast/tothehouristheminutehand?Explainyourthinking.

4 Ontheclockwiththehourhandonly,isthehourhandjustpastthehouror closertothenexthour?Explainyourthinking.

5 Wheremighttheminutehandbeifyouaddedittothisclock?Explainyour thinking.

Measurement Time Time Time TimeTime Time Time Time Time TimeTime Time Time Time Time Time TimeTime Time Time Time Time Time Time Time TimeTime Time Time Time Time Time Time Time TimeTime Time Time Time Time Time Time TimeTime Time Time Time Time Time Time Time TimeTime Time Time Time Time Time Time Time TimeTime Time Time Time Time Time Time Time TimeTime Time Time Time Time Time Time Time TimeTime Time Time Time Time

Inthischapterwearegoingtolookat time. Thebasicunitofmeasurementfortimeisthe second (s).

• Thereare60secondsin1minute.

• Thereare24hoursin1day.

• Thereare60minutesin1hour.

• Thereare7daysin1week.

Wecanalsousefractionswhenwerecordtime.

Thereare30minutes inhalfanhour.

Thereare15minutes inaquarterofanhour.

10A Recordingtime

Therearetwowaysofrecordingthetimeofday.

Usinga 12-hourclock

Whenweusethe12-hoursystemthedayisbrokenupintotwo12-hourblocks. 11:30a.m.meanseleventhirtyinthemorningand11:30p.m.meanseleventhirtyin theevening.

Wewrite‘a.m.’toshowthatwemeanthemorning.Theletters‘a.m.’comefromthe Latinwords‘antemeridiem’meaning‘beforenoon’.

Wewrite‘p.m.’to showthatwemeantheafternoonorevening.Theletters‘p.m.’ comefromtheLatinwords‘postmeridiem’meaning‘afternoon’.

Midnightiswrittenas 12:00a.m.andmiddayiswrittenas12:00p.m.

Usinga 24-hourclock

Whenweuse 24-hourtime,alltimesaremeasuredfrommidnightonedayuntil midnightthenextday.So11:30a.m.iswrittenas1130and11:30p.m.iswritten as2330.

Wedonotusea.m.orp.m.withthe24-hourclock.Midnightiswrittenas0000and middayiswrittenas1200.

Twenty-four-hourtimeismostoftenusedwhenitisimportanttoavoidconfusion aboutatimethatcouldeitherbemorningorevening,forexample,inthearmed forces,byairlinesforflighttimesandinhospitals.

Thesemorningtimesarerecordedinthiswayusingthe24-hourclock:

6:00a.m.iswrittenas06006:05a.m.iswrittenas0605

7:00a.m.iswrittenas07007:05a.m.iswrittenas0705

8:00a.m.iswrittenas08008:05a.m.iswrittenas0805

Using24-hourtimewerecordtheseafternoontimesinthisway: 1:00p.m.iswrittenas13001:05p.m.iswrittenas1305 2:00p.m.iswrittenas14002:05p.m.iswrittenas1405 3:00p.m.iswrittenas15003:05p.m.iswrittenas1505

12-hour

Example1

Grace’sbedroomclockshows1745.Whattimeisthatin12-hourtime?

Solution 1745isafter1200soitisafternoon.1745is5hoursand45minutesafter1200, soitis5:45p.m.

Example2

JustintoldColbythetimewas7:20p.m.Whattimewasthatin24-hourtime?

Solution ‘p.m.’meansitisafternoon.In24-hourtime,noonis1200,so7:20p.m.is7hours and20minutesafternoon,whichis1920.

10A Wholeclass LEARNINGTOGETHER

1 Workwithapartnertodrawatimelineshowing24hoursfrommidnighton 1daytomidnightthenextday.Writeinthe12-hourtimesalongthebottom ofthetimeline.Writethe24-hourtimesalongthetopofthetimeline.

Usethetimelinetohelpyouconvertthese12-hourtimesto24-hourtime. 8a.m.

2 Usethetimelineyoumadewithyourpartner.Onepersonsaysatimein either12-houror24-hourtimeandtheotherpersonconvertsittotheother timeformat.Repeat,swappingroles.

3 Convertthese24-hourtimesto12-hourtimes.(Remembertowritea.m.orp.m.)

10A Individual APPLYYOURLEARNING

1 Changethese12-hourtimesto24-hourtimes.

2 Changethese24-hourtimesto12-hourtimes.

3 Changethese24-hourtimesto12-hourtimes.

4 Writeeachofthesetimesin12-hourand24-hourtime.

Twoo’clockintheafternoon a Teno’clockinthemorning b Eleveno’clockintheevening c Twenty-threeminutespast5inthemorning d

5 Putthesetimesinorder,startingatmidnight.

3:45p.m.midnight13107:15p.m.5:05a.m. noon232010:25p.m.04502055

6 Dadputontheroastat4:30p.m.Itfinishedcooking2hourslater.Whattime diditfinishin24-hourtime?

7 Thecarclockshowed19:45.Whattimeisthatin12-hourtime?

8 Write20minutesto6inasmanydifferentwaysasyoucan.

10B

Timeduration

Sometimesweneedtoknowhowmuchtimehaspassedfromthestarttotheendof anevent,thisis timeduration.Forexample,whenIammakingacakethatneeds 35minutestocook,IneedtoknowwhattimeIshouldtakeitoutoftheoven. Thereareanumberofwaysofdoingthis.

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Onewayistocountonfromonetimetoanother:

Iputthecakeintheovenat2:50p.m.

Buildupfrom2:50p.m.to3:00p.m. = 10minutes

35minutes 10minutes = 25minutestogo

Iadd25minutesto3:00p.m.andget3:25p.m.

SoIwouldtakethecakeoutat3:25p.m.

Example3

a Sharnilefthomeat8:15a.m.towalktoschool.Shearrivedatschoolat 8:40a.m.Howlongdidittakehertowalktoschool?

b Whenwalkinghomefromschool,Sharnileftat3:30p.m.andinsteadof walkingstraighthomeshewenttotheshops.Shegothomeat4:28p.m. Howlongdidittakehertowalkhomefromschool?

Solution

a Bothtimesarebetween8:00a.m.and9:00a.m.

Sowesubtract15from40.Ittakes25minutes.

b Sharnistartsat3:30p.m.

Buildupfrom3:30to4:00 = 30minutes

Buildupfrom4:00to4:28 = 28minutes

Add30minutesand28minutes = 58minutes

IttookSharni58minutestowalkhomefromschoolviatheshops.

Timetables

Weneedtimetablestohelpusknowwhenwehavetodothings. Wehavetimetablesatschoolandforbuses,trainsandtelevision.

Calculating elapsedtime helpsustounderstandandusetimetables.

Example4

ThisisaYear5∕6timetable.HowmuchtimeisspentdoingMathseachweek?

Time

9:00–10:00 English English Maths English Maths

10:00–11:00 English English English English English

11:00–11:20 RECESS

11:20–12:30 Society Maths English Music Library

12:30–13:30 LUNCH

13:30–14:30 Music PE Art Maths Science

14:30–15:15 Maths Health Art Italian PE

Solution

Monday:14:30–15:15 = 45minutes

Tuesday:11:20–12:30 = 70minutes

Wednesday:9:00–10:00 = 1hour

Thursday:13:30–14:30 = 1hour

Friday:9:00–10:00 = 1hour

Total:3hours115minutes,whichis4hours55minutes

10B

Wholeclass LEARNINGTOGETHER

1 a Startat3:00p.m.Counton3hours.Whattimeisitin24-hourtime?

b Startat2210.Counton40minutes.Whattimeisit?

c Whattimeisit10minutesafter2:55p.m.?

d Whattimeisit3hoursbefore12:00p.m.?

e Whattimeisit3hoursbefore2:10p.m.?

2 a Calculatetheamountoftimeyouspendeachdayatschool.Howmuchis thisperweek?

b Calculatetheamountoftimeyouspendinclasseachdaywhenyouare atschool.Howmuchisthisperweek?

c Howmuchof1weekdorecessandlunchtimetakeup?

10B Individual APPLYYOURLEARNING

1 FourfriendsenteredtheSouthernDistrictFunRun.TheFunRunstartedat 11:30a.m.Herearethetimeswheneachofthefriendsfinishedtherun. CalculatehowlongeachofthemtooktofinishtheFunRun.

Name Finishingtime

Anton 1156

Sienna 1227 Ryan 1238 Georgia 1304

a Whoranthefastesttime?

b Whocamesecondamongthefriends?Howlongdidthatpersontake?

c Whatwasthedifferenceinthetimesoftheslowestandthefastestrunner?

2 Ittakes3minutestocookanegginboilingwater.Fionaputsanegginto boilingwaterwhentheclockisshowing3:58p.m.Whenshouldshetakeit out?

3 Add45minutesontothesetimes. 1725 a 11:55a.m. b 0504 c 2123 d 2339 e

4 Whatisthetime50minutesbeforethesetimes? 3:30p.m. a 1:27p.m. b 12:42a.m. c 1821 d 0026 e

5 Howmuchtimeuntilmidday?

d 1104 e 0731 f

6 Kyliewenttosleepat9:38p.m.Shewokeupat6:56a.m.Howlongwas Kylieasleep?

10C Reviewquestions–Demonstrateyourmastery

1 Changethese12-hourtimesto24-hourtimes.

a 6a.m.

b 1:25p.m.

c 3:38p.m.

d 11:59p.m.

2 Changethese24-hourtimesto12-hourtimes.

a 1400

b 0432

c 1358

d 1845

e 2127

3 Writeeachofthesetimesin12-hourand24-hourtime.

a Fouro’clockintheafternoon

b Fiveo’clockinthemorning

c Twelveminutespastfiveintheevening

d Twenty-sevenminutestosixinthemorning

e Quarterpasteightintheevening

4 TarrynsetherDVDplayerclockbyherwatch,thenwentoutsidetospray-paint hercar.Shelookedatherwatchwhenshestartedpaintingthecar.Itsaidthetime was11:30a.m.Whenshefinished,theclockonherDVDplayersaid1708.How longdidittaketopaintthecar?

5 Add25minutesontothesetimes.

a 1925

b 10:55p.m.

c 0738

d 2123

6 Whatisthetime1hourand20minutesbeforethesetimes?

a 10:30a.m.

b 11:15p.m.

c 2:02a.m.

d 1433

10D Challenge–Ready,set,explore!

MrFunnyman’sclock

MrFunnymandecidedtochangethewaywemeasuretimeontheclock.Heinvented aclockwithadecimalface.

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Thefirstdigittellshowmanytenthsofamorningorafternoonhavegoneby.Call theseFH(forfunnyhours).

1FHisthesameas 12 10 hours.

Eachfunnyhourisdividedinto10funnyminutes.

So1FM= 1 10 FH= 12 100 hours.

Wecanconvertfunnyhoursandfunnyminutestostandard12-hourtime.

1FH= 12 10 hours

=1hour + 2 10 hour

=1hour + 12minutes

1FM= 1 10 FH

= 1 10 × 1hour+ 1 10 × 12minutes

=6minutes+ 1 10 × 10minutes+ 1 10 × 2minutes

=6minutes+1minute+ 2 10 minute

=7minutes+12seconds

So1.4infunnytimecanbeconvertedtostandard12-hourtime.

1FH=1hour+12minutes

4FM=4× (7minutes+12seconds)

1FH4FM=1hour+40minutes+48secondsona12-hourclock

Challengequestions

1 Givethe12-hourtimeforeachofthefollowingtimesonthefunnyclock:

2 MrFunnyman’sfriend,MsRegular,callsintosayhello.Thetimeonherwatchis shownonthe12-hourclockbelow:

Whattimeisthatonhisfunnyclock?

3 Writeyourschoolstarttime,finishtimeandthetimeyoustartedeatinglunchin funnyhoursandfunnyminutes.

4 Writethreetimesusing12-hourtimeandconvertthemtofunnyhoursandfunny minutes.

10E

FirstNationspeopleand mathematics

ObservationsoftheSunandMoon

InAustralia,manyAboriginalgroups,includingEuahlayi(NewSouthWalesand SouthernQueensland),Yolŋu(NorthernTerritory),Warlpiri(NorthernTerritory)and Wirangu(SouthAustralia),recognisedthatinatotalsolareclipse,theMoonseemsto covertheSuntotally.ThisisbecausetheSunis400timewiderthantheMoonand also400timesfurtheraway!.

InYolŋuknowledge,thestoryoftheSunwoman(walu)andtheMoonman(ngalindi) isusedtodescribethis phenomenon.InWarlpiritradition,theMoonisdescribedasa malefigurewhocrossestheskytomeettheSun.TheEuahlayialsoexplainaneclipse asatimewhentheSunandMoonalign.SomeWirangupeoplehaveexplainedthat,in asolareclipse,theSunandMoonbecame guri-arra,meaning’husbandandwife together’.

Activity1:CyclesoftheMoonandeclipsetiming

TheEuahlayi,YolŋuandWarlpiripeoplespredictsolareclipsesusingmooncycles.The Mooncompletesafullcyclein29.5days.

Giventhatthereare12mooncyclesinalunaryear,calculatehowmanyextradaysare addedafter10yearsifweuseasolarcalendaryearof365.25days.

Howwouldthisaffectthealignmentoflunarandsolarcalendarsoverthepast decade?

Activity2:PredictingtheeclipsewithMoonphases

TheYolŋustorydescribestheMoon,Ngalindi,asmovingthrougharepeatingcycle. ThenewMoonisalmostinvisiblefor3days,whileNgalindiremainsdead.Herises again,growingroundandfat.ThesetwostagesarepartofthewaxingMoonphase, lastingabouttwoweeks.WhentheMoonisfull,Ngalindi’swivesattackhim,andhe startstoshrinkagain.ThefullmoonandshrinkingphaseiscalledthewaningMoon. ThetotalMooncyclelasts29.5days.

• ForhowmanydaysistheMoonvisibleduringthewaxingmoonphase?(The waxingphaseincludesthenewmoonwhentheMoonisn’tvisible)

• Howlongisthewaningmoonphase,includingthefullmoondays?

Inthediagramshowingmoonphases,thefirstthreearewaxingmoons.Thecentre showsthefullmoonandthelastthreearewaningmoons.

Seehttps://science.nasa.gov/moon/moon-phases/forfurtherexplanations.

Activity3:Durationoftheeclipse

TheWirangupeopleexplainedthatduringasolareclipse,theMooncoverstheSun. Twominutesafterthestartoftheeclipse, 1 8 oftheSuniscovered.Afteranothertwo minutes,theamountcoveredhasdoubled.Thefractioncoveredcontinuestodouble every2minutes.HowlongdoesittakefortheSuntobefullycovered?

Activity4:PredictingtheEclipseDay

FirstNationsastronomersofDjugun-YawuruCountry, RoebuckBay,WesternAustralia,haveadeepunderstandingof celestialmotions.Theyareabletopredictsolareclipses.They observethatthesolareclipsehappensevery18months(about 548days),plusorminus10days.

Ifaneclipseisobservedon1stJanuary2000,calculatethe rangeofpossibledatesforthenexteclipse.

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Activity5:Moonphaseandtide

YolŋuknowledgetellsusthatthetidesareconnectedwiththeMoonphase.They observethatspringtides(highesttides)occurduringthefullornewmoon.Local factors,suchastheshapeofthecoastline,canchangethetimingofthetides.These observationshelpYolŋupeoplemakeaccuratepredictions.

Galileo’searlymodelofthetidesdidnotworkwellbecauseitdidnotconnectthetides totheMoonanditpredictedonlyonetideeachday.TheYolŋupeople’sclose observationsoftheMoonandlocalconditions,letsthempredictthetidesaccurately.

AYolŋuobserverpredictsaspringtideat8:00p.m.duringafullmoon.Theynotethat thetidalcycle,thetimebetweensuccessivehightides,is12hoursand25minutes. Localcoastalconditionscausea15minutedelayatthislocationcomparedtothe prediction,butdoesnotchangethelengthofthetidalcycle.Calculatethetimeofthe nexttwospringtidesobservedatthelocation.

Usefulskillsforthistopic

• identifyinganglesintheenvironment

• drawingstraightlineswitharuler

Vocabulary

Rightangle

• Acuteangle • Obtuseangle •

Reflexangle

• Straightangle • Protractor • Horizontal

• Arms • Vertex •

• Vertical • Parallel • Intersection

Revolution

• Perpendicular • Lines

Linesegment

• Angles •

1 Lookaroundyourclassroom.Howmanyrightanglescanyoufind?Seeifyoucan find3differentrightanglesandwritedowntheirlocations.Explainwhythistype ofanglemightbeusefulinthatparticularlocation.

2 Howmanyanglessmallerthanarightanglecanyoufind?Findatleast3andwrite downthelocationofeach.Explainwhythistypeofanglemightbeusefulinthat particularlocation.

3 Howmanyangleslargerthanarightanglecanyoufind?Findatleast3andwrite downthelocationofeach.Explainwhythistypeofanglemightbeusefulinthat particularlocation.

Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles

Lines and angles areeverywhere.

Thedoors,desksandwindowsinyourclassroomallhavelinesandangles.

Whentwolinesmeet,theymakeanangle.Whenadoorisajarorawindow opensinorout,anangleisformed.

11A Lookingatlines

Inmathematicsthewordlinealwaysmeansastraightline.Itdoesnotincludecurves suchascirclesandsquiggles.

Linesgoonforever.Wecannotdrawsomethingthatgoesforeverinbothdirections, sowedrawpartofalinecalleda linesegment andimagineitgoingonforever. Therearedifferenttypesoflines.

Horizontallines

Lookatthisgymnast’sbalancebeam. Thebeamis horizontal.Whenyougoto thebeachandlookoutatthesea,you canseethehorizon.That’swherethe word‘horizontal’comesfrom.

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Wecanthinkofthetopandbottom edgesofapieceofpaperas representinghorizontallines.

Abuilderusesaspiritleveltomakesuresomething likethetopofadoorframeishorizontal.Whenthe airbubbleisinthecentreofthegauge,thetimber ishorizontal.

Verticallines

Abuilderusesaplumblinetomakesureawallisvertical.Theplumbline isa vertical line.

Averticalline,likeaplumbline,goesfromtoptobottom.Ifthetopand bottomedgesofapieceofpaperarehorizontal,thenwecanthinkofthe sideedgesasvertical.

Parallellines

Twoormorelinesthatarealwaysthesamedistanceapartandnevermeetareknown as parallel lines.

Wethinkofthelinesgoingonforeverinbothdirections.

Wedrawasmallarrowoneachlinetoshowthelinesthatareparallel.Iftherearetwo groupsofparallellines,wedrawtwoarrowsononeset.

Intersections

Whentwolinescrossormeetatapointwesaytheyintersect.Thepointwherethe linesmeetiscalledan intersection.Forexample,tworoadsmeetorcrossatan intersection.

Thewordintersectcomesfrom‘inter’,meaning‘between’,and‘sect’,meaning‘cut’. Therearemanywayslinescanintersect.

11A Wholeclass LEARNINGTOGETHER

Workwithapartnertocopyandcompletethischartwith2examplesof wheretheselinescanbeseenfrominsidetheclassroom.

11A Individual APPLYYOURLEARNING

1 Drawandlabelashapethatcontainsatleastoneofeachofthefollowing. Canyoufindmorethanoneshape?

• Asetofparallellines

• Ahorizontalline

• Averticalline

2 Makeamodel.Usematchsticksandplasticinetodesignandbuildabridge thatcansupportatoycar.Useatleastoneofeachofthekindsoflines mentionedinthissection.Identifythelinesonthepartsoftheconstruction. Labelthelinesonyourdesign.

11B Angles

Herearetwolinesmeetingatapointnamed O.Wecall thetwolinesthatmaketheanglethe arms ofthe angle.Thepointwherethearmsoftheanglemeetis knownasthe vertex

Thelinesmaketwoangles.Wecan shadetheangle between thelinesorthe angle outside thelines.

Tomeasureanangleweseehowmuch wehavetoturnoneofthelinesthrough theshadedareatogettotheotherline. Wemarktheanglewearemeasuring withacurvedarrow.

Wemeasureanglesindegrees.

Therearemanytypesofangles.

Revolution

Turningthroughacompletecircleisaturnwithanangleof360degrees.Wewritethis as360°.A360° turnisalsocalleda revolution

Youcanseeinthediagramthatwhenweturnthrough360°,thearmsfinishupresting ontopofeachother.

Ifyouwanttoknowwhythereare360° inarevolution,investigatethehistoryof Babylonianastronomy.

Vertex

Straightangles

Halfafullturniscalleda straightangle becausethetwoarmsoftheanglemakea straightline.Astraightangleisequaltohalfof360°,whichis180°

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Rightangles

Takeapieceofpaperwithonestraightedgeandfolditalongthatedge.Byfoldinga straightangleinhalf,youmaketwoanglesequalto90°.A90° angleisknownasa rightangle andisone-quarterofafullturn.

Thepieceofpaperwiththerightangleinthecornercanbeusedtofindrightangles aroundyourclassroom.Holdyourrightangleineachcornertoseeiftheangleyou havefoundisarightangle.

Herearesomerightanglesyoucanprobablyfind.Lookat cupboards,doors,desks,books...Youmayevenlosecount, becausetherearesomany!

Wemarkrightangleswithasmallsquareinthecornertoshowthatthearmsofthe angleareat90° toeachotherlikethis:

Whentheanglebetweentwolinesisarightangle,wesaythelinesare perpendicular toeachother.

4th

Vertex

Acuteangles

An acuteangle islessthan90°

Acutemeans‘sharp’.Anangleisacuteifitlooks‘sharp’.

Obtuseangles

An obtuseangle isonethatisbetween90° and180° .

Theword‘obtuse’means‘blunt’–itistheoppositeof‘sharp’.

Reflexangles

Ananglelargerthan180° iscalleda reflexangle.Theword‘reflex’means‘turned back’or‘bentback’.

ChrissieboughtaroundchocolatecakeforJemma’sbirthday.Jemmacutaslice, cuttingfromthecentreoutwards,likethis:

Thisgavetwopiecesofcakeandtwoangles.Thelargerpiecehasananglelargerthan 180°.Itisareflexangle.

Thesmallerpiecehasananglelessthan90° andisanacuteangle.

Labeltheseangles.

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Afullturnis360° andiscalledarevolution. Astraightangleishalfafullturnandisequalto180

Arightangleis 1 4 ofaturnandisequalto90

Arightangleishalfastraightangle.

Angleslessthan90° areacuteangles.

Anglesmorethan90° butlessthan180° areobtuseangles.

Anglesmorethan180° butlessthan360° arereflexangles. Twolinesareperpendiculariftheanglebetweenthemisarightangle.

11B Wholeclass LEARNINGTOGETHER

1 Makeyourownangleestimator:

a Youwillneedtwocirclesofpaperofthesamesize.Eachshouldbeadifferent colour.

b Rulealineoneachcirclefromtheouteredge,alongtheradius,tothecentreof thecircleandcutalongthisline.

c Connectthetwocirclesthroughtheradiuscutlines.

d Youwillbeabletotwistyourpapertocreatedifferentsizedanglesto: showarightangle i showanacuteangle ii showastraightangle iii showanobtuseangle iv

e Couldyoulabeldifferenttypesofanglesastheyarerevealedononeofthe papercircles?

2 Whatdidwediscoverinthisactivity?Copyandcomplete. Halfafullturnis________° One-quarterofafullturnis________°.Thisisalsocalleda________angle.

3 Startwithadisplayclockwiththehandspointingto12:00.Movethehandsso theyareperpendicular(forexample,3:00).Canyoufindothertimeswhenthe handsoftheclockareperpendiculartoeachother?Howmanycanyoumake?

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4 Takeasquarepieceofcolouredpaper.Asmallkindergartensquarewilldo.We havelabelledthefourcorners A, B, C and D inthediagrambelowsothatwehave anameforeach.

Foldthesquareoversothetopside AD fallsexactlyon AB.Makeaneatcrease andthenunfoldthepaper.Itshouldlooklikethis:

a Arethe2markedanglesthesameordifferent?Discussyouranswerandgive yourreasonstoyourclassmates.

b Whatfractionofarightangleisthisangleat A?

Ifarightangleis90°,howmanydegreesisthisangle?

5 Herearesomeshapes.Copytheshapesandlabeltheseangles.

Anyreflexangles a Anyacuteangles b

Anyobtuseangles c Allangles d

11B Individual APPLYYOURLEARNING

1 Copyandcompletethesesentences.

a Anacuteangleislessthan______°

b Astraightangleis______° .

c Arightangleis______ofacompleteturn.

d Anobtuseangleisbetween______° and______°

e A______angleismorethan180° butlessthan360°

2 Whattypeofanglearethesemarkedangles?

3 Useyourpencilandrulertodrawtheseangles.Markthenumberofdegreeson eachone.Thefirstoneisdoneforyou.

a Aquarter-turn 90°

b Threequarter-turnsoneaftertheother

c Ahalf-turn

d Three-quartersofacompleterevolution

e Acompleterevolution

4 Find(atleast)6anglesinthisdiagram.Copythediagramandlabeltheangles youfind.

11C

Measuringanglesusing aprotractor

Howdowemeasuretheanglemadewhentwolinesintersect?

Ifweusearulertomeasurethedistancebetweenthearms,themeasurementcould bethesame,butweknowthatoneangleis90° andtheotherisanacuteangle, whichislessthan90° . Also,ifyoumovetherulerupordowntheangle,thelengthchanges. Wecannotusearulertomeasuretheanglemadewhentwolinesintersect. A protractor isusedtomeasureangles.

Aprotractorhastwosetsofnumbers.Onesetofnumbersisformeasuringangles fromtheright.Theothersetofnumbersisformeasuringanglesfromtheleft. Herearetwoangles.

Tomeasurethemweputthecentrepointofthe0° lineoftheprotractoronthevertex oftheangleandreadalongthescale.Thetwomarkedanglesareboth50° .

Measuringacuteangles

Tomeasureanacuteangleweplacethecentrepointofthe0° lineonthevertexand readthescalewheretheotherarmlies.

Thisangleis40° .

Measuringobtuseangles

Thisangleis30° .

Bothoftheseanglesaregreaterthanarightangle.Thismeansthattheiranglesare greaterthan90°.Theyareboth110°

Measuringreflexangles

Tomeasureareflexangle,youfirstneedtorotatetheprotractor.

Thisgivesyoutheshadedpartoftheangle (55°).Tofindthefullsizeoftheangle,you nowneedtoadd180° tothenumberofdegreesshownontheprotractor:

180° + 55° = 235°

Example2

Drawa55° angle.

Solution

First,drawonearmoftheangle.

Thenplaceyourprotractoratthevertexandmarkadotat55°.Nowjointhedot andthevertexwithalinetomakethesecondarmoftheangle.Labeltheangle. Yourdotsdonotneedtobequitesolarge.

11C Wholeclass LEARNINGTOGETHER

1 Whatisthesizeofeachoftheseangles?

2 Draweachoftheseangles.Askapartnertocheckyourdrawingsusingaprotractor.

3a Draw2linesmeetingatavertex.Hereisoneexample:

b Measureeachangle. c Whatshouldthesumofthe2anglesbe?

4 Usingtheoppositeendsoftheprotractor,draweachoftheseanglesintwoways.

1 Measureeachoftheseanglesusingaprotractor.

2 Measuretheanglesmarkedwithletters.

a Whatisthesumoftheangles A, B and C?

b Whatisthesumoftheangles P, Q, R and S?Whatisthesumoftheangles F, G and H?

c Whatdidyounoticeaboutthesumoftheanglesaboutapoint?

11D Reviewquestions–

1 Drawapictureofahousethatcontainsatleastoneofeachofthefollowingkinds oflines:

• asetofparallellines.

• 2linesthatareperpendicular

• averticalline

• anacuteangle

2 Whattypeofanglearethesemarkedangles?

3a Copythepicturebelowandname5differentkindsofangles.

b Measureeachofthe5anglesusingaprotractorandrecordthemeasurements onyourangles.

4 Draweachoftheseangles.Askapartnertocheckyourdrawingsusingaprotractor.

Flaginvestigation

1 Selectatleastsixflagsfromaroundtheworldanddrawthemintoyourworkbook.

2 Foreachflaglabelthefollowing:

a Anylines-parallel,perpendicular,vertical,horizontal,etc.

b Anyangles-acute,obtuse,right,etc.

3 Couldyougroupflagswithsimilarfeaturestogether?

4 Whatnamecouldyougivetoeachgroupofflags?

Usefulskillsforthistopic

• identifyingandnaming2Dand3Dshapes

• recognisingdifferentshapesandobjectsintheenvironment

• identifyingandnamingcubes,rectangularprismsandsomeotherpolyhedra

• identifyingandcreatingsymmetricalshapesbyrecognisinglinesofsymmetry

• understandinghowshapescanfittogethertocoverasurfaceorcreateapattern

Vocabulary

Polygon

• Quadrilateral

• Pyramid

• Prism

• Net

• Polyhedron

• Vertex

• Polyhedra

• Reflection

• Symmetry

• Vertices

• Rotation

• Transformation

• Translation

• Three-dimensional(3D)

• Two-dimensional(2D)

• Sides

• Tesselation

•

1 Iamtwo-dimensional.Ihave3verticesand3sidesthatareallthesamelength. WhatamI?

2 Iamthree-dimensional.Ihave6faces,8verticesand12edges.Myfacesareall thesameshapeandsize.WhatamI?

3 Iamtwo-dimensional.Ihave4rightanglesand2pairsofsidesthatarethesame length.WhatamI?

4 Iamthree-dimensional.Ihave4faces,4verticesand6edges.Myfacesareall thesameshapeandsize.WhatamI?

Shapesareallaroundus,fromcirclesinaclocktothecubesinabuilding. Understandingshapeshelpsusmakesenseoftheworldandsolveeveryday problems.Inthischapter,welookattwo-dimensionalandthree-dimensional shapes.

Two-dimensionalshapesarealsoknownas polygons.Three-dimensionalshapes orobjectsareeverythingaroundus;youareathree-dimensionalshape,andacar isathree-dimensionalshape,too.

Knowingthepropertiesoftheseshapeshelpsusidentifyandusethemindifferent contextssuchasart,designandconstruction.

A polygon isatwo-dimensionalshapeenclosed bythreeormorelinesegmentscalled sides Exactlytwosidesmeetateachvertex,andthe sidesdonotcross.

Polygonsarenamedaccordingtothenumberof sidesthattheyhaveorthenumberofanglesthat theyhave.

Polygonshavenothickness,buttherearesolidobjectsthatarelike two-dimensionalshapeswiththickness.Canyoufindsomeinyourclassroom?

Whenwegoshoppingweseealotofthree-dimensionalshapeswithspecial mathematicalnamesandproperties.

Chocolatecomesinboxesthatarerectangularprismsandtriangularprisms.

Soupandpotatochipsaresometimespackagedincylinders.

Shapescanbemanipulatedandusedindifferentways.Bylearningaboutshapes, symmetry,tessellation,transformationsandenlargements,wegainvaluableskillsthat helpusinmanyareasoflife,fromsolvingpuzzlestodesigningbuildingsinengineering andarchitecture.

12A 2Dshapes–Trianglesandquadrilaterals

Whatisatriangle?Thinkofwordsthatstartwith‘tri’.Atriathlonisathree-eventrace andatripodisathree-leggedstandforkeepingacameraortelescopesteady.The prefix‘tri’means‘three’.Soatrianglehasthreeangles.Italsohasthreestraightsides. Trianglescanbesortedaccordingtothelengthsoftheirsidesoraccordingtothesizes oftheirinteriorangles.

Equilateraltriangles

Atrianglewithallofitssidesthesamelengthiscalled equilateral.‘Equilateral’comes fromtwoLatinwordsmeaning‘equal’and‘sides’.Herearesomepicturesof equilateraltriangles.

Isoscelestriangles

Atrianglewithatleasttwosidesthesamelengthiscalled isosceles,fromtwoGreek wordsmeaning‘equal’and‘legs’.Everyequilateraltriangleisisosceles,butthereare isoscelestrianglesthatarenotequilateral.Herearesomepicturesofisoscelestriangles. Whichoneisequilateral?Whichonesareisoscelesbutnotequilateral?

Ifatrianglehasexactlytwoanglesthesame,thenithastobeisosceles,butcannotbe equilateral.Youcanseeinthepicturesabovethatthetriangleinthemiddleandthe oneontherighthaveexactlytwoanglesequal.

Scalenetriangles

Theonlyotherthingthatcanhappenisthatallofthesidesofthetrianglehave differentlengths.Wecallthesetriangles scalene,fromaLatinwordmeaning‘tomix thingsup’.

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Herearesomepicturesofscalenetriangles.

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Ifallthreeanglesinatrianglearedifferent,thenthetrianglehastobescalene.Drawa fewtoconvinceyourselfthisistrue.

Right-angledtriangles

Whenoneoftheanglesinatriangleis90°,wecallita right-angledtriangle.Hereare someright-angledtriangles.Whichonesareisoscelesandwhichonesarescalene?

Canaright-angledtrianglebeequilateral?Trytodrawone. Canyouseewhytherecannotbetworightanglesinatriangle? Drawsomediagramstohelpexplain.

Whatisaquadrilateral?

InLatin,‘latus’means‘side’andtheprefix‘quadri’means‘four’,soa quadrilateral isa shapewithfoursides.Ithasfourverticesalso. Therearemanydifferentkindsofquadrilaterals;somehavespecialnames.Weknow twokindsofquadrilateralsalready.Rectanglesandsquareshavefoursides.

Rectangle

Arectangleisaquadrilateralinwhichalltheanglesarerightangles. Theoppositesidesofarectanglehavethesamelength.Thesesides areparalleltoeachother.

Propertiesofarectangle

1 Allanglesarerightangles.

2 Oppositesidesareparallel.

3 Oppositesideshavethesamelength.

Square

Asquareisaveryspecialkindofrectangle.Allofitssideshavethe samelength.

Parallelogram

Aparallelogramisaquadrilateralwithoppositesidesparallel. Itlookslikea‘pushedover’rectangle.

Rectanglesandsquaresarespecialkindsofparallelograms. Theyhavefourrightanglesaswellasoppositesidesparallel.

Trapezium

Atrapeziumhastwosidesthatareparallel. Youmighthaveseenatableatschoolwiththisshape. Thepluraloftrapeziumistrapezia.

Rhombus

Arhombusisaparallelogramwithfourequalsides.Thinkofa rhombusasasquarepushedsideways.

Asquareisaspecialkindofrhombus.Ifyouhavearhombuswithfour rightangles,itisasquare.

Lookatapackofcardsandfinda‘diamond’card.Canyousee thatthediamondisarhombus?

Adiamondisarhombusdrawnvertically.

Kite

Akitehastwopairsofadjacentsidesequal.

Soarhombusandasquarearespecialkindsofkite.

12A Wholeclass LEARNINGTOGETHER

1 Drawandthencutoutasmanydifferenttypesoftriangles,quadrilateralsand otherpolygonsasyoucanfromwhatyouhavelearntsofar.Labeleachshape andmakeapostertodisplayyourwork.

12A Individual APPLYYOURLEARNING

1 Draw:

a asquarewith5cmsides

b atrapeziumwithabaseof6cmandthesideoppositeitsbaseequalto4cm

c aparallelogramwithatleastoneangleequalto130°

d arectanglewithonepairofsidesequalto1cmandtheotherpairlongerthan yourleftthumb.

2 Copyeachshapeanddrawalineinsideeachtoformtworight-angledtriangles.

3 Drawarhombuswithatleastonerightangle.Whatdoyounotice?

4 Constructquadrilateralsusingthesidesandanglesshown.Measurethemissing sidesandmissinganglesandmarkeachonyourdrawing.

c Whatisthesumoftheanglesineach?

d Whatdoyounoticeaboutthemissingsideinpart a?

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5 Drawashapethathasfoursidesof4cmandthe angleat M asshown.(Youmayneedtousetrialand errortogetthesidestomeet.)Thefirstonehasbeen doneforyou:

12B

2Dshapes–Otherpolygons

Inthissectionwelookathowshapeswithmorethanfoursidesarenamed.Asbefore, thenameofeachshapetellsussomethingaboutitsproperties.

Pentagons

TheGreekprefix‘penta’means‘five’and‘gon’means ‘angle’.Soapentagonhasfiveangles.Italsohasfive verticesandfivesides.Herearetwopentagons.

Regularpentagons

Regularpentagonshavefiveequalanglesandfiveequalsides. Eachangleis108° .

Themarksonthesidesinthediagramindicatethattheside lengthsareallthesame.

Hexagons

TheGreekprefix‘hexa’means‘six’and‘gon’ means‘angle’.Soahexagonhassixangles.It alsohassixverticesandsixsides.Thisisa non-convexirregularhexagon.

Regularhexagons

Regularhexagonshavesixequalanglesandsixequalsides. Eachangleis120

Othertwo-dimensionalshapes

Two-dimensionalshapes arenamedaccordingtothenumberofsides.Wecould startthelistbelowbycallingaone-sidedshapeamonogonandatwo-sided shapeadigon.

Butwhatwouldtheylooklike?Tryforyourself.Doyouagreethatone-sidedshapes andtwo-sidedshapesdonotmakeanysense?

Wehavealreadydiscussedathree-sidedshape–whichwecallatriangle–butitcould alsobecalledatrigon.Afour-sidedshapeisknownasaquadrilateral,butitcouldbe calledatetragon.

Aregularpolygonhasallsidesequalandallanglesequal.

12B Wholeclass LEARNINGTOGETHER

1 a Drawasketchofaregularpentagon.Nowdrawlinestoshowhowyou couldcutthepentagoninto5isoscelestriangles.

b Drawaregularpentagon.Nowdrawlinestoshowhowyoucouldcutthe pentagoninto3triangles. Canthepentagonbecutinto3trianglesinanotherway?

c Areanyofyourtrianglesspecial,suchasequilateral,isoscelesorscalene?

12B Individual APPLYYOURLEARNING

1 Iamashape.WhatshapeamI?

a Ihave6equalsidesand6equalangles.

b Ihave12sides.

c Ihavethesamenumberofsidesasanoctopushaslegs.

d Ihave5sides.

e Ihave10sides.

f Myprefixmeans5andtherestofmynameisthesameas10–sided.

2 Drawortracetheseshapestocompletethequestionsbelow.

a Drawarectangle.Drawalinetoshowhowyoucouldcuttherectangleinto 2right-angledtriangles.Inhowmanywayscanyoudothis?

b Drawasquare.Nowdrawalinetoshowhowyoucouldcutthesquareinto 2rectangles.Howcanyoumakethemequalrectangles?

c Drawasquare.Nowdrawlinestoshowhowyoucouldcutthesquareinto 3equalrectangles.

d Drawarhombus.Nowdrawalinetoshowhowyoucouldcuttherhombus into2equaltriangles.Inhowmanywayscanyoudothis?

e Drawortracearegularhexagon.Nowdrawlinestoshowhowyoucouldcut thehexagoninto6equilateraltriangles.

f Drawasquare.Nowdrawalinetoshowhowyoucouldcutthesquareinto onetriangleandoneirregularpentagon.

g Drawortracearegularhexagon.Nowdrawalinetoshowhowyoucouldcut thehexagonintooneisoscelestriangleandoneirregularpentagon.

3 Lookatthepolygonsbelow.

a Howiseachpolygonthesame?

b Canyoufindapolygonthatdoesnotbelong?Explainwhyitdoesnotbelong.

3Dshapes–Polyhedra

Many three-dimensionalshapes havespecialnames–forexample,cubesand pyramids

A polyhedron isathree-dimensionalobjectwithflatfacesandstraightedges.The facesarepolygons.Theyarejoinedattheiredges.Theword‘poly’meansmany,and theword‘hedron’meansface.

Thepluralofpolyhedronispolyhedra,sowecanhaveonepolyhedronandtwoor morepolyhedra.

Whenwedescribepolyhedra,thepropertiesweareinterestedinarethefaces,vertices andedges.

Afaceofapolyhedronistheshapethatmakesuponeofits flatsurfaces.

• Thefacesofacubeareallsquares.

An edge ofapolyhedronisalinewheretwofacesmeet.

• Acubehas12edges.

A vertex ofapolyhedronisthepointatwhichthreeormore edgesmeet.Thepluralofvertexis vertices

• Acubehaseightvertices.

Polyhedrahavespecialnamesdependingonthenumberoffaces thattheyhave.Therearesomesimilaritieswiththenamingof polygons.

Tetrahedrons

Thesmallestnumberoffacesapolyhedroncanhaveisfour. TheGreekprefix‘tetra’meansfour.Atetrahedronhasfour vertices,fourfacesandsixedges.

Atetrahedronisalsocalledatriangular-basedpyramid.

Pentahedrons

Herearetwodifferentpentahedron.‘Penta’meansfive. Youmightknowthispentahedronasasquare-based pyramid.Ithasfivevertices,fivefacesandeightedges.

Thispentahedronhassixvertices,fivefacesandnineedges.Itis calledatriangularprism.

Regularpolyhedra

Forpolyhedra,‘regular’meansthatallofthefacesareidenticalregularpolygonsand thatthesamenumberoffacesmeetateachvertex.Theword‘regular’inmathematics meansfollowingarulelikethis.

Uncorrected 4th

Acubehassixfaces,allofthemidenticalsquares.Threefacesmeet ateachvertex.Itisaregularpolyhedra,whichiscalleda hexahedron.‘Hex’meanssix.

Aregulartetrahedronhasfourfaces.‘Tetra’meansfour.The fourfacesareidenticalequilateraltriangles.Threefacesmeetat eachvertex.

4 Tetrahedron

5 Pentahedron

6 Hexahedron

Thisexampleisalsoknownas atriangular-basedpyramid.

Thisexampleisalsoknownas asquare-basedpyramid.

Thisexampleisalsoknownas acube.

Heptahedron

Octahedron

Thisexampleisliketwo square-basedpyramidsjoined togetheratthesquarefaces.

Nonahedron

12C Wholeclass LEARNINGTOGETHER

1 Practisedrawingsketchesofpolyhedra.Showtheedgesthatyoucannotsee withadottedline.Completethestatementsforeach.

a Startwithacube.

STEP1: Drawasquare.

STEP3:

Jointheverticesthatyou canseewithsolidlines.

STEP2: Draw2linesatrightangles toeachotherasshown:

STEP4:

Connecttothevertexthat youcannotseewithdotted lines

Acubehas faces, edgesand vertices.

b Sketchatetrahedron.Startwiththefronttriangularface.Atetrahedron has faces, edgesand vertices.

c Sketchapentahedronthatisasquare-basedpyramid.Startwiththesquare base.Asquarepyramidhas faces, edgesand vertices.

d Sketchapentahedronthatisatriangularprism.Startwithatriangleface. Atriangularprismhas faces, edgesand vertices.

12C Individual APPLYYOURLEARNING

1 Namethefollowingsolids.

a b c d

2 Eachoftheitemsbelowhastheshapeofoneofthepolyhedra.Count thenumberoffacesandusethelistofprefixestohelpyounameeach polyhedron.

tetra = 4penta = 5hexa = 6hepta = 7

octa = 8nona = 9deca = 10

3 Thesepolyhedrahavebeendrawnsothateachface,edgeandvertexcan beseen.Nametheshapeandcompletethestatementaboutfaces,edges andvertices.

a This has faces, edgesand vertices.

b This has faces, edgesand vertices.

c This has faces, edgesand vertices.

12D Nets

Prisms

A prism isapolyhedronwithabaseandatopthatarethesame.Allofthesidefaces arerectanglesperpendiculartothebase.Thisisalsoknownasarightprism.

Aprismisnamedaccordingtotheshapeofitsbase. Theoneinthediagramontheleftisarectangular prism,becauseitsbaseisarectangle.

Arectangularprismisalsoahexahedronbecauseit hassixfaces.

Cylinders

Thisthree-dimensionalshapeiscalleda cylinder.Ithasacircular baseandtop.

Everycross-sectionisacircleofthesamesize.

Cylindersarenotprismsbecausetheydonothaverectangular sidefaces.

Cylindersarenotpolyhedrabecausetheydonothavepolygonal faces.

Netsofthree-dimensionalshapes

A net islikeanunfoldedsolid.Everypolyhedroncanbecutintoanet.

Whenwe‘unfold’acube,sixsquaresarejoinedtogether.Thenetmusthavesix squaresbecausethecubehassixsquarefaces.

Belowisthemostfamiliarnetofacube.Belowisanetforasquareprism.

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12D Wholeclass LEARNINGTOGETHER

1 Youwillneedvariousmodelsofprismsandpyramids.Countthenumberof facesforeachshape.Placetheshapesinincreasingorderaccordingtothe numberoffaces.

2 Makethesemodelsofprismsandotherpolyhedrausingtoothpicksforthe edges,andtinyballsofclayorplasticineforthevertices.Ifyouuseclay,let yourconstructionsdryonawindowsillandhandlethemgently.

a Use12toothpicksand8verticestomakeacube.

b Use8toothpicksand5verticestomakeasquarepyramid.

c Use18toothpickstomakeahexagonalprism.

d Makeapentagonalprism.

e Use4verticesand6edgestomakea .(Complete)

f Makeaheptagonalpyramid.

3 Useconstructionequipmenttomake4different3Dshapes.

a Nameeachshape.

b Drawasketchofeachshape.

c Describeeachshapeintermsoffaces,verticesandedges.

d Flattenoutthepiecesoftheshapesothatthepiecesarestilljoined together.Thiswillmakeanetoftheshape.Sketchthenet.

4 Whatmightthenetofeachshapelooklike?Sketchit.

5 a Drawaheptahedronthatisnotapyramid.

b Drawanoctahedronthatisnotapyramid.

1 Namethebaseofeachprism.

2 Matcheach3Dshapetoitsnet.

3 Takeaboxandopenitupsoitisflat.Labelthedifferentfaceswiththeir2Dname.

4 Designapackageforanewbrandofcereal.Createanetfromcard,decorateit, andassembleit.

a Howdoesunderstandingthenethelpyoutocreateanaccurateand functionalbox?

b Howdothefoldsandflapscontributetothestructuralintegrityofthecereal box?Whatshapesmakeupthefoldsandflaps?

c Willyourboxholdthecerealsecurely?Whatimprovementscanyoumaketo thenettoimprovethedesign?

12E Symmetryof2Dshapes

Inmathematics,whenthepiecesofatwo-dimensionalshapematchupexactlyacross astraightline,wesaytheshapeissymmetricalabouttheline. Forexample,thistriangleissymmetricalaboutthereddottedline:

Innature,wesee symmetry inanimalsandinplants. Thelineiscalledalineofsymmetry.

Whenwesaythatsomethingissymmetrical,wemeanthatitisidenticalonbothsides ofthelineofsymmetry.Thedrawingofthetreeontheleftisanexampleofsymmetry.

Symmetric Asymmetric

Theoppositeofsymmetricalisasymmetrical,asshowninthepictureofthetreeon theright.

Ashapecanhavemorethanonelineofsymmetry. Theshapebelowhastwolinesofsymmetry.

Theshapesbelowhavefourlinesofsymmetry.

Imaginefoldingashapealongalineofsymmetry.Thetwohalvesthenmatcheach otherexactly.Theimageisreflectedintheline.Wecallthelinethe axisofreflection or the axisofsymmetry

Acirclehasinfinitelymanylinesofsymmetry!It wouldnotbepossibletodrawthemall.

1 Createapictureusingyourclasssetofpatternblocksorusetriangle-grid papertodrawonethatincludeshexagons,trapezia,trianglesandrhombuses. Askyourpartnertomakeitsreflection.Hereisoneexample.

2 Usetrianglegridpaperandcreateapicturethathas: onelineofsymmetry a twolinesofsymmetry b threelinesofsymmetry. c

1 a Draw5regularpolygonsofdifferentsizes. b Markinthelinesofsymmetrywithadottedline.

2 Copyeachdiagram,thencompletethemissingpartsofeachshape.The dottedlinesarelinesofsymmetry.

3 Drawashapethathasthefollowing: Morethanonelineofsymmetry a Nolinesofsymmetry b Onlyonelineofsymmetry c

12F

Transformations andtessellations

Weseepatternsallaroundus.Manypatternsaremadebyshapesfittingtogether. Rotation,reflectionandtranslationaresomeofthedifferentwayswecantransforma two-dimensionalshape.Thesearecalled transformations.

Rotation

Arotationofashapeaboutapointiswhentheshapeisturnedthroughanangle aboutthepoint.

Theword‘image’isusedtolabeltheshapeafterrotation.

Thisshapehasbeenrotatedclockwisethrough90° aboutthepointmarkedwitha reddot.

Wecanrotateanticlockwiseaboutapoint.

Thisarrowhasbeenrotatedanticlockwisethrough90° . Example1 Howhasthisshapebeenmoved?

Reflection

A reflection isatransformationthatflipsafigureaboutaline.Thislineiscalledtheaxis ofreflection.Agoodwaytounderstandthisistosupposethatyouhaveabookwith clearplasticpagesandatriangledrawn,asinthefirstdiagrambelow.Ifthepageis turned,thetriangleisflippedover.Wesayithasbeenreflected;inthiscasetheaxisof reflectionisthebindingofthebook.

Thisshapehasbeenreflectedintheverticalline.

Translation

Whenwetranslateashape,weslideit.Wecanslideitleftorright,upordown. Translations movetheshapewithoutrotatingit.

Thisshapehasbeentranslated horizontally.

Thisshapehasbeentranslated vertically.

Image

Tessellation

A tessellation isatilingpatternthatfitstogethertwo-dimensionalshapeswithnogaps oroverlaps.Thetessellationcancontinueinalldirections. Forexample,wecouldstartwithanequilateraltriangle.

Wecanrotateit180° andshiftitsothetrianglesfittogetherperfectly.Thetilingcan continuehorizontallyandvertically.Wesaythattheequilateraltriangle tessellates.

Circlesdonottessellatebecausewecannotrotateandshiftthemtofillupthewhole spacewithoutgapsoroverlaps.

Itispossibletotessellatetwoormoreshapes.

Thetessellationbelowusesregularhexagonsandequilateraltriangles.

12F Wholeclass LEARNINGTOGETHER

1 Useyourclasssetofshapesorcutoutsomeofyourown.Taketurnsgiving instructionstoyourpartnertotranslateashapeindifferentways.

2 Lookaroundtheschoolfortessellatingpatterns.Takedigitalphotographsof themanddescribetheshapesused.Drawinthelinesofsymmetry.

12F Individual APPLYYOURLEARNING

1 Useyourclasssetofpatternblocksorusetriangle-gridpapertodrawandcoloura tessellatingpatternthatfillsa10cm × 10cmspaceonthepageanduses: onlytriangles a onlyhexagons b onlytrapezia c hexagonsandtriangles. d

2 LookingatthepatternsyoucreatedinQuestion 1,answerthequestionsbelow.

a Canthisshape,ortheseshapes,betessellated?

b Doesyourpatternhaveanylinesofsymmetry?

c Isyourshapetessellatedthroughreflection,translationand/orrotation?

3 YoucanquicklycreatedesignsusingacomputerandaprogramlikeMicrosoft Wordorsomethingsimilar.

a Openanewdocumentonyourcomputer.

b FindtheShapesiconinthemenubar,clickonitandchoosea2Dshapeyou thinkyoucantessellate.

c Clickanddragtheshapetoyourpage.Torepeattheshape,continuetocopy andpastetheshape.Youmayneedmorethanoneshapetocreatea tessellation.

d Repeatuntilyourpatterniscomplete.

4 LookingatthepatternsyoucreatedinQuestion 3,answerthequestionsbelow.

a Canthisshapebetessellated?Didyouneedmorethanoneshapetocomplete thetessellation?

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b Doesyourpatternhaveanylinesofsymmetry?

c Isyourshapetessellatedthroughreflection,translationand/orrotation?

12G Reviewquestions–

1 Usearulerandaprotractortodraw:

a atrianglewith1sideoflength5cm

b ascalenetrianglewith1sideequalto5cm

c aright-angledtrianglethatisnotisosceles

d atrianglewith1angleequalto30°

e aquadrilateralwithnorightangles

f arectanglewith1sideequalto6cm

g arhombuswith1angleequalto45°

2 Drawasketchofeachofthesepolyhedra.Showtheedgesthatyoucannotsee withadottedline.

Rectangularprism a Octahedron b Triangularprism

c Rectangularpyramid d

3 Namethepolyhedraandcompletethestatementaboutfaces,edgesandvertices.

a This has faces, edgesand vertices.

b This has faces, edgesand vertices.

c This has faces, edgesand vertices.

4 Namethese3Dshapes.

5 DrawanetforeachoftheshapesinQuestion 3

6 Copytheseshapesanddrawintheirlinesofsymmetry.

12H Challenge–Ready,set,explore!

Polyiamondschallenge

Hereisanequilateraltriangle.Wearegoingtojoinequilateraltrianglesofthesame sizealongtheirsidestomaketwo-dimensionalshapescalledpolyiamonds.‘Poly’ means‘many’.

Yourteacherwillprovideyouwithtrianglegridpapertodrawyourpolyiamonds.You cancolourandcutthemoutifyouwish.

Onetriangleiscalledamoniamondbecausemonomeansone.

Twotrianglesmakeadiamond.Dimeanstwo.

Challengequestions

1 Whatdoyouthinkashapemadewiththreeequilateraltrianglesiscalled?

2 Ashapemadewithfourequilateraltrianglesiscalledatetriamond.Polyiamonds thatarereflectionsorrotationsofeachotherareconsideredthesame.

a Drawtwomoretetriamonds

b Drawthetetriamondsthatcanbefoldedtomakeatriangularpyramid.

3 Hereisonepentiamond. Drawthreemore.

4 Hexiamondsusesixequilateraltriangles.Thisoneiscalledthesphinxbecauseit looksliketheEgyptianSphinx.Drawthetwelvehexiamonds.

5 Sallyusesidenticaltriangles,eachwithaperimeterof9cm,tomakeall 12hexiamonds.Whatistheperimeterofeachhexiamond?

Usefulskillsforthistopic

• collectingandorganisingdataandpresentingitasadisplay

• interpretingdatafromadisplay

Vocabulary

x-axis(horizontal)

• y-axis(vertical)

Mode

Survey

Poll

• Countdata

Categoricaldata

Pictograph

Columngraph

Measurementdata

Dotplot

Linegraph

1 Whatdoyounotice? 2 Canyouidentifyanytrendsinthegraph? 3 Whatmightthisgraphberepresenting?

4 Whatfurtherinformationmightyouneedtointerpretthisgraph?

5 Ifyouweretoaddlabelstothe x and y-axis,whatwouldtheybe? 6 Howdoyouthinkthedatamighthavebeencollectedforthisgraph?

Whenwegatherinformation,wearecollectingdata.Sometimeswecancollect dataaboutpeople’sopinions.Sometimesthedatawecollectmightbeabout physicalfeaturessuchaseyecolourorheight.Wecanorganisethedatathatwe collectintotablesordiagramsandwecangraphitindifferentways.

Collectingandstudyingdatainthiswayiscalled statistics.Peoplewhogather andanalysestatisticsarecalledstatisticians.Theword‘statistics’hasthesame originastheword‘state’becausetheoldestuseofstatisticswastohelp governmentsmakedecisions.

Thestatisticalprocess

Whenweplanastatisticaldatainvestigation,weneedtodecidetheproblem wearegoingtoinvestigateandposesomequestionsthatwemightlike answersfor.

Forexample,ifthestudentcouncilwantedtomakesome suggestionsaboutchangingtheschooluniform,wemight ask:

• WhatuniformpiecesaremostpopularamongYear5 students?

• Whichcoloursarepreferredfromthechoicesavailablefor schoolT-shirts?

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• Whomakesthedecisionsaboutbuyinguniforms?

Then,wethinkaboutwhatdataweneedtocollectsothatwecananswerthose questions.Therearelotsofwaystocollect,organiseandpresenttheinformation, sotherearemanychoicestobemade.

Finally,welookatthedatawhenithasbeenorganisedandpresentedintables,charts andgraphsandinterpretthatinformationinordertomakesomeconclusionsand recommendations.Inourschooluniformexample,wemightmakesomesuggestions totheschoolstaffaboutthetypesofT-shirtsthatYear5studentspreferfromthe informationwehavecollected.

Thestatisticaldatainvestigationprocesswillbeexplainedinmoredetailinalater sectionofthischapterwhereyouwilluseittocarryoutyourowndatainvestigations.

Typesofdata

Therearedifferenttypesofdata.Foreachtypetherearedifferentwaystopresentthe dataanddifferentthingstoconsiderwhencollectingandrecordingthedata.

Thereisdatathatwecancount.Weget countdata whenweinvestigatesituations suchas:

• thenumberoftreesindifferentbackyards

• thenumberofgoalsscoredinanetballmatch

• thenumberofjellybeansinapacket.

Thereisdatathatwecanmeasure.Herearesomesituationswhereyoumightcollect measurementdata:

• theheightofstudentsinyourclass

• theageofstudentswhentheyfirstrodeabikewithouttrainingwheels

• theamountofwaterleftineveryone’sdrinkbottlesafterlunch.

Thereisdatathatbelongsincategories.Sometimesthereisachoicetobemadeabout whichcategorythedatabelongsto. Categoricaldata includes:

• typesofhouses

• coloursofcars

• typesofhairstyles.

14A Posingquestions andcollectingdata

Onewayofcollectingdataistoaskquestions.Thisiscalledconductinga survey or takinga poll.Weneedtoaskclearquestionstogetaccuratedata.Whenweare conductingasurvey,wealsoneedtothinkaboutthepeoplewhowillbeaskedthe questions.Willthepeopleinterviewedbeabletogiveustheinformationweneed?

Forexample,ifwewantedtofindoutaboutthefavouriteholidaydestinationfor retiredpeople,wewouldnotaskschoolchildrenbecausetheyarenotretired.We wouldaskretiredpeople.

Tallymarks

Whenwecollectdata,wecanuse tallymarks.Eachstrokestandsforoneitem,and thefifthstrokeismadeacrossagroupoffour.

Atallyoffiveiswrittenlikethis: ✚✚ ||||

Teniswrittenastwobundlesoffive: ✚✚ ||||

Thenwecountbyfivestoworkouthowmanythereareinthetally.

Example1

Thisdatatablewithtallymarksshowsthepreferencesofstudentsandteachers whousetheschoolcanteen.

a Whatisthemostpopularcanteendrink?

b Whatistheleastpopularcanteendrink?

Solution

Whenwelookatthedatatable,wecanseethemostpopulardrinkisfruitjuice (21tallymarks)andtheleastpopulardrinkismilk(11tallymarks).

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Two-waytables

Tablesanddiagramscanhelpusunderstanddata.

Wecanusethemtogroupdataindifferentways.Wecanrecordopinionsfrom differentgroupsusinga two-waytable.Wesummarisethetallymarksbywritingthe numberthatourtallymarksrepresent.

HereisthedatafromExample1showninatwo-waytable.

Favouritedrinks

Supposewecollectdataaboutstudentswholikeswimmingandstudentswholike athletics.Somestudentslikejustone,somestudentslikebothandsomestudentsdo notlikeeither.Wecanuseatwo-waytabletoshowthisdata.

Likesswimming Doesnotlikeswimming

Likesathletics Jason+Ali Rebecca+Jules

Doesnotlikeathletics Simone+Luke Flavia

Example2

Mumaskedthefamilywhichvegetablestheylike:carrotsorpeas.Dadlikescarrots andpeas,Melialikesjustcarrots,JoshualikesjustpeasandEmmadoesn’tlike carrotsorpeas.Mumlikesjustcarrots.Presentthisdatainatwo-waytable.

Solution Atwo-waytablecanshowthisdata. Likespeas Doesn’tlikepeas

Likescarrots Dad Melia+Mum Doesn’tlikecarrots Joshua Emma

14A Wholeclass LEARNINGTOGETHER

1 Makethistwo-waytableonthefloorwithmaskingtapeandlabels.

Likesfootball Dislikesfootball

Likesnetball

Dislikesnetball

Studentswritetheirnamesonapieceofcardboardandplacethenamesinthe boxthatbestreflectstheirpreferences.Howmanystudentslikebothfootball andnetball?Howmanystudentsdon’tlikeeithernetballorfootball?

2 Usethesurveyresultstoanswerthequestionsbelow.

Favouriteballgames

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a Whatisthemostpopularballgameoverall?

b Whatistheleastpopularballgameoverall?

c Howmanystudentsweresurveyed?

d Whatisthemostpopularballgameamonggirls?

e Whatisthemostpopularballgameamongboys?

3 Youareorganisingaspecialclasslunchandneedtoestablishwhattoorder.

a Createaquestionthathelpsustocollectcategoricaldata.

b Createanumericalquestionthathelpsustocollectdataonamounts offood.

14A Individual APPLYYOURLEARNING

1 Usethetabletocompletethequestionsbelow.

a Howmanypeoplelikemangoesbutdon’tlikegrapes?

b Howmanypeoplelikegrapesbutdon’tlikemangoes?

c Howmanypeoplelikemangoesandgrapes?

d Howmanypeopledon’tlikeeithermangoesorgrapes?

e Howmanypeopleweresurveyed?

2 Thesportsteacherwantedtobuynewequipmentforthestudentstouseat playtime.Hecouldbuyonly2typesofequipment.Eachchildintheclasswas surveyed;theycouldonlyvoteonceeach.

Fourboysvotedfortennisballs.

Sevengirlsvotedforskippingropes. Twogirlsvotedforfootballs.

Sixboysvotedforbasketballs.

Oneboyvotedforskippingropes.

Threegirlsvotedfortennisballs.

Fiveboysvotedforfootballs. Fourgirlsvotedforbasketballs.

a Drawatwo-waytabletorecordthisdata.

b Howmanystudentsweresurveyed?

3 Youarecollectingdataontheamountofwaterstudentsdrinkduringtheday atschooloveraweek.

Create2questionsthatwillenableyoutocollectmeasurementdata.

14B Representingand interpretingdata

Agraphhelpsusorganisetheinformationwehavecollectedandmakesiteasierto makeconclusionsaboutourdata.

Mode

Howcanwefigureoutwhichitemisthemostpopular, themostcommon,orthefavourite?Allthese questionsareaskingthesamething.Theywantto knowwhichvalueshowsupthemostoften.Thisvalue iscalledthe mode.

Forexample,Tomsurveyedagroupofpeopleabout theirsportandwrotedowntheresults.

Tomarrangedhisdataintoafrequency table.Atallyandfrequencytableisawayto organisedatabyusingtallymarkstocount howoftensomethingoccursandanumber (frequency)toshowthetotal.

InTom’ssurvey,soccerwasthesportthat peoplesaidmostfrequently,sosoccerwas themodeforhissurvey.

Thereisaneasywaytorememberthis. Mode istheFrenchwordfor‘fashion’,anditis alsothemostfashionable(ormostpopular)valueinadataset.

Sometimestwovaluesareequallypopular,andalltheothersarelesspopular.Inthis case,wetakebothvaluestobethemode.

Pictographs

A pictograph usessymbolsto showthenumberofitemsin thesamecategory.

Ifwearesurveyingalarge numberofpeoplewecanuse onepicturetorepresenta numberofpeople.Inthe examplebelow,thepictograph showsdatacollectedabout favouritetelevisionstations.

Onetelevisionrepresents10people.Halfatelevisionrepresentsfivepeople.The key tellsushowmanypeopleeachpicturerepresents.

Favourite television station

Example3

Schoolchildrenweresurveyedtofindouttheirfavouritesubjectatschool. 55studentsvotedforMaths.40studentsvotedforArt. 65studentsvotedforEnglish.20studentsvotedforMusic. 15studentsvotedforScience.70studentsvotedforSport. Representthisdatainapictograph.

Solution

Wecandrawapictographwitheachpicture representing10people.

Favourite subject

Columngraphs

A columngraph usescolumnsofdifferentlengthstorepresentdifferentquantities.The columnscanbeeitherverticalorhorizontal.Columngraphsarealsoknownasbar graphsorbarcharts.

Numbersalongoneaxisshowthenumberrepresentedbyeachcolumnonthegraph. Thenumbers,measurementsorcategoriesbeingrepresentedarewrittenalongthe otheraxis.

Herearetwocolumngraphsforthesamedatacollectedabouttypesofbreadrollssold ataschoolcanteen.

MathsArt

Rolls sold at the canteen

Thescaleonacolumngraphgoalongtheaxisinthesame-sizedsteps. Wecanreadthegraphstofindoutinformation.Inthegraphsonthepreviouspage:

• themostpopularfillingisham

• thereweresevenhamrollssoldandonlytwotomatorolls.

Example4

Asurveywastakentofindhowmanydifferentkindsofvehiclespassedthelocal schoolbetween8:00a.m.and9:00a.m.Herearetheresultsofthesurvey.

Vehiclesdrivingpast

a Presentthisdatainabarchart.

b Howmanymorecarspassedtheschoolthanmotorbikes?

Solution

Vehicles driving past a Therewere10morecarsthan motorbikespassingbytheschool.

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Dotplots

A dotplot isusedforcountdata,whereonedotdrawnaboveabaselinerepresents eachtimeaparticularvalueoccursinthedata.Ifavalueoccursthreetimes,thereare threedotsinalineabovethatvalue.

MsApap’sYear5studentscollecteddataaboutthenumberofhourstheyspentonthe computerinoneweek.Partsofhourswereroundeduptowholehours.Atfirstthe teacherwrotethenumbersontheboardasalist: 14,20,13,13,14,11,12,20,13,15,11,13,14,14,15,16,14,14

Thelistdidnottellthemverymuch,sotheytalliedthenumberoftimeseachvalue occurredandorganisedthedataintoafrequencytable.

Numberofhoursofcomputerusebychildrenin5A

Thentheyorganisedthedatainto adotplot,placingonedotabove thelinetorecordeachtimethe numberbelowthelineoccurredin thedata.Thedotsmustbe carefullylinedupsothattheyare spacedevenlyandcanberead easily.

11121314151617181920

Fromthedotplotwecanseethatthemostfrequentlyoccurringvaluewas14.This meansthatthemostfrequentlyoccurringnumberofhoursofcomputerusewas14. Dotplotsareusefulwhenwewanttoseewhatthedatahastosayveryquickly.

Linegraphs

Linegraphs aremadebyputtingpointson agraphandthendrawinglinestoconnect them.Weoftenuselinegraphstoshow dataliketemperature,wherethenumbers goupanddownovertime.Theline connectingthepointshelpsusseethese changes.

Thistableshowstemperaturesrecordedin PortlandononedayinJanuary.

Beforewecanplotanydata,weneedtocreatetheaxes.Inthisexample,the x-axis willshowthetimes,andthe y-axis willshowthetemperatures.Toplotthefirstpiece ofdata,weneedtodrawapointwhere7a.m.and28◦ Cmeetonthegraph.Repeat thisstepfor9a.m.and32◦ C,thenfortherestofthedata.

Ifwedrawadotforeachpieceofdata,wegetaseriesofdotslikethis:

Remembertonamethe x-and y-axestoshowwhatisbeingrecorded.

Joiningthepointstomakealinegraphisusefulinthisexample,asitgivesanideaof thelikelytemperaturebetweenthetimesthatthedatawasmeasured.

Wholeclass

1 a Surveystudentstofindtheireyecolour.Copyandcompletethedatatable.

Eyecolour

b Createalargepictographthatrepresentstheoccurrenceofdifferenteye coloursinyourclass.

c Drawalargecolumngraphtorepresentyourdataabouteyecolour.

d Whichcolouristhemostpopular?

Blue Brown Green Hazel Other

2 HereissomeinformationClaracollectedaboutthepocketmoneyshesavedover sixweeks.

Clara’spocketmoneysavings

a Drawalinegraphtorepresentthisdata.

b Whichweekdidshemaketheleastmoney?

c HowmuchmoneydidClaramakealtogether?

3 a Rolla6-sideddie25times.Createadotplotforthisdata.

b Whichnumberwasrolledmostoften?

1 a Drawapictographwithakeyandpicturestorepresentthisdata.

Moneyraisedforcharity

Writetheanswerstothesequestionsaboutyourpictograph.

b Whichclassraisedthemostmoney?

c HowmuchmoneywasraisedbyYear6altogether?

d Whichyearlevelraisedthemostmoney?

e Howmuchmoneywasraisedaltogether?

f Isthisthebestgraphforrepresentingthisdata?Whattypeofgraphmightbe betterandwhy?

2 a Drawacolumngraphtoshowthisdata.

Favouriterestaurants

b Write4questionsyoucouldaskaboutthegraph.Make1ofthesequestions aboutthemode.

Thesevehiclescheckedintotheshoppingcentrecarpark.

a Changetherepresentationofthis dataintoalinegraph.

b Theshoppingcentremanager needstoknowthebusiesttimesin thecarpark.Whatquestioncould theyask?

c Howmanycarshadusedthe carparkbefore12noon?

d Whichrepresentationofthedata doyoupreferandwhy?

4 MatthewandChristopherrecordedthetimeittakestoridetoschooleachdayin wholeminutes.

a Createadotplotforthisdata.

b Whatisthemostfrequentlyoccurringtimetakentoridetoschool?

c Onthedaysthattheygetgreenlightsalltheway,thetriptoschooltakes MatthewandChristopherlessthan10minutes.Ondayswherethetriptakes lessthan10minutes,MatthewandChristopherarriveearly.Onhowmany daysdidtheyarriveearly?

5 TheLancasterfamilyhavewatertanksastheironlysourceofwater.

At4a.m.,thewatertankshad300litresinit.Afteritrainedfrom5a.m.until 6a.m.,thevolumeofwaterinthetankwas400litres.

Between7a.m.and8a.m.thefamilywokeandgotreadyfortheday.Theyused 90litresforshowers,35litresforflushingthetoilet,4litresforwashingthedishes and1litreforcookingbreakfast.

At8a.m.theLancasterswenttoworkandschool. Itrainedfrom1p.m.to2p.m.andthetankreceived70litresofwater.

At4p.m.theLancastersreturnedhome.Theyused1litretomakecoffeeandtea.

At6p.m.theywateredtheirgarden,using74litresofwater.

a Copyandcompletethefollowingtable.

b Presentthisdatainalinegraphbyplottingthepointsandjoiningthemwith linesegments.

c Whatwasthevolumeofwaterinthetankat8a.m.?

d Howmuchwaterwasinthetankat6p.m.?

14C Reviewquestions–

1 IntheTanfamily,differentpeoplelikedifferentfruitsfordessert.Mumlikeskiwi fruitandstrawberries.Anhlikesneither.Dadlikesstrawberriesbutnotkiwifruit, andJedlikeskiwifruitbutnotstrawberries.

Drawatwo-waytabletorepresentthisdata.

2 SlipperyBanksSchoolchildrenwereaskedabouttheirpreferenceforthecolourof thenewschoolT-shirt.

SchoolT-shirtcolours

a Drawacolumngraphtoshowthisdata.

b Howmanychildrenvoted?

c Whichcolouristhemostpopular?

3 Thefootballcanteenhasthefollowingfoodonitsmenu.Themanagerrecorded thesalesforSaturdayandSunday.

a Calculatethetotalforeachfood itemoverthe2days.

b Whichfooditemsoldthemost?

c Whichfooditemsoldtheleast?

d Which2itemstogethersoldthe sameamount?

e Hotdogscomeinpacketsof20. Howmanypacketswouldhavebeen needed?Howmanyhotdogswere leftover?

f Whatwouldthemodebe?

4 Jamescountedthenumberofpencilsineachclassmember’spencilcase. 12,10,11,12,12,9,8,12,11,10,11,12,10,9,8,9,10,12,11,12,11,10

a Drawadotplotforthisdata.

b Whatisthemostfrequentlyoccurringnumberofpencils?

5 VanessawouldliketotrackhowlongshewatchesTVovertheweekandifthere aredaysthatshewatchesmoreTVthanothers.

• Monday,shewatched2hoursofTV.

• Tuesday,shewatchedTVbetween4p.m.and8p.m.

• Wednesday,shewatched3hoursofTVmorethanMonday

• Thursday,shedidnotwatchTV

• Friday,shewatchedbetween1700and2100

• Saturday,shewatchedTVfrom0900to1100and1500to1900

• Sunday,shewatchedanhourofTVmorethanonSaturday.

a Copyandcompletethefollowingtable.

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b Presentthisdatainalinegraphbyplottingthepointsandjoiningthemwith linesegments.

c DoesshewatchmoreTVatthebeginningoftheweekortheendoftheweek?

d Writedowntwothingsthatyounoticeaboutthisgraph.

Thestatisticalinvestigation

Itisimportanttogetgoodatusingstatisticsanddatasoyoucanunderstandhowthey areusedinnewspapersandonTV.Knowinghowtouseandunderstandstatisticsis importantinmanyjobs,likebeingaweatherforecasteroramarketresearcher. Statisticshelpusunderstandtheworldaroundus.

Thebestwaytounderstandhowdataiscollectedand shownistodosomedatacollectionandpresentation activitiesyourself.Inthischapter,wehavetalkedabout differentwaystoorganise,presentanddiscussdata. Now,wearegoingtoputitalltogethersowecanplan datacollection,investigationandinterpretation activitiesinanorganisedway.Wewillcallthisthe StatisticalInvestigationprocess.Eventhoughthere aremanywaystocollectandorganisedata,wewill usethesesteps.

Theplan

Thefirststepistoplantheinvestigation.Decideonthetopicandthinkabout questionsandhowdatawillbecollected.Forexample,iftheinvestigationwasonhow muchfruitYear5studentswereeating,questionscouldinclude:“Whatisyour favouritefruit?”and“Howoftenwouldyoueatiteachweek?”.Datacouldbe collectedthroughaclasssurvey.

Collectingdata

Oncewehavedecidedwhattocollectandhow,youcanstartcollectingdata.Inthe exampleabove,asurveywasdecidedupon.Tallymarkscouldhelprecordfavourite fruitsandafrequencytablecouldbeusedtorecordhowoftenthefavouritefruitis eaten.

Presentthedatainawaythatmakesiteasytounderstand,suchasusingagraph. Abargraphordotplotcouldshowfavouritefruitsfortheexampleaboveoradotplot forfrequency.Youcanthenmakestatementsaboutthedata,includingthemode-for example,’Applesarethefavouritefruitinourclass.’

Discussresults

Inthissection,findingscanbediscussed,andstatementscanbemadeasaresult.The investigationmightalsoresultinnewquestionsbeingasked:

1 Whatcanwesayaboutthetypeoffavouritefruit?

2 Whichdisplaysshowtheinformationclearly?Couldadifferentdisplayhavebeen better?

3 Whatarethelimitationsofmyinvestigation?

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Conductyourowninvestigation

GatherdataonhowYear5studentsusetheirtimeoutsideofschool,includingscreen time.Rememberyouwillneedtofollowthesesteps:

1 Plan: Whatquestionscanyouasktofindthenecessaryinformation?

2 Collectdata: Willyousurveytheclass?Howwillyoukeeptrackoftheanswers?

3 Presentdata: Howwillyoupresentyourfindings?Whatstatementscanyoumake aboutthemodeandotherfindings?

4 Discussresults: Whatstatementscanbemadeaboutthisinvestigation?What furtherquestionsdoyouhave?

Usefulskillsforthistopic

• identifyingoutcomesthataremorelikely,lesslikelyorequallylikely

• decidingwhetheraneventwillbeimpossible,possible,ordefinitelygoingtohappen

Vocabulary

Possibilities

• Probability • Certain • Likely

Unlikely

Impossible

Imagineyouareplanningapicnic.

1 Whatdoyouthinkistheprobabilitythatitwillrainonthedayofyourpicnic?

2 Isitimpossible,unlikely,likelyorcertain?

3 Explainyourreasoningandwhatfactorsmightaffectyourdecision.

Probability

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Probability isthestudyofhow likely or unlikely differenteventsaretohappen.It helpsusunderstandandpredicteveryday occurrences,fromtheweathertosimple games.Forinstance,inachance experimentlikeflippingacoin,thereare twoequallylikelyoutcomes:headsortails. Thismeanseachoutcomehasthesame chanceofhappening.

However,notallexperimentshaveequallylikelyoutcomes.Imaginerollinga six-sideddiewherefivesidesarepaintedblueandonesideispaintedred.Inthis case,landingonthebluesideismorelikelythanlandingontheredside,showing thatsomeoutcomescanbemoreprobablethanothers.

15A Probability

Ifwewanttomeasurethelengthofsomething,wemightusemetresorkilometres. Whatifwewanttomeasurethechanceofaparticulareventoccurring?

Untilnowyouhaveprobablyspokenofthechanceofsomethinghappeningusing wordssuchas‘likely’or‘unlikely’,withsomeeventsbeing‘certain’or‘impossible’. Inmathematicsweusethe wordprobabilitytodescribethechanceofanevent takingplace.

ThechanceofthePrimeMinisterwalkingintoyourclassroominthenext5minutesis notverylikely,butitisnotimpossible.Hereisawordscaletoshowprobability.

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Writingprobabilities

Wecanmeasureprobabilityonascalefrom0to1,where0meansthatthereisno chanceofaneventoccurring,and1meansthatthechancethattheeventwilloccuris certain.Wecandescribethechanceofaneventhappening,suchastheresultofaspin onaspinner,andwriteitasafraction.

Onthisspinnerthepossibleeventsarethefourdifferentcolours:red,green,blueor orange.

Itiscertainthatwewillspinoneofthefourcolours,sothechance ofspinninganycolouronthespinneriscertain.Onaspinnersplit intofoursegmentsofequalsize,thechanceofspinningjustoneof thecolours,suchasred,isequallylikely.Wesaythatthechanceof spinningredisoneoutoffour.Thisisbecausethereisonlyone waytospinred,butfourdifferentcoloursonthespinnerandeach eventisequallylikely.

Theprobabilityofspinningredonthisspinnerisoneinfour,or 1 4 .Thenumeratoris thenumberofwaystheeventmayhappen,andthedenominatoristhenumberof equallylikely possibilities

Describingprobabilities

• Impossibledescribesa0chanceofaneventhappening,suchasseeingapolarbear onBondiBeachinsummer.

• Notverylikelyisarounda25%chanceofaneventhappening,ora1in4chance, forexample,akoalawanderingintoyourclassroom.

• Equallylikelyisa50/50chanceofaneventhappeningora50%chanceofseeinga kangarooorkoalainanAustralianwildlifepark.

• Highlylikelyisarounda75%chanceofaneventhappening;forexample,aflywill crossyourpathduringsummer.

15A Wholeclass

1 Matcheachspinnertotheprobabilityof landingongreen:certain,highlylikely, equallylikely,notverylikely,orimpossible.

LEARNINGTOGETHER

2 Thereare50sweetsinajar.Halfareingreenwrappersandhalfareinorange wrappers.Youpickout10sweetsandfindyouhave7greenand3orange. Oftheremainingsweets,whichcolourareyoumorelikelytodrawout?Can youbecertainwhichcolouryouwilldrawoutnext?Discussyourreasoning withapartner.

15A Individual APPLYYOURLEARNING

1 Usingastandard6-sideddienumberedfrom1to6:

a Listtheeventsthatarepossible.

b Ifyourolledthedieonce,whatistheprobabilityofrollinga1?Writeyour answerasafraction.

c Ifyourolledthedieonce,whatistheprobabilityofrollinga4?Writeyour answerasafraction.

d Ifyourolledthedieonce,whatistheprobabilityofrollinga7?Writeyour answerasafraction.

e Ifyourolledthedieonce,whatistheprobabilityofrollinganevennumber? Writeyouranswerasafraction.

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f Ifyourolledthedieonce,whatistheprobabilityofrollinganumberless than6?Writeyouranswerasafraction.

g Ifyourolledthedieonce,whatistheprobabilityofrolling1,2,3,4,5or6? Writeyouranswerasafraction.

2 Angeliquehasabagof10coloureddiscs.Inthebag,thereare5bluediscs,3pink discsand2yellowdiscs.Angeliquepicks1disc.Writingyouranswersasfractions, whatistheprobabilitythatitwouldbe: yellow? a pink? b blue? c

3 a Whatistheprobabilityoftossingaheadifyoutoss1coin?Writetheansweras afraction.

b Whatistheprobabilityoftossing2headswhentossing2coins?

Completethistabletohelpyouworkoutthedifferentcoincombinationsthat arepossible.

4 Fromadeckof52playingcards,predictthelikelihoodofthefollowingusingthe language:certain,highlylikely,equallylikely,notverylikely,orimpossible.

Drawingaheart a Drawingaking b

Drawinganumber c Drawingaredorblackcard d

Takeadeckof52cardsandshuffle.Drawacardandrecordwhatitis,thenreturn ittothedeck.Repeatthis30times.Compareyourresultstoyourpredictions. Wereyourpredictionsaccurate?Whymightyourresultshavediffered?

Goingfurther:Createabargraphtovisualiseyourresultsandwriteastatement aboutyourfindings.

Chancechallenge

Youwillconductrepeatedchanceexperiments, includingthosewithandwithoutequallylikely outcomes,observeandrecordtheresults,and analysethedata.

YourclassishostingaProbabilityCarnival!

Youwillcreateandrundifferentcarnivalgamesthat involvechanceexperiments.Eachgamewillhelp youexploretheconceptsofprobabilityandchance.

Coinone

Tail

1 Designingthegame

Createatleasttwodifferentcarnivalgamesthatinvolvechanceexperiments.One gameshouldhaveequallylikelyoutcomes,andtheothershouldhaveoutcomes thatarenotequallylikely.

Examplesofgames

a Equallylikelyoutcomes:Aspinnerdividedintofourequalsections(red,blue, green,yellow)

b Notequallylikelyoutcomes:Abagwith3redcounters,2bluecountersand 1greencounter

c Otherresourceideas–adeckofcardsordice

2 Conductingtheexperiment

a Foreachgame,conductrepeatedtrialstoobservetheoutcomes.Aimto conductatleast20trialsforeachgame.

b Recordtheresultsofeachtrialinatableorchart.Forexample,keeptrackof howmanytimeseachcolourisspunonthespinnerorhowmanytimeseach counterisdrawnfromthebag.

3 Analysingthedata:

a Calculatetheexperimentalprobabilityofeachoutcomeforbothgames.For example,ifyouspunthespinner20timesandlandedonred5times,the experimentalprobabilityoflandingonredis5∕20or25%.

b Discussanydifferencesandpossiblereasonsforthem.

4 Communicatingfindings:

a Prepareapresentationorreporttoshareyourfindingswiththeclass.Include thedesignofyourgames,thedatayoucollectedandyouranalysisofthe results.

b Explaintheconceptsofequallylikelyandnotequallylikelyoutcomes,andhow theyaffectedtheresultsofyourexperiments.

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Algorithms

Analgorithmislikearecipe.Inbothcasestherearealistofingredientsandthen stepstofollowinagivenorder.Thinkaboutmakingacakeforexample.Ifyou havetheincorrectingredients,ortoomuchortoolittle,thenyouwon’tbeableto finishthecake.Also,youhavetofollowthestepsinorderandcookthecakefor thecorrectamountoftime,otherwisethecakewillnotbindtogetherorcook properly.

Amuddled-upcake

Hereisarecipeforathree-colouredspongecake.Unfortunately,everythingisout oforder.

• Takethecaketinsoutoftheoven,removethecakesfromthetinsandallowto cool.

• Putthecakemixtureintotheovenfor20to25minutesuntilthecakeisgolden brown.

• 375gbutteratroomtemperature.

• Mixthebutterandsugaruntilsmoothandcreamythenslowlyaddintheeggs, beatingcontinuously.

• Mixtheflourinwiththeotheringredientstoasmoothconsistency.

• Spreadthecreamoricingmixtureontothelayersoneatatime.

• 375gcastersugar.

• 375gselfraisingflour.

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• Pourthemixtureintothepans.

• Add4dropsofbluefoodcolouringtoonethirdofthemixtureand4dropsofred foodcolouringtoanotherthirdofthemixture.

• Greasethree20cmroundcaketinsandlinethebaseswithbakingpaper.

• 6eggs.

• 3tbspofmilk.

• redandbluefoodcolouring.

• whippedcreamoricingmixture(pre-prepared).

• Separatethemixtureequallyintothreebowls.

• Pre-heattheovento180◦ Celsius.

Function

Anotherimportantwordthatwewilluseandneedtounderstandis function.Let’s lookatsomeexamples.

Herearetwoexamplesoffunctionasaverb:

Ifwesaythatacarorsomeothermachineis functioning properly,wemeanthatitis workingproperly.

Manytoysanddevicesneedbatteriesforthemto function Usingfunctionasanounwecouldsay:

The function ofahoseistocarrywaterfromonepoint,thetap,toanother,thegarden.

The function ofpowerlinesistocarryelectricityfromthepowerstationtohomesand businesses.

Inmathematics,wealsousethewordfunction.Mathematicalalgorithmsareusually madeupofanumberoffunctionsappliedtogiveninputs.Tohelpyouunderstand mathematicalfunctions,wearegoingtousesomefunctionmachines.

16A What’smyrule?

Functionmachinesaregiveninputs,liketheingredientsinacake,andthencarryouta setofinstructions,therecipe,toproduceanoutput,thecake.Inthediagrambelow youcanthinkoftheblueballastheingredientsandthemachineasthekitchen.

Functionmachinescanbeusedinmathematicsaswell.Theyprovidetheorderin whichtheinstructionsneedtobefollowedtogetthedesiredanswer.Theinput,blue ball,couldbeanumberandtheoutput,inthiscaseayellowball,wouldbetheresult ofperformingoperationsonthatnumber.

Themathematicaloperationsthatyoualreadyknowarethefouractions(orfunctions) ofaddition(+),subtraction( ),multiplication(× or ∗)anddivision(÷ or/).Thereare alsootheroperationsthathavesymbols,likethesquarerootsymbol √ ,thatyouwill learnaboutoverthenextfewyears.

Inthefollowingexercisewewillworkwithfunctionmachines.

Anaccompanyingactivitysheetwillguideyoutoanonlineprojectfeaturingdigital functionmachines.

Ourmachineswillhavethreeessentialpartsas:

1 aninput,inthiscasetheblueball.

2 anactionoroperationtoperform,thegreytube.

3 anoutputoranswer,theyellowball.

Thiscanalsobedrawnasaflowchart.

Functionmachinefigure.

Example1

Aninputnumberisfedintothefunctionmachineandreturnsanoutputasshown. Whatmightbethefunction?

input number is 19

19 + 8 = 27sotherulecouldbeadd8. Thiscanbewrittenas: add8 plus8or +8

Theoutnumberisbiggerthan theinsotheoperationismost likelyadditionormultiplication. Ofthese,adding8isthesimplest waytoarriveattheanswer.

16A Individual APPLYYOURLEARNING

1a Afunctionmachinesaddsaparticularnumbertoanynumberthatisinput.The firsttwo‘out’valueshavebeenrecorded.Completetherestofthetable. in 3 5 7 9 11 13 out 12 14

b Writedowntheoperation(action)performedbythisfunctionmachine.Thatis, whatnumberisbeingaddedtoeachnumberinthe‘in’row?

2a Afunctionmachinessubtractsaparticularnumberfromanynumberthatis input.Thefirsttwo‘out’valueshavebeenrecorded.Completetherestofthe table. in 10 12 14 16 18 20 out 7 9

b Writedowntheoperation(action)performedbythisfunctionmachine.Thatis, whatnumberisbeingsubtractedfromnumberinthe‘in’row?

3 Functionmachinescanalsobedrawnasflowcharts.Forthefollowingflowcharts, writedownwhatthefinalanswer,theoutput,willbe.

4 Fortheflowchartsbelow,theoperationismissing.Usingjustaddition,subtraction, multiplicationordivisionforeachone,determinewhattheruleis.

16B Beingafunction machine

Itispossibleforpeopletoactasfunctionmachines.Todosohowevermeansthe personactingasthefunctionmachinemustperformexactlythesamesequenceof operationsoneachinputnumber.

16B Wholeclass LEARNINGTOGETHER

Workingwithoneortwopartnersdevelop:

1 Afunctionmachineformultiplyinganinputnumber

2 Afunctionmachinefordividinganinputnumber

3 Afunctionmachineforeithermultiplyingordividinganumbereachtimethe programisrun.

Whenyouhaveyourcompletedyourfunctionmachines,testthemouton othermembersoftheclass.

16C Puttingfunctionmachines together-Twostepmachines

Whenyoulookatmosttoolsandmachinesintherealworld,theyoftenhaveseveral stepsinvolvedingettingtothefinalgoal.Eachofthesestepsperformsasimple functionandthentheresultismovedalongtothenextstep.

Usefulcalculationsinmathematicsaresimilar.Itisveryraretojustwanttoaddor multiplynumberstogether.

Therearemanymoreoccasionswhereweneedtoaddandmultiplynumberstogether inaparticularsequence.In thisexercise,wearegoing tomodelsuchbehaviourby puttingpairsoffunction machinesfromExerciseA together.

Considerwhathappenswhenwejoina multiplicationmachinewithanaddition machine.Wecanimagineittolooklikethis.

Similarly,wecoulduseaflowchart.

So,taking4astheinputnumberweget:

16C Individual APPLYYOURLEARNING

Usetheflowchartsbelowtodeterminetheoutputnumbersforeachofthese functionmachines.

1a + 8 × 5 4

b × 5 + 8 4

2a 9 × 3 18

b × 3 9 18

3a + 15 ÷ 5 10 b ÷ 5 + 15 10 4a 9 ÷ 3 27

÷ 3 9 27 5a × 3 ÷ 4 8

÷ 4 × 3 8

6a Foreachpairofmachinesinthequestionsabove,whatdoyounotice abouttheoperations?

b Whichpairisdifferenttotheothers?Explainwhytheorderdoesnotmatter forthatpairofoperations.

c Whyisitimportanttobecarefulabouttheordereachoperationisdone inwhenitcomestowritingalgorithms?

16D Puttingthefunction machineinreverse

Likecars,itispossibletoputourfunctionmachineintoreverse.Whenweputacar intoreverse,ittravelsbackward.Whenwerunatvormovieclipinreverse,broken platesbecomewholeagain,fallenbuildings‘magically’rebuildthemselves.

Inasimilarway,puttingafunctionmachineinreversetakestheanswer,undoesthe originaloperationsandgivesyouthestartingnumber.

Example2

Thefunctionmachineshownisknowntoadd9totheoriginalnumber.Ifthe answeris15,whatwasthestartingnumber?

Solution

Let’suseadiamondshapetostandfortheinputnumber.Thatmeanswecanwrite theruleas:

+ 9 = 15

Usingasimplediagramwehave:

Forthisexamplewehave:

Reversingtheprocessmeansstartingwiththeoutputnumberandchangingeachof theactionsintoitsopposite.Forthisexample,theoppositeofaddingissubtracting sowesubtract9.

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My output number is 15

Noticethatthearrowsindicatingthedirectionthattheballismovingthroughthe machinehavealsobeenreversed.

9 Input 15

So,wewritethisas15 9meaningthattheinputnumberis6.

Or,if ◊ + 9 = 15, then ◊ = 15 9 ◊ = 6

16D Individual APPLYYOURLEARNING

1 Foreachofthefollowingfunctionmachines,writeouttheflowwhentheyareput inreverse.

2 Forthefunctionmachinesbelow,theinputnumberismissing.Usethereverse flowdiagramtoworkoutwhateachinputnumberis.

3 Forthetwostepfunctionsbelowworkbackwardtodeterminetheinputnumbers.

16E Walkingaround

Algorithmscanbeusedwithmorethanjustnumbers.Theycanalsogivedirectionsfor movingobjectsinspaceorformovingaroundobjects.Takethesimplecaseofwalking aroundasquare.

Theinstructionscouldbe:

• Takeonestepforward.

• Turn90◦ totheleft.

• Takeonestepforward.

• Turn90◦ totheleft.

• Takeonestepforward.

• Turn90◦ totheleft.

• Takeonestepforward.

• Turn90◦ totheleft.

Thisalgorithmtakeseightlines.Anotherwayofwritingdownthesameprocessis:

• Repeatthefollowingfourtimes.

• Takeonestepforward.

• Turn90◦ totheleft.

Athirdwayofdescribinghowtowalkaroundasquareisdifferentagain.Followthe followinginstructionswhilealwaysfacingthesameway,forexample,alwaysfacing thefrontoftheclassroom.

• Takeonesteptotheright.

• Takeonestepforward.

• Takeonesteptotheleft.

• Takeonestepbackward.

16E Individual APPLYYOURLEARNING

Considereachofthealgorithmsforwalkinginasquareabove.

1 Whichonemakesthemostsensetoyou?Explainwhy.Discussthisingroups orasaclass.

2 Thesecondalgorithmismuchshorterthanthefirst.Whenwouldhavinga shorterdescriptionbeuseful?

3 Whatisthekeydifferenceinthethirdalgorithmcomparedtothefirsttwo? Doyourecognisethemovesdescribed?

4 WritealgorithmsforwalkingouttheshapeofthelettersL,M,N,VandZ.

5 Foreachofthefollowingfigures,writeoutanalgorithmforwalkingaround itandhaveapartneractuallyfollowyourinstructionstomakesureitworks. Havingalargeprotractorwillbehelpful.

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16F Maps

Manynewcars,andalmostallmobilephones,haveGPSnavigationthesedays.This technologycombinesmapswithalgorithmstodetermineeitherthepathwiththe shortestdistancebetweentwoplaces,ortheshortesttraveltime.Inthisexercise,you aregoingtobetheGPSnavigator.

Hereisamapshowingthehousesofthreefriends,Anh,BobandCarol.Thereare threeroutesthatAnhcantaketogettoBob’shouse.

Usingcompassdirectionsandthedistancescaleprovidedwriteoutinstructionsin

3 Whichpathisthelongest?

RememberthatpartoftheroleofGPSistochoosebetweenthedifferentpossible routes.

4 IfAnhtravels1kmevery3minutes,howlongwouldittakehimtotravelalong eachroute?

5 Ifeachchangeofdirectionisatanintersectionandadds30secondstothetrip, whichpathwouldyourecommend?

6 Ifthereareroadworksalongthebluepathcausinga30minutedelay,whichpath wouldyourecommendnow?Again,givereasonswhy.The30seconddelayper intersectionstillapplies.

7 Usingthegridmapbelowandonlytravellingalongtheroads,answerthe followingquestions.

a CanyoufindapaththatstartsatAandtravelstoeachoftheotherpoints? Writedowntheinstructionsyouwouldgivetoafriendforthemtofollow.

b NowfindandwritedownasetofinstructionsthatwillstartatA,thenpass throughB,thenCandendupatD.

c Aroute’crossesoveritself’ifitusesthesamestreetmorethanonceorifthe pathcrossesaroadithasalreadytravelledon.Doesyourpathcrossoveritself? FindanddescribeanotherpathA-B-C-Dthatdoesnotcrossoveritselfand writedowntheinstructionsforit.

d ArethereanyintersectionsyoucouldmarkasEonthemapthatyoucannot reachwithoutcrossingoveranexistingpath?Canyounowfindanewpath A-B-C-D-Ethatdoesnotcrossoveritself? Thisisthesortofproblemthattransportcompanieshavetosolveona regularbasis.

16G Algorithmicthinking problems

Inthissectionyouwilllookatsomequestionsthatinvolvealgorithmicthinking.The questionsaretakenfromtheComputationalandAlgorithmicThinking(CAT) competition,formerlyknownastheAustralianInformaticsCompetition(AIC).The competitionisrunbytheAustralianMathematicsTrust.Itisnotintendedthatyouuse codingtosolvethesequestions.

1 Lotusbirds

Lotusbirdssteporjumpfromlilypadtolilypadinsteadofswimminginthewater. AlotusbirdstartsonthelilypadmarkedwithanX.Thenitmoves10times accordingtothesequenceofarrows.

X ↑←↓→↑←↑→↓↓

Eachmoveistothefirstlilypadinthatdirection.

Afterthese10moves,onwhichlilypaddoesthelotusbirdfinish?

2 MazeBot

MazeBotisarobotdesignedtofinditswaythroughamaze.Themazeconsistsof rooms,eachwithanumber.

• MazeBotwillalwaystraveltotheneighbouringroomwiththehighestnumber.

• MazeBotwillnevertraveltoaroomithasalreadybeenin.

Forinstance,ifitstartedinroom30inthismaze,MazeBotwouldmovetoroom 26,then22,then12.

Inthefollowingmaze,MazeBotstartsinroom50atthecentre.Whichroomdoes itendupin?

Alaserisfiredintoaroomwithdouble-sidedmirrors.Thelaserreflectsoffseveral mirrorsandthenexitstheroom.

Inthisroom,thelaserreflectsoffmirrorsthreetimes.

Inthisroom,thelaserreflectsoffmirrorsseventimes.

Howmanytimeswouldthelaserreflectoffamirrorbeforeexitingtheroom below?