ICE-EM Mathematics 4ed Year 6 – uncorrected sample pages by cambridge_aused

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CambridgeUniversityPress&AssessmentisadepartmentoftheUniversityofCambridge. WesharetheUniversity’smissiontocontributetosocietythroughthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org

©TheUniversityofMelbourneonbehalfoftheAustralianMathematicalSciencesInstitute(AMSI)2017,2026

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Firstpublished2017 FourthEdition2026 2019181716151413121110987654321

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ISBN978-1-009-76084-3

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ICE-EMMathematicsFourthEdition isaseriesoftextbooksforstudentsinYears5to10throughout AustraliawhostudytheAustralianCurriculumV9.0anditsstatevariations.

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

Highlightsofthe ICE-EMMathematicsFourthEdition seriesinclude:

• updatedandrevisedcontenttoprovidecomprehensivecoverageoftheAustralianCurriculumV9.0and itsstatevariations,inasingletextbookforeachyearlevel

• anewdesigntoprovidestudentswiththebestpreparationforsuccessinseniorhighschoolsubjects,such as SpecialistMathematics and MathematicalMethods (MathematicsExtension and Advanced Mathematics inNSW)

• newcontenttohelpconnectmathematicallearningtoFirstNationsPeoples’knowledge andcultures

• AMSI’sextensiveonlinesupplementarycontentsuchasincludingworkedsolutions,videoexplanations andtheAMSICalculateteacherandstudentresources

• 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.Thetextbooksareclearlyandcarefullywritten,andcontainbackground 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 RowenaBallandDr.HongzhangXufromthe MathematicsWithoutBorders programattheAustralian NationalUniversity.Therearequestionsonastronomyandeclipses,songlines,fishingpractices,animal tracking,gameplaying,kinshipstructuresandfiremanagement,whichwillenablestudentsandteachersto learnabouttheculturesofFirstNationsPeoples,inamathematicalcontext.

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

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

SeonaidChioistheHeadofTeachingandLearningatGrimwadeHouse,MelbourneGrammarSchool.With over20yearsofteachingexperienceandmorethanadecadeofleadingteachingandlearning,shebringsa depthofknowledgeincurriculumdesign.Herleadershipspansarangeofschoolsacrossthreedifferent countries,enrichingherapproachwithdiverseeducationalperspectives.Sheispassionateabout empoweringteacherstobuildstudentconfidenceandcuriosity,particularlyinmathematics,through collaborativepractice,explicitteaching,andreflectivedialogue,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.

GarthGaudry

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

JacquiRamagge

JacquiRamaggeisExecutiveDeanofSTEMattheUniversityofSouthAustraliaandisPresidentofthe AustralianCouncilofDeansofScience.Aftergraduatingin1993withaPhDinMathematicsfromthe UniversityofWarwick(UK),sheworkedattheUniversityofNewcastle(Australia),attheUniversityof 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.

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AuthorsofFirstNationscurriculumcontent

RowenaBall

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

HongzhangXu

DrHongzhangXuisanAdjunctResearchFellowattheAustralianNationalUniversity(ANU)andasenior ecohydrologistattheMurray–DarlingBasinAuthority.HehasworkedattheMathematicalSciencesInstitute ANU,asapost-doctoralresearcher,investigatingAboriginalandTorresStraitIslandermathematicsand sciences.Hisworkisbroadlyreadandcitedfrequently,andheregularlyreceivesinvitationstocommenton popularissuesfrommajormedia,suchasCNN,ABC,TheConversation,Bloomberg,andNatureNews.

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 Foundation,fortheirfinancialsupportaspartoftheChooseMATHSproject.

Wehopethatyouenjoyusingthistextbookandthatithelpsyouprogressalongyourownmathematical journey.

MichaelEvansandTimMarchant, AustralianMathematicalSciencesInstitute, September2025

Theauthorandpublisherwishtothankthefollowingsourcesforpermissiontoreproducematerial:

Everyefforthasbeenmadetotraceandacknowledgecopyright.Thepublisherapologisesforanyaccidental infringementandwelcomesinformationthatwouldredressthissituation.

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,ratherthanseparatingthemout.Studentsare encouragedtodevelopandapplyUnderstanding,Fluency,Problem-solvingandReasoningskillsinevery exercise.

Questiontags

Thequestionsineachchapteraretagged.Thetagsareintendedasaguidetoteachers.Theyshouldbe regardedasawayofencouragingstudentprogress.

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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.

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

InteractiveTextbook

TheInteractiveTextbookistheonlineversionoftheprinttextbookandcomesincludedwithpurchaseofthe printtextbook.Itisaccessedbyfirstactivatingthecodeontheinsidecover.Itiseasytonavigateandisa valuableaccompanimenttotheprinttextbook.

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 Gettingready

Usefulskillsforthistopic

• understandingplacevalueofnumbersto1000000andbeyond

• anunderstandingofeachdigit’spositionwithinanumberanditsplacevalue

• recognisetheroleofzeroinplacevaluenotation

• recognisethevalueofanumbercanberepresentedonanumberline

Vocabulary

Digits • Placevalue

Integers

Negativenumbers

Positivenumbers

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 Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage

Australiaisacountryofextremetemperatures.Thehighestrecordedtemperaturewas closeto50degreesCelsiustakenin1960atOodnadatta,SouthAustralia.Thelowest recordedtemperaturewas–23degreesCelsiustakenin1994atCharlottePass,NSW.

Whatistherangebetweenthesetemperatures?

Howcouldyoushowyourthinkingtoproveyouranswer?

Shareideaswithapartnerandyourclass.

Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers Positiveandnegative wholenumbers wholenumbers

Thischapterbeginsbylookingatwholenumbers;thenitlooksatnegativewhole numbers.

Thewholenumbersaresometimescalledthe‘countingnumbers’.Thewhole numbersarethenumbers0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,andsoon.1canalways beaddedtoanywholenumberandgetanotherwholenumber.Thatiswhythe listofwholenumbersissaidtobeinfinite–itneverends.

Negativewholenumbersareusedtodescribenumbersbelowzero, suchastemperatures.Thebasicunitofmeasurementfortemperature isthe degreeCelsius

Degreesareshownusingthedegreesymbol ◦ .Celsiusisabbreviated as C.Thesymbolandtheabbreviationarecombinedinto ◦ C,sotwelve degreesCelsiusiswritten12◦ C.Thetemperatureatthetopof MtKosciuskoinwintercanbeaslowas 18◦ C.

A thermometer isusedtomeasuretemperature.Therearemany differenttypesofthermometers.Eachtypeofthermometerhas differentscales.

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1A Placevalue

The digits 0, 1, 2, 3, 4, 5, 6, 7, 8and9canbeusedtowrite:

• a5-digitnumbersuchas89371

• a6-digitnumbersuchas450672

• a1-digitnumbersuchas2

• anywholenumber.

Eachplaceinanumberhasaspecialvalue. Forexample,inthenumber3427568:

• the‘3’means3millions

• the‘4’means4hundredsofthousands

• the‘2’means2tensofthousands

• the‘7’means7thousands

• the‘5’means5hundreds

• the‘6’means6tens

• the‘8’means8ones.

3427 56 8

Thevalueofadigitchangesifitisinadifferentplace.Ifwetakethenumber3427568 (onthepreviouspage)andchangethepositionofthedigitstomake8674325:

• the‘8’means8millions

• the‘6’means6hundredsofthousands

• the‘7’means7tensofthousands

• the‘4’means4thousands

• the‘3’means3hundreds

• the‘2’means2tens

• the‘5’means5ones.

926714538isreadasninehundredandtwenty-sixmillion,sevenhundredand fourteenthousand,fivehundredandthirty-eight.

Example1

Writethevalueofthe8ineachnumber.

Solution

a In28774,the8isinthethousandsplace,soithasthevalueof8thousands or8000.

b In289,the8digitisinthetensplace,soithasthevalueof8tensor80.

c In18693002,the8digitisinthemillionsplace,soithasthevalueof 8millionsor8000000.

d In781721,the8digitisinthetensofthousandsplace,soithasthevalueof 8tensofthousandsor80000.

2 Writethenumber.

a 23hundreds,5tensand3ones

b 4thousands,5hundredsand18ones

c 128thousands,43tensand9ones

d 34ones,18hundredsand16millions

e 17ones,1tenthousandand23hundreds

f 4hundredsofmillions,26tensofthousandsand6hundreds

1A Individual APPLYYOURLEARNING

1 Writeeachnumber.

a Onehundredandfivethousand,twohundredandforty-nine

b Thirteenmillion,sevenhundredandninety-eightthousand,fivehundred andsixty-two

2 Writeeachnumberinwords.

3 Writethevalueofeachhighlighteddigit.

4 Copyandcompletethisplace-valuechart.

1B Thenumberline

Anumberlinehelpsusmakesenseofnumbers.Tomakeanumberline,drawalineon thepage.Thearrowsshowthatthelinecontinuesinthesamewayforever.

Youcanmakenumberlinesoutofstringortape,ordrawthemonpaper.Theycanbe usedtoshowanynumber,fromtheverysmallestnumbertothelargestnumberyou canthinkof.

Showwhere150wouldbeonthisnumberline.

Markinstepsof100.Thenumber150ishalfwaybetween100and200.

Numberlinescanalsobeusedtocomparenumbers.

Numbersgetlargermovingtotherightonthenumberline.So700islargerthan200 becauseitliesfurthertotheright.

Numbersgetsmallermovingtotheleftonthenumberline.So200issmallerthan500 becauseitliesfurthertotheleft.

Useanumberlinetoshowthat113islargerthan108.

Solution

Placebothnumbersonthenumberline.113istotherightof108,so113isthe largernumber.

1B Wholeclass LEARNINGTOGETHER

1 Drawthesenumberlinesinyourworkbook.

a Markin0and1000,thenmark100, 250and777.

b Markin0and1000000,thenmark350000,500000and896000.

1B Individual APPLYYOURLEARNING

1 Drawanumberline,thenmark0and1000onit.Uselargedotstomark100, 400, 825and940.

2 Drawanumberline,thenmark0and10000onit.Uselargedotstomark 1000, 2500and9999.

3 Whatnumbersdothedotsonthesenumberlinesshow?

1C Integers

Ifyouhaveeverbeentothesnoworusedafreezertokeepfood,thenyouwillalready befamiliarwithnegativenumbersbeingusedtodescribetemperaturesbelowzero. Youmightnothavethoughtaboutnegativenumbersinamathematicalsense.Inthis sectionwearegoingtoextendthenumberlinetoincludenegativenumbers.

Thisnumberlineshowspositivenumbers.Thenumberscontinuetotherightofthe numberlineindefinitely.

Butwhathappenstotheleftof0onthenumberline?

Thenumberstotheleftofthezeroareknownasthe negative numbers.Allofthe negativeandpositivewholenumbers,togetherwithzero,arecalledthe integers. Thisnumberlineshowsnegativeandpositivewholenumbersandzero.

Everyintegerexceptzerohasasymboltoshowifitispositive (+) ornegative (−) Since 0isequalto + 0,weusuallywrite0withoutasymbol.Apositiveintegercan bewrittenwithorwithouttheplussign,so +3isthesameas3.

Wecanuseintegerstodescribetemperatures.Whenthetemperatureisbelow freezing(whichis0◦ C),thetemperatureiswrittenasanegativenumber,suchas 4◦ C. Theword‘minus’isoftenusedinweatherreports.Forexample, 5◦ Cmightberead outas‘minusfivedegreesCelsius’.

Whenthetemperatureisabovezero,wedonotusuallyincludethe + symbol.For example,twenty-twodegreesiswritten22◦ C.

3rd sample pages

Asanexample,Mickcheckedthethermometeratthebackdooroftheskichalet. Itshowed5◦ C.Overnight,thetemperaturedropped8degrees.Inthemorningthe temperaturewas 3◦ C.Thenextnight,thetemperaturewas 7◦ C,whichwas evencolder.

Thediagrambelowshowsabuildingthathas3floorsabovethegroundfloorand 3carparksbelowtheground.Itislikeaverticalnumberline.

Thegroundflooriszero.Imaginethatyouareinaliftgoingdownfromthefirstfloor tocarpark3.

Thissequenceofintegersgoingfrom1to 3is:1, 0, 1, 2, 3. Ifyouparkyourcarincarpark3andgouptothethirdfloor,yougothroughthis sequence: 3, 2, 1, 0, 1, 2, 3.

Integersarealsousedinfinances.Apositivenumberwouldrepresentmoneyaperson hasintheirwalletorinabankaccountforexample.Anegativenumberwould representmoneytheyowe,oradebt.

Ifyouaddmoneytoabankaccountyouaredepositing,butifyoutakemoneyout youarewithdrawing.[Insertimageofbankstatementhere]

3rd

Wecandecidewhetheranintegerislargerthananotherjustaswedidwithwhole numbers.

Thesamerulesapply.

• Numbersgetlargermovingtotherightonanumberline.

• Numbersgetsmallermovingtotheleftonanumberline.

Usingtheserules,weseethat:

3islessthan5

1islessthan0

8islessthan 4

Example4

Writethenumberthatis5lessthan2.Useanumberlinetohelpyou.

Solution

Startfrom2andtake5stepsalongthenumberlinetotheleft.

5lessthan2is 3. Example5

Useanumberlinetoplacetheseintegersinorder,fromsmallesttolargest.

Marktheintegersonthenumberline.

1C Wholeclass LEARNINGTOGETHER

1 Drawaladderwitheachrunglabelledfrom9to 9,asshown. Therungsgoingupfromzeroarepositiveintegers,andthe rungsgoingdownfromzeroarenegativeintegers.Usethe laddertoactoutthesesituations.

a Startfrom5.Moveup4rungs. Whichrungareyouon?

b Startfrom0.Movedown3rungs. Whichrungareyouon?

c Startfrom3.Movedown10rungs. Whichrungareyouon?

d Startfrom 2.Movedown3rungs. Whichrungareyouon?

e Startfrom 3.Moveup5rungs. Whichrungareyouon?

2 Theschoolcafeteriabought60applesfor50centseachtoselltostudents. Copythetablebelowtofindanswerstothesequestions:

a Calculatethetotalinitialcostonthetablebelow.

b Thecafeteriadecidedtoselltheapplesfor $1eachinthehopeofmaking aprofit.Overtheweektheyonlymanagedtosell20applestostudents. Calculatetheincomethecafeteriageneratedonthetable.

c Didthecafeteriamakeaprofitorlossonthesaleofapplesthisweek? Tocalculatewemustsubtractthecostsfromtheincome.Iftheincome islessthantheoutput,thereisalosswhichisrepresentedasanegative number.Iftheincomeishigherthanthecosts,thereisaprofit.Record yourworkingsonthetable.

1C Individual APPLYYOURLEARNING

1 Drawanumberlineandmarktheseintegersonit.

2, 4,

2 Whattemperaturedoesthisthermometershow?

a Ifthetemperaturedroppedby2◦ Cwhatwouldthe temperaturebenow?

b Ifthetemperaturethenroseby10◦ C,whatwouldthe temperaturebenow?

3 At3∶00a.m.,thetemperatureatMtMacedonwas 2◦ C.By6∶30a.m.,the temperaturehadrisenby3◦ C.By10∶15a.m.,thetemperaturehadrisenafurther 6◦ C.Whatwasthetemperatureat10∶15a.m.?

4 Listtheintegersthatare:

a smallerthan2andlargerthan 3

b largerthan 10andsmallerthan 2

5 Puttheseintegersinorder,smallesttolargest.

6 Puttheseintegersinorder,largesttosmallest.

a 9, 3, 4, 2, 0

b 111, 99, 56, 99, 56

c 77, 136, 0, 3, 2

7 Thissequenceisgoingdownbytwos:4, 2, 0, 2, 4, 6.Writethenext 5integersinthesequence.

SAMPLEPAGES

8 Theschoolcaféspent $80on40fruitjuicebottles.Bytheendoftheweek,they hadsoldall40juicebottlesfor $3each.Didthecafemakeaprofitorloss?Explain youranswer.

Uncorrected 3rd sample pages

9 WhenPatrickwokeat7∶00a.m.,thetemperaturewas12◦ C.Itroseby 1degreeinthefirsthour,2degreesinthesecondhourand3degreesinthethird hour.Thetemperaturecontinuedtoriseinthesamepatternuntil1:00p.m.,when acoolchangearrived.Thetemperaturehaddroppedby17degreesbythetime Patrickcamehomefromschool.WhatwasthetemperaturewhenPatrickarrived home?

1D Reviewquestions–Demonstrateyourmastery

1 Writetheseinnumbers.

a Twohundredandtenthousand,fivehundredandsixty-three b Elevenmillion,ninehundredandsixty-seventhousand,threehundredand twenty-four

2 Writethesenumbersinwords. 23178 a 914207 b

3 Writethevalueofthehighlighteddigit. 359267 a

4 Copyandcompletethisplace-valuechart.

5 Drawanumberlinewith0and1000markedonit.Uselargedotstomark 300, 700, 860and975.

6 Drawanumberlinewith0and10000markedonit.Uselargedotstomark 1111, 3500and9000.

Writethenumbersshownbythereddotsonthesenumberlines.

8 Eachletteronthisnumberlinerepresentsanumber.Writethenumbers.

9 Drawanumberlineandmarktheseintegersonit. 17, 5, 2, 21, 7, 3, 2

10 Listtheintegersthatare:

a between0and 5

b smallerthan3butlargerthan 10

c smallerthan 22butlargerthan 33

11 Writetheseintegersinorder,smallesttolargest.3, 6, 8, 9, 0, 33, 133, 132

12 Writetheseintegersinorder,largesttosmallest.89, 2, 66, 101, 4, 45, 6

13 Calculatethenewtemperatureafterthesetemperaturechanges.

a Startat8◦ C.Takeoff15◦ C.Whatisthetemperature?

b Startat 6◦ C.Takeoff12◦ C.Whatisthetemperature?

c Startat 18◦ C.Addon19◦ C.Whatisthetemperature?

14 Duringthewinter,thetemperatureinMelbournecandropovernight.Itwas9◦ Cat 6p.m.,butbymidnightthetemperaturehaddroppedby12degrees.Whatwas thetemperatureatmidnight?

15 Theshoppurchased50t-shirtsat $7eachtosellintheirstore.T-shirtsweresold for $10each.Theshoponlysold20t-shirts.Didtheshopmakeaprofitorloss? Explainyouranswer.

16 TugofIntegersGame

Thisisagamefor2playerstopracticeyourunderstandingofmovingalonga numberline. Youwillneed: 2six-sideddiceand2counters(oneforeachplayer).

Steps:

• Drawanumberline(horizontalorvertical)withnumbers 20to20onit.

• Decidewhowillstartonnegative20andwhowillstartonpositive20.Youmust startatoppositeendsofthenumberline.Youraimistoreachtheoppositeend towhichyoustart,soifyoustarton 20youaimtogetto +20,andviceversa.

• Tobeginoneplayerrollsthe2dice.Youmayadd,subtract,multiplyordivide thesenumberstogetanewnumber.Forexample,ifyourolleda6anda3,you couldaddthemtogethertoget9.Or,youcouldsubtract3from6toget3.Or, youcouldmultiply6times3toget18.Or,youcoulddivide3into6toget2. Yourresultingnumberishowmanyspacesyoumustmoveonthenumberline.

• Tobegin,thepersononpositive20movestotheright,andthepersonon negative20movestotheleft.Afterthefirstmoveyoucanchoosewhichway youwillmoveonthenumberlinebystatingwhetheryouwilladdyournumber orsubtractitfromyourcurrentposition.

• Towinthegameyoumustreachthenumberontheoppositeendexactly.

• Playanewgame,startingattheoppositeendfromyourlastgame.

1E Challenge–Ready,set,explore!

Flipnumbers

Thinkofa5-digitnumber,thenwriteitdown.Thisisyour‘flipnumber’.Youcannot showanyoneelseyourflipnumber–youcanonlytellittothembackwards.For example,ifyourflipnumberis14165,youwouldsay‘five,six,one,four,one’.You canonlyaskaclassmatefortheirflipnumberonce.Sorecordyourclassmates’flip numbersinasecretplace(likea‘lifttheflap’inyourmathsworkbook).

UNCORRECTEDSAMPLEPAGES

Five, six, one, four, one

Challengequestions

1 Writedownhowmanyones,tens,hundreds,thousandsandtensofthousandsare inyournumber.

2 Writedownhowmanyones,tens,hundreds,thousandsandtensofthousandsare inyourneighbour’sflipnumber.

3 Countupfromyourflipnumberbysevens.Writethefirst30numbersyoucome tointhesevenscountingpattern.

4 Countdownfromyourflipnumberbyfours.Write30numbersinthefours countingpatternyoucometo.

5 Addyourflipnumbertotheflipnumberofthepersonnexttoyou.

6 Addyourflipnumbertotheflipnumbersoftherestofthetableyouusuallysit with.

7 Whatisthebiggesttotalyoucouldgetbyaddingthreeflipnumbers?Whatisthe lowestpossiblesumofthreeflipnumbers?

8 Subtractyourflipnumberfrom100000.

9 Subtractyourflipnumberfromaclassmate’sflipnumber.Isyouranswerpositive ornegative?

10 Estimatetheresultwhenyoumultiplyyourflipnumberby3.Dothemultiplication. Whatisthedifferencebetweenyourmultipliedflipnumberandyourestimate?

11 Multiplyyourflipnumberby10,thenby100andthenby1000.Describethe pattern,thenkeepthepatterngoing.

12 Divideyourflipnumberby10,by100,by1000andby10000.Whatdoyou notice?

13 Imagineyourflipnumberisanumberofseconds.Howmanydays,hours,minutes andsecondswoulditbe?

14 Imagineyourflipnumberistheheightofabuildingincentimetres.Howmany metrestallisit?

15 ImagineyouhaveajarcontainingyourflipnumberofSmarties.Howmanypeople couldyougive239Smartiesto?Wouldyouhaveanyleftover?

CHAPTER

Usefulskillsforthistopic

• anunderstandingoftheplacevalueofnumbersto1000000andbeyond

• fluencywithadditionandsubtractionfactswithin20

• recordingadditionandsubtractionequationsonaverticaldiagram

• efficientmentalstrategiesforadditionandsubtractione.g.partitioning

Vocabulary

sum

digit

partitioning

total

algorithm

calculate

difference

decomposition

compensation

1 Lookattheequationsbelow.Wouldyouuseamentalstrategyoravertical algorithmtosolveeachequation?Explainyourthinking.

2 Takeafewminutestosolveeachequationindependently.Shareyourstrategies andsolutionswithapartner.

+ 23 a

+ 127 + 640 c

998 e

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

UNCORRECTEDSAMPLEPAGES

Additionandsubtractionaretwoofthemostfrequentlyusedmathematical operations.Weuseadditionandsubtractioneveryday,andnotjustatschool.

Whenweuseadditiontoworkoutthe sum oftwoormorenumbers,thistellsus the total amountornumberofthingswehave.

Forexample,duringasoccergame,Handel scoresfourgoals,Uyenscorestwogoals,and Brodiescoresninegoals.Thesumofthegoals is4 + 2 + 9 = 15goalsintotal.

Wecanusesubtractiontofindouthowmany itemsareleftwhenwe takeaway one numberofitemsfromanother.

Forexample,if44carswereparkedina supermarketparkinglot,and14ofthem weredrivenaway,therewouldbeatotalof 44 14 = 30.Weareleftwith30,whichis the difference between14and44.

Wecanalsousesubtractiontofind differencessuchashowmuchtallerone personisthananother,or howmuch more moneyweneedtobuya $10item ifweonlyhave $7.

2A Mentalstrategiesfor addition

Awell-chosenmentalstrategyallowsyoutodoadditionmorequicklyinyourhead thanusingpencilandpaper.

Orderdoesnotmatterforaddition

Theorderinwhichwedoadditiondoesnotmatter.Theanswerwillbethesameno matterwhichorderweuse.Forexample:

Thisstrategyworkswellforthreeormorenumbers.Carefullychoosingtheorderin whichyouaddthenumberscansavealotoftime.

Forexample:

Makeuptoaten

Takesomeofonenumberandaddittotheothernumbertomakeupatenora multipleof10.

Example1

Add34and26. Solution

Addfromtheleft

Whenweaddfromtheleft,westartwiththelargestpartsfirst.Addthedigitswiththe sameplacevalue,startingfromtheleft.

Example2

a Peterhas87marbles.Alihas55marbles.Howmanymarblesdotheyhave intotal?

b Lisahas3428cardsinhercollection.HerbrotherMichaelhas2236cards.Find thetotalnumberofcards.

Solution

a 87 + 55 = 80 + 50 + 7 + 5 (Addtens,thenones) = 130 + 7 + 5 = 142

b 3428 + 2236 = 3000 + 2000 + 400 + 200 + 20 + 30 + 8 + 6 (Addthousands,thenhundreds,thentens,thenones)

= 5000 + 400 + 200 + 20 + 30 + 8 + 6

= 5600 + 20 + 30 + 8 + 6

= 5650 + 8 + 6 = 5664

Compensation

Addmorethanisneeded,thensubtracttheextrayouaddedon.

Example3

Add35and28.

Solution 35 + 28 = 35 + 30 2 = 65 2 = 63 (Adding28isthesameasadding30and subtracting2.)

Practiseyourmentalstrategiestofindoutwhichstrategiesworkbestforyou.Youwill needtovaryyourstrategiestosuitthenumbersyouareworkingwith.

2A Wholeclass LEARNINGTOGETHER

1 Completeeachadditionmentally.Discussthedifferentstrategiesforeach addition.Isthereabestone?Whyisitbest?

+ 85

+ 29 c

2 Workinpairs.Youwillneed2dice.Player1rollsthe dicetocreatea2-digitnumber.Player2then mentallyaddsanumberfromthetableontheright tothenumberrolled.Forexample: 5and3 Thatmakes53.

1 Whatnumberdoyouneedtoaddtoeachofthesetomakeatotalof50?

2 Georgianeedstosave $300foraPlaystation.Howmuchmoredoessheneedto saveifshealreadyhas:

3 Completetheseadditionsmentally.Choosethebeststrategy.

4 FarmerKimlooksafter93sheep,75cowsand57goats.Howmanyanimalsdoes shelookafter?

5 Simon’sgrandfatherlovesgardening.Hehas65potsofroses,37potsofdahlias and43potsofpetunias.Howmanyplantsdoeshehaveintotal?

6 a HowmanydaysarethereinSeptember,OctoberandNovembercombined?

b Isyouranswertopart a moreorlessthanthetotalnumberofdaysinMarch, AprilandMay?Byhowmuch?

7 Thisiscalleda‘magicsquare’.Eachrow,eachcolumnandeachdiagonalshould adduptothesametotal.

Copythesemagicsquares,thenwritethemissingnumbers.

8 Makeamagicsquareofyourown.Seeifyourpartnercansolveyourmagicsquare.

2B Thestandardaddition algorithm

Theadditionalgorithmislikearecipefordoingaddition.Analgorithmworksmost efficientlyifitusesasmallnumberofstepsthatapplyinallsituations.

Theadditionalgorithmcanbeusedtoadd37,48,and56.

Setoutthenumbersoneundertheother, accordingtotheirplacevalue.

Startwiththeonesdigits.

Wesay,‘7onesplus8onesplus6onesis 21ones.Thatisthesameas2tensand1one’.

Writethe‘1’intheonescolumnandcarrythe 2tenstothetenscolumn.

Nowaddthetensdigits.

Wesay,‘3tensplus4tensplus5tens,plusthe 2tenscarriedfrombefore,is14tens.Thatis thesameas1hundredand4tens’.

Writethe‘4’inthetenscolumnand,asthere arenohundredstoadd,writethe‘1’inthe hundredscolumn.

37 + 48 + 56 = 141

Thesumof37,48and56is141.

Theadditionalgorithmcanbeextendedtoaddnumbersofanysize.Allyouneedto doisaddthecolumns,startingfromtherightandcarryingwhenneeded.

Example4

a Findthesumof345and267.

b Findthesumof3526, 988, 469and85.

Solution

a 345 + 21 61 7 612

Thesumof345and267is612.

b 3526 988 469 + 22 82 5 5068

Thesumis5068.

(Remembertoputthedigitsinthe correctplace-valuecolumns.)

(Addtheones,carrying2tensintothe tenscolumn.Addthetens,includingthe carriedtensfrombefore.Addthe hundreds,carryingwherenecessary. Thenaddthethousands.)

Theadditionalgorithmcanbeusedtoaccuratelyfindthesumoftwo ormorenumbers.

2B Individual APPLYYOURLEARNING

1 Usetheadditionalgorithmtodothesecalculations.

2 Intheirlastfourmatches,theSmithtonRangersnetballteamscored47goals, 38goals,52goalsand49goals. Whatwasthetotalnumberofgoalsscored?

3 Hamishboughta1964MiniMinorfor $2775.Hespent $875onrepairsand $388 onnewtyres.Howmuchdidhespendaltogether?

4 TheRivertonNewsagentssold357newspapersonMonday,289newspaperson Tuesday,336newspapersonWednesdayand427newspapersonThursday.What wasthetotalnumberofnewspaperssold?

5 AndrewhelpedwiththestocktakeattheWasherCompany.Hehadtocountthe numberofwashersinfivedifferentboxes.Hecounted1455, 1327, 1604, 1298 and1576washersinthefiveboxes.

Hethenaddedthemtogetherforagrandtotalof7030. WasAndrew’sadditioncorrectornot?Ifnot,whatshoulditbe?

6 a Whichnumberis3775morethan8370?

b Whatisthesumof2613and3621?

c Add23to6985.

d Writethenumberthatis1805morethan99.

UNCORRECTEDSAMPLEPAGES

e Whichnumberisaddedto6937tomake10000?

f Whatisthesumof693, 271, 596and703?

7 Completetheseadditions.

8 Themissingdigitsaremarkedwitha ★. Writethemissingdigitstomakeeachadditioncorrect.

Mentalstrategiesfor subtraction

Whenwesubtract,weareeither‘takingaway’onenumberfromanother,or‘adding on’togetfromonenumbertoanother.

Takingaway

Asubtractionsuchas36 19meansstartingat36andgoingback19steps.Ona numberlineitlookslikethis. 0

Wegoback10stepstogetto26,thentakeafurther9stepsandarriveat17: 36 19 = 17

Addingon

Usingtheadding-onstrategytocalculate36 19meansasking‘whatdoIaddto19 togetto36?’

Onestepgetsusto20,then10moreto30and6moreto36.

1 + 10 + 6 = 17,sowehavetoaddon17.

Subtractabitatatime

Breakthenumberyouaretakingawayintoseparatepiecesandtakeawayonepiece atatime.

Example5

Subtract38from95. Solution

Builduptothelargernumber

Addontothesmallernumbertobuilduptothelargernumber.Keeptrackofwhat youhaveadded.

Example6 a 82subtract35 b 643 285

Solution

a 82 35

35 + 5 = 40

40 + 40 = 80

80 + 2 = 82

82 35 = 47

b 643 285

285 + 15 = 300

300 + 300 = 600

600 + 43 = 643

643 285 = 358

(5hasbeenadded.)

(Atotalof45hasbeenadded.)

(Atotalof47hasbeenadded.)

(15hasbeenadded.)

(Atotalof315hasbeenadded.)

(Atotalof358hasbeenadded.)

Addthesametobothnumbers

Addingthesamenumbertobothnumbersdoesnotchangethedifferencebetween them.

Example7

Subtract47from93byaddingthesameamounttobothnumbers.

Solution

93 47 = 96 50 = 46 (Add3tobothnumbers.)

Individual APPLYYOURLEARNING

1 Calculatethedifferencebetween:

a 64and27

b 45and28

c 39and83

d 322and128

2 Usethe‘builduptoalargernumber’strategytocalculatethesesubtractions.

63 37 a 82 45 b

71 38 c 462 178 d

444 199 e 279 86 f

3 Flora’sFlowersstartedthedaywith125bunchesofflowers. Florasold87bunches.Howmanybuncheswereleft?

4 Mario’sfamilysetofffromTownsvilleonajourneyof265kilometres. Theydrove148kilometresbeforestopping. Howfardidtheyhavelefttodrive?

5 Katetheelectricianhad227metresofcableleftonaroll.Kateused89metreson hernextjob.Howmuchcablewasthereleftontheroll?

6 HarcourtElectricalwasadvertisingatelevisionfor $425.Ifitcutthepriceby $57, whatwouldbethenewprice?

7 a Whatisthedifferencebetween644and397?

b Whatis2047takeaway1804?

c Calculate1743lessthan2962.

8 a Writethesenumbers:98, 97, 96, 95, 94, 93, 92, 91. Nowreversethedigitsofeachnumberandsubtractthesmallernumberfrom thelargernumber;forexample,98 89 = 9. Dothisforeachnumber.

b Whatdoyounoticeaboutallofyouranswersto a?

c Trythiswithother2-digitnumbers.Doyougetthesameresults?

2D

Subtractionalgorithms

Inthesubtractionalgorithms,weworkfromrighttoleft.Wesubtractthedigitsone columnatatime.Herearetwodifferentsubtractionalgorithms.

Tradingordecomposition

‘Trading’isbasedontheideathat10onesisthesameas1ten;that10tensisthe sameas1hundred;andsoon.

Startintheonescolumn.Therearenotenough onestotake9away.

Trade1tenfor10onesinthetopnumber. Crossoutthe4andwritea3toshowthatthere are3tensleft.Writea1totheleftofthe2to showthattherearenow12ones.

12onestakeaway9ones = 3ones.

Nowworkinthetenscolumn. Therearenotenoughtenstotake5tensaway. Trade1hundredfor10tens.Crossoutthe3 andwritea2toshowthatthereare2hundreds left.Nowwritea1inthetenscolumntoshow thattherearenow13tens.

13tenstakeaway5tens=8tens.

Nowforthehundredscolumn. 2hundredstakeaway1hundred = 1hundred.

Equaladdition(borrowandpayback)

Thisusestheideathatifyouaddthesameamounttotwonumbers,thedifference betweenthemthenumbersremainsthesame.

Forexample,findthedifferencebetween342and159.

Startwiththeonesdigits.Therearenotenough onestotake9away.

Add10tobothnumbersbyadding10onesto the2intheonescolumnand1tentothe5in thetenscolumn.

12onestakeaway9ones = 3ones.

Nowworkinthetenscolumn.Therearenot enoughtenstotake5 + 1 = 6tensaway.

Add100tobothnumbersbyadding10tensto the4inthetenscolumnand1hundredtothe1 inthehundredscolumn.

14tenstakeaway6tens = 8tens.

Nowworkinthehundredscolumn. 3hundredstakeaway2hundreds (1 + the1addedbefore) = 1hundred.

342 159 = 183

Example8

Findthedifferencebetween6043and2796.

1 Workinpairs.Person1doessubtractions a–d andPerson2checksthe answers.ThenPerson2doessubtractions e–h andPerson1checksthe answers. 7383 4269 a

2567 b

3784 c

4629 e

2 Ingroups,taketurnstorollfourdicetogeta4-digitnumber.Forexample, ifyourolled2,4,6and3,twoofthenumbersyoucouldmakeare6342 and2634.

Eachpersonstartswith100000pointsandtakesturnstosubtractthenumber fromtheirtotal.Thefirstpersonbelow50000pointswinsthegame.

1 Calculatethesesubtractions.

2 Workouttheanswerstothese.

3 TheOrangeCompanytook8435casesoforangestothemarket.Itsold 3768casesoforanges.Howmanycaseswereleft?

4 Alibraryhas5043fictionbooksand2706non-fictionbooks.Howmanymore fictionbooksaretherethannon-fictionbooks?

5 TheLovelyEggCompanynormallysells9750eggseachweek.Ithasalready sold7995eggsthisweek.Howmanymoreeggsdoesitneedtosell?

6 MrMcDuffearned $56044lastyear.Hesaved $8675.Howmuchdidhe spend?

7 Thereare7746sheeponHelen’sfarm.Helensells4975sheep.Howmany areleft?

8 Adams’Appleshasacontracttosupply9250applestothemarket.Ithas alreadypicked3878apples.Howmanymoredoesitneedtopick?

9 TheFabulousFishFarmhad12125smallfishinalargepond.Itsold5850fish. Howmanywereleft?

Boththeadditionandsubtractionalgorithmscanbeextendedtolargernumbers.You areonlyeverdealingwithonecolumnofsingledigitsatatime.

Addition

Startattheright-handsideandaddeachcolumninturn,movingfromtherighttothe left.Remembertorecordanycarrynumbersinthenextcolumn.

Example9

Findthesumof53482, 48677, 21953and30945.

Toaddwholenumbersofdifferentlengths,linethemupaccordingtotheirplacevalue. Theorderyouwritethemindoesnotmatter.Aslongasthedigitsandthecarry numbersareinthecorrectcolumn,andyouradditionisaccurate,youwillgettheright answer.

Example10

Findthesumof7,43468,62,6504and793.

Writethenumbersoneundertheother,accordingtotheirplacevalue.Thenumberto besubtractedgoesunderneath.Startattheright-handsideandsubtracteachcolumn inturn,movingfromtherighttotheleft.Tradewhereverneeded.

Example11

Findthedifferencebetween70204and31627.

Solution

Usetradingorequaladditiontofind70204 31627.

Trading 70204 − 31627

−

Thedifferencebetween70204and31627is38577.

2E Individual

1 Calculate:

a 17755 + 26426

b 29216 + 13278

c 66009 35228

d 91334 48675

2 Completetheseadditions.

a 43600 + 65 + 6897 + 378

b 3801 + 66224 + 89

c 55214 + 899

d 276 + 88 + 46354 + 4683

3 Calculatethesesubtractions.

a 36221 18365

b 40832 26338

c 54361 − 28979

d 17055 9648

e 29342 4366

APPLYYOURLEARNING

4 Usethealgorithmstocalculatethesemoneyamounts.

a $3266.75 +$2845.65

b $5504.30 +$4385.85

c $4423.20 −$1758.95

d $8000.00 −$3365.55

5 TheestimatedresidentpopulationoftheseAustraliancitiesinJune2024wasas follows.

Source:AustralianBureauofStatistics(2023–24),Regionalpopulation,ABSWebsite,accessed February2026

a HowmanymorepeoplelivedinLauncestonthanAliceSprings?

b WhatwasthedifferenceinpopulationbetweenMountIsaandBendigo?

c WhatwasthetotalnumberofpeoplelivinginAlbany,DubboandWhyalla in2024?

d HowmanyfewerpeoplelivedinWhyallathaninDubbo?

e Whatwasthetotalpopulationofallthetownslistedinthetablein2024?

f Howmanyshortof400000isyouranswertopart e?

6 a Take19748from33925.

b Subtract45968from100000.

c Findthedifferencebetween1795936and3857339.

d Writethenumberthatis142587937morethan372959475.

7 Addtenthousandfourhundredandfifty-sixtotwenty-threethousandand eighty-eight.

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8 Tamara’sbrotherhasexactlyfiftythousanddollarsinhisbankaccount.Ifhebuysa newcarfor $36895andpaysforitoutofhisbankaccount,howmuchwillbeleft inhisaccount?

9 TheMathsyTheatrehas13seatsinthefirstrow,15seatsinthesecondrow, 17seatsinthethirdrow,andsoon.Howmanyseatsareinthetheatreifthereare 15rowsinall?

10 Aspidercaught175fliesinherwebinoneweek.Eachdayshecaught7moreflies thanshedidthedaybefore.Howmanyfliesdidthespidercatchoneach individualday?

11 Forwhichnumbersbetween1and209dothedigitsofthenumber addto8?

1 Tomneedstosave $500.Howmuchmoredoesheneedtosaveifhehas:

2 Calculatetheseadditions.

3 Thereare347boysand638girlswhowanttoplayNewcombballforinterschool sports.Howmanychildren,intotal,wanttoplayNewcombball?

4 Inamagicsquare,eachrow,eachcolumnandeachdiagonalshouldadduptothe sametotal.Workoutthenumbersmissingfromeachmagicsquare.

5 Whatnumberis4967morethan7084?

6 Completetheseadditions.

a 4375 + 619 + 241 + 98

b 643 + 879 + 4277 + 67

7 Themissingdigitsaremarkedbelowwitha ★. Writethemissingdigitstomakeeachadditioncorrect.

8 Mentallycalculatethesesubtractions.

9 Usethe‘addingon’methodtodothesesubtractions.

10 TheNewtownnewsagentshad237newspapersdeliveredinthemorning.They sold98newspapers.Howmanynewspapersdidtheyhaveleft?

11 Helenwantstodrivethe873kilometresfromMelbournetoSydneyinoneday.She stoppedforlunchafterdriving467kilometres.Howfardoesshehavelefttodrive?

12 Amobilephonewaspricedat $187.Whatwasthenewsalepriceifthepricewas cutby $29?

13 Whatisthedifferencebetween691and487?

14 Stellahasajobpickingoranges.Lastweek,shepicked9387oranges.Sofarthis weekshehaspicked4629oranges.Howmanyorangesdoessheneedtopickto equallastweek’snumber?

15 Trevorearned $7279fromhiswheatharvestinthefirstweekand $6825inthe secondweek.HowmuchlessdidTrevorearninthesecondweek?

16 Usethealgorithmstocalculatetheseamountsofmoney.

a $5379.45 +$1947.75

b $6207 60 +$2782 95

c $4526.20 −$3764.85

d $7000.00 −$5279.15

Themarketexpedition

Youareabouttoembarkonaquesttopurchasevariousitemsfromalocalmarket.The marketisknownforitsexoticgoodsandchallengingtransactions.

Inyourwalletis$50.00.Youmustnavigatethemarket,buyingitemsandensuringyou donotrunoutofmoney.

Marketstallsanditems:

1 Fruitstall

• Dragonfruit:$4.75each

• Starfruit:$3.60each

• Durianfruit:$5.25each(Buy2,get1free)

2 Spicestall

• Saffron:$10.00per1gram

• Cinnamon:$1.20perstick

• Cardamon:$3.00per10grampacket

3 Artifactstall

• Moodring:$12.50

• Ancientcoin:$7.45

• Crystalball:$9.98(Buy2,get50%offyoursecondpurchase)

Yourmission:

Usingyour$50.00allowance,planyourpurchasestocollectasmanyuniqueitems aspossiblewithoutexceedingyourbudget.Youmustbuyatleasttwoitemsfrom eachstall.

Youmayusedigitaltoolstokeeptrackofyourspendingandperformthenecessary calculations.

Remember:

• addandsubtractdecimalsaccuratelytofindthetotalcostofyourpurchases

• useestimationandroundingtocheckthereasonablenessofyouranswers

• consideranyspecialoffersordiscountsatthestallsandcalculatehowtheyimpact yourtotal.

Compareyourpurchaseswithapartner.Whohasboughtthemostuniqueitems?

CHAPTER

Usefulskillsforthistopic

Recallofmultiplicationfactsto12 × 12.

Vocabulary

Primenumbers

Triangularnumbers • Factors • Multiples

1 Drawalineacrossthewhiteboard.Placesomenumbers aboveandsomebelowthelinebasedonasecretrule (e.g.multiplesof6above,othersbelow).Don’trevealthe ruleyet.

2 Askstudents,‘Whatdoyounotice?’.Invitethemtoaddanumberusingsticky notes.Discussanyreasonablebutincorrectguesses(e.g.odd/even).

3 Buildontheconversationwithreasoningquestions:

a Whydidyouchoosethisnumber?

b Couldyouaddanegativenumberabovetheline?

c Coulda3-or4-digitnumberfit?

d Wherewould4527go?Explainwhy.

e Couldanumberendingin5goabovetheline?

Repeattheactivitywithdifferentrules(e.g.primes,squarenumbers)throughout theweek.

Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers Specialnumbers

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Likemanysubjects,mathematicsusesspecificlanguagewhenreferringtodifferent concepts.Specificvocabularycanbeusedwhenreferringtodifferentsetsof numbers,suchasoddandevennumbers.Somenumbershavespecialproperties thatmakethemuniqueandmaybeusefulwhenitcomestosolvingproblems.

Inthischapterwewillinvestigatevocabularyusedtodescribesomespecialsetsof numbers.Wewillexplorefactorsandmultiples,examininghowtheyrelatetoone another.Additionally,wewilllookintoprimeandcompositenumbers, distinguishingbetweenthosethatcanonlybedividedbyoneandthemselvesand thosethathaveadditionaldivisors.Furthermore,wewillinvestigatesquare numbers,whicharetheproductofanintegermultipliedbyitself,andtriangular numbers,whichcanberepresentedasdotsarrangedintheshapeofatriangle. Bytheendofthischapter,youwillhopefullyhaveadeeperunderstandingofthe vocabularyassociatedwiththesespecialnumbersandhowtoapplythis knowledgetosolvemathematicalproblemseffectively.

3A Multiplesandfactors

Multiples

Thenumberswegetwhenweskip-countbyanynumberarecalledthe multiples of thatnumber.Forexample,ifweskip-countinfours,thenumberswegetarecalledthe multiplesof4.

Multiplesaretheproductyougetwhenyoumultiplyonewholenumberwithanother wholenumber.Herearethemultiplesof4upto36.

Thenumbers shadedingreen showthe multiplesof4up to48.

Wecanalsoshowmultiplesbydrawingrectangulararrays.

Factors

1 × 6 = 6cherries,becausethereis1rowof6cherries.

2 × 6 = 12cherries,becausethereare2rows,eachwith6cherries.

3 × 6 = 18cherries,becausethereare3rows,eachwith6cherries.

Factorsandmultiplesarerelated.Thenumberswemultiplytogethertogetamultiple orproductarecalledthe factors ofthatnumber.

Wecanusethefollowingarraysof24studentstosee theconnection.

Iftheylineupinfourrows,eachwithsixstudents,we canseethat4 × 6 = 24.

24isa multiple of4and4isa factor of24.

24isa multiple of6and6isa factor of24.

Thestudentscouldbearranged intworows,eachwith12 students,andso2 × 12 = 24.

24isa multiple ofboth2and 12,and2and12areboth factors of24.

Thestudentscouldbearrangedinotherways. Iftheylineupinthreerows,eachwillhave eightstudents,andso3 × 8 = 24.

24isa multiple ofboth3and8,and3and8 areboth factors of24.

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Theycouldevenjuststandinonelineof24students,because1 × 24 = 24. 24isa multiple ofboth1and24,and1and24areboth factors of24.

Allofthesedifferentarrangementscometothesametotal.Thisshowsusthat24has manyfactors:1, 2, 3, 4, 6, 8, 12and24.

3A Wholeclass LEARNINGTOGETHER

1 Createarraystoshowthat:

a 7, 14and28arefactorsof28

b 10, 20and40arefactorsof40

2 Whichofthefollowingisamultipleof9?

3 Completethesesentencesbyadding‘is’or‘isnot’intheblankspaces.

a 24_____amultipleof12

b 63_____amultipleof8

c 7_____afactorof49

d 11_____afactorof78

e 26_____amultipleof3

f 6_____afactorof72

4 a Listallthefactorsof32.

b Listallthefactorsof28.

c Whatarethecommonfactorsof28and32?

5 Twonumbershave3and4ascommonfactors. Whatcouldthetwonumbersbe?

3A Individual APPLYYOURLEARNING

1 Writethefirst12multiplesof7,withoutlookingatamultiplicationtable. Skip-countbyseventocheckyouranswers.

2 Michaelbuiltthiswallinhisgarden.Whatisthetotalnumberofbricks Michaelused?Workthisoutmentallywithoutcountingonebrickatatime.

3 Whichofthesenumbersaremultiplesof8? 233648100076

4a Writefour9-digitnumbersthataremultiplesof10.

b Writetwo9-digitnumbersthatarenotmultiplesof10.

5 Whichofthesenumbersaremultiplesof5butnot10? 225100939550351005

6 Copyandcompletethesesentencesbywriting‘factor’or‘multiple’.

a 3isa______of36because36isa______of3.

b 4isa______of8because8isa______of4.

c 3is not a______of7because7isnota______of3.

d 16is not a______of5,so5isnota______of16.

e 8isa______of4,so4isa______of8.

f 19is not a______of2,so2isnota______of19.

7 Usethedigits5, 6, 2, 5, 0, 1, 0, 4tomake:

a a5-digitnumberthatisevenandamultipleof5

b an8-digitnumberthatisoddandnotamultipleof5

c anumberthatisamultipleof5and100

d threedifferentnumbersthathavemorethan5digitsandaremultiplesof 2, 5and10.

3B Primeandcompositenumbers

Somenumbershaveonlytwofactors,1andthemselves.Forexample:

2hasonlytwofactors:1and2

3hasonlytwofactors:1and3

7hasonlytwofactors:1and7.

Thesearecalled primenumbers. Aprimenumberisanumberwithonlytwofactors,itselfand1.

Sometimesnumbershavemorethantwofactors.Forexample:

4hasthreefactors:1, 2and410hasfourfactors:1, 2, 5and10

Numberswithmorethantwofactorsarecalled compositenumbers. Thenumbers0and1arespecialnumbersbecausetheyareneitherprimenor composite.

Primefactorisationofwholenumbers

Wecanalwayswriteacompositenumberasaproductoftwonumbersotherthan1 anditself.Wecanneverdothisforprimenumbers.

Weknowthatthenumbers4, 9and21arecompositebecause:

4 = 2 × 29 = 3 × 321 = 3 × 7

Wecankeepdecomposingcompositenumbersuntilwehavewrittenthemasa productofprimes.

Forexample,36 = 4 × 9.Wecanfindfactorsfor4and9:

4 = 2 × 2and9 = 3 × 3

Sowecanwrite36as:

36 = 2 × 2 × 3 × 3

Wecannotfindanymorefactorsbecauseallofthefactors2, 2, 3and3areprime. 2 × 2 × 3 × 3iscalledthe primefactorisation of36.

Thestandardwaytowritetheprimefactorisationofanumberistoputtheprime factorsinincreasingorderfromlefttoright.

Example1

Findtheprimefactorisationof45.

Solution

Firstwrite45asaproduct:45 = 9 × 5

5isaprimenumber.9isnotaprimenumber:9 = 3 × 3

So,45 = 3 × 3 × 53and5areprime.

Sometimestheprimefactorsarenotsoobvious,especiallywhenthenumbersarelarge.

Example2

Findtheprimefactorisationof54.

Solution

Weknowthat:

54 = 6 × 9

Wecanwrite6as2 × 3and9as3 × 3.

Sotheprimefactorisationis54 = 2 × 3 × 3 × 3.

Example3

Findtheprimefactorisationof117.

Solution

117isanoddnumber,so2isnotafactor.

Divide117by3.

3 39 ) 11 27117 = 3 × 39

39isdivisibleby3. (39 = 3 × 13)

So,117 = 3 × 3 × 133and13areprimenumbers.

Theprimefactorisationis:117 = 3 × 3 × 13

Whenwewriteanumberasaproductoftwonumbers,thosetwo numbersarefactorsofthefirstnumber.Forexample,7 × 8 = 56,so 7and8arefactorsof56.

Aprimenumberisanumberlargerthan1thathasonlytwofactors, itselfand1.

Compositenumbersarelargerthan1andhavemorethantwofactors.

Tofindtheprimefactorisationofanumber,writeitasaproductof primes.

Thestandardwaytowritetheprimefactorisationistoputtheprime factorsinincreasingorderfromlefttoright.

3B Wholeclass LEARNINGTOGETHER

1 100chartprimesearch

a Thisactivityinvolvesworkingoutwhichnumbersbetween1and100areprime numbers.Remember:1isnotaprimenumber.Anyothernumberlargerthan1 isprimeifithasonlytwofactors:itselfand1.

Youwillneeda1–100numberchartliketheonebelow.Crossout1,asitisnot aprimenumberoracompositenumber.

b Thefirstprimenumberis2.Circleit,thencolourallthemultiplesof2because theyarecompositenumbers.Yourchartshouldlooklikethis.

c Gotothefirstnumberafter2thathasnotbeencolouredin.Itis3.Circleit, because3isprime.Nowcolourallthemultiplesof3thatarenotalready coloured.

d Nowgotothenextnumberthathasnotalreadybeencolouredin.Circleit.Itis prime.Colouritsmultiples.Repeatthisstepandcontinueuntilyoucannotgo anyfurther.

e Whenyouhavefinished,thenumbersthathavecirclesaroundthemarethe primenumberslessthan100.Listtheseprimenumbers.

f Findthenextprimenumberafter97.

2 Thenumbers12, 18, 33, 98, 196and333arecomposite.Showthisbywritingeach numberasaproductoftwonumbersinwhichnofactoris1.

3 Onewaytofindtheprimefactorisationofa numberistodrawafactortree.

Inthisexample,24isfirstsplitintoitslargestand smallestfactors,12and2(leavingout24and1). Theneachoftheseissplitintoitsfactorsuntilthey cannotbesplitanyfurther.

Itisnotalwaysnecessarytostartwiththesmallest andlargestfactorotherthanthenumberitselfand 1.Sometimesitiseasiertostartwithafactor youknow.

Eventhoughallthreefactortreesaredifferent,theyallgiveustheprime factorisationfor24,thatis:

24 = 2 × 2 × 2 × 3

Drawfactortreesforthesenumbers.Writedowntheprimefactorisations.

a 36

b 100

c 520

4 Findtheprimefactorisationof280.

3B Individual APPLYYOURLEARNING

1 Copythesentencethatdescribesthearray,thenwritethemissingnumbers and/orwords.

Thisarrayshowsthat3 × = 39.So_____and_____arefactors of_____.

Thisarrayshowsthat7is not a_______of22.

2 Whichoftheseareprimenumbers? 23111423374557629099

3 Listtheprimenumbersbetween: 40and50. a 10and20. b 40and60. c

4 Listtheprimenumberslessthan100thatcontainthedigit3.

5 Whyiseveryprimenumberoddexceptfor2?

6 Findtheprimefactorisationofthesenumbers.

7 Whichnumberhastheprimefactorisation:

8 Trueorfalse? Samwrotethisstatementinhismathsbook. Theprimefactorisationof1140is1140 = 2 × 2 × 2 × 5 × 57 Isthiscorrect?Ifnot,whatcorrectionsdoesSamneedtomake?

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9 Writesixcompositenumbersgreaterthan30andlessthan60thatdo not haveanyevendigits.

10 Thesumoftwoprimenumbersis60.Whatmightthenumbersbe?

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3C Squareandtriangular numbers

Squarenumbers

Somenumbersarecalled squarenumbers becausetheycanberepresentedbyasquare array.Forexample,4isasquarenumberbecauseitcanbeshownlikethisarray.

Asquarenumberistheproductofanumbermultipliedbyitself:3 × 3=9,4 × 4=16, andsoon.

Anothernameforsquarenumbersis perfectsquares.Herearethefirstfour perfectsquares:

Wewritesquarenumbersinaspecialway.Wecandraw25as asquarearraywith5rowsof5stars.

Wesay‘5times5is5squared’or‘5tothepowerof2’.Thisis written:25 = 5 × 5 = 52 . Wecandraw36asasquarearraywith6rowsof6stars.

Wesay‘6times6is6squared’or‘6tothepowerof2’. Thisiswritten:36 = 6 × 6 = 62

Triangularnumbers

A triangularnumber canbedrawnusinganarrangementofdotsintheshapeofa regulartriangle.Herearethefirstfourtriangularnumbers: 1 3

Canyouseethepattern?Anotherrowhasbeenaddedtothetriangleeachtime.Each newrowcontainsonemorethanthepreviousrow.

Countthenumberofdotsononesideofeachtriangle.Canyouseeapattern?The firsttriangularnumberhas1dot,thesecondhas2dotsoneachside,thethirdhas 3dotsoneachside,andsoon.

3C Wholeclass

LEARNINGTOGETHER

1 Copyandcompletethesesentences.Thefifthonehasbeendoneforyou.

a The1stsquarenumberis____ × = =

b The2ndsquarenumberis____ × = =

c The3rdsquarenumberis____ ×

d The4thsquarenumberis____ ×

e The5thsquarenumberis5 × 5 = 52 = 25

f The6thsquarenumberis___ ×

g The7thsquarenumberis____ ×

h The8thsquarenumberis____ × = =

i The9thsquarenumberis____ × = =

j The10thsquarenumberis____ × = =

k Whatdoyounoticeaboutthedifferencebetweenconsecutiveperfectsquares?

l Canyouusearraystoexplainwhatishappening?

2 Drawarrangementstoshowthefirst9triangularnumbers,thencopyand completethetablebelow:

Triangularnumber Numberofdots

1sttriangularnumber 1

2ndtriangularnumber 3

3rdtriangularnumber

4thtriangularnumber

5thtriangularnumber

6thtriangularnumber

7thtriangularnumber

8thtriangularnumber

9thtriangularnumber

3 Howmanyextradotsdoesittaketogofrom:

a thefirsttriangularnumbertothesecond?

b thesecondtriangularnumbertothethird?

c thethirdtriangularnumbertothefourth?

d thefourthtriangularnumbertothefifth?

4 Withoutdrawingadiagram,writehowmanydotsyouaddwhenyougofromthe fifthtriangularnumbertothesixth,andfromthesixthtriangularnumbertothe seventh.

5 Workoutthe10thtriangularnumberfromthepatternyouhavediscovered.What isthe15thtriangularnumber?Whatisthe20thtriangularnumber?

Individual APPLYYOURLEARNING

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1 Listthesquarenumbersfrom1squaredto20squared.

2 Lookatthefinaldigitinyourlistofsquarenumbers.Writewhatyounoticeabout thepattern.

3 Thepatternwillnothelpyoudecidewhichnumbersaresquarenumbers,butit cantellyouwhichnumbersarenotsquarenumbers.Whatwillthenumbersthat aredefinitelynotsquarenumbershaveastheirfinaldigits?Whydoyouthinkthis happens?

4 AccordingtothetestidentifiedinQ3,whichofthesenumbersisdefinitelynota squarenumber? 1883 a

c 2025 d

5 Answerthefollowinginyourworkbook:

a Whatisthelargest2-digitsquarenumber?

b Whatisthelargestsquarenumberlessthan200?

c Writethenextsquarenumberafter900.

d Whatisthelargest3-digitsquarenumber?

e Writethreesquarenumbersthatendinthedigit0.

f Writethreesquarenumbersthatendinthedigit5.

g Writethreesquarenumbersthatendinthedigit1.

6 Listthefirsttentriangularnumbers.Whatdoyounoticeaboutthesumwhenyou addtwoconsecutivetriangularnumberstogether?Canyouexplainwhythis happens?

7 Copyandcompletethistableinyourworkbook:

8 Oneisbothasquarenumberandatriangularnumber.Whatisthenextnumber thatisbothsquareandtriangular?

9 Canyoufindthenextnumberthatisbothsquareandtriangular?

1 Drawfactortreestofindtheprimefactorisationofthesenumbers.

2 Findtheprimefactorisationofthesenumbers.

3 Writethenumberthatistheproductofeachprimefactorisation.

4 Copyandcompletethistableinyourworkbook:

5 Whatdoyounoticeaboutsquarenumbersandthenumberoffactorstheyhave? Couldyouexplainwhythismaybe?

3E Challenge–Ready,set,explore!

Factory-y

Beforewebegin,let’sestablishsomegroundrules.Everynumbercanbedividedby1. Forexample,33 ÷ 1 = 33.Everynumbercanalsobedividedbyitself.Forexample, 33 ÷ 33 = 1.Thismeansthateverynumberhasatleastonepairof factors –itself and1.

Somenumbershaveonlythesetwofactors.Theyare prime numbers.Thenumber31 isprime.Itcanbedividedonlybyitselfand1.Nothingelsedividesintoitwithout leavingaremainder.

Numbersthatarenotprimehavemorethantwofactorsandarecalled composite numbers.33isanexample.33 ÷ 3 = 11.Soitiscompositewithfactors3and11 (aswellas1and33).

Remembertheexception:1isneitherprimenorcomposite.

Thenumber72isacompositenumber.Thefactorsof72are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36and72.Becausefactorscomeinpairs,whenyoufinda factorofacompositenumberyouareimmediatelyrewardedwithanotherfactor.For example,91canbedividedby7andwhenyoudothistheansweris13.So13isalsoa factorof91.

Challengequestions

1 Thislistcontains15numbers,alllessthan100.Sevenofthemareprimeandthe othereightarecomposite.Sortthemintotwogroups: 462765295743836779305371398751.

2 Makealistofallthenumbersthatwilldivideexactlyinto30withoutleavinga remainder.

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Thereasonthat30hassomanyfactors(didyoufindall8?)isthatsomeofits factorshavefactorsthemselves.Forexample,30 = 5 × 6and6 = 3 × 2.

3 Dothesamefor84.Youshouldbeabletofind12factors,including1and84.

4 63hasonly6factors.Seeifyoucanfindthem.

Usetheideasabovetocompletefurtherquestions:

5 Anynumberthathas15asafactorcanalsobedividedbyboth and .

6 Anynumberthathas26asafactorcanalsobedividedbyboth and

7 Anynumberthatcanbedividedbyboth7and11alsohas asafactor.

8 Anynumberthatcanbedividedbyboth17and alsohas51asafactor.

9 Anynumberthathas12asafactorcanalsobedividedby , , and

10 Anynumberthathas42asafactorcanalsobedividedby ,3, ,7, and .

11 Anynumberthathas90asafactorcanalsobedividedby , , , 6, ,10, ,18, and

12 Thenumber7383haseightfactors.Twohave4digits,twohave3digits,twohave 2digitsandtheothertwohave1digit.Findthefactorsof7383.Hint:Finda 1-digitfactorfirst.Divideandconquer!

13 Althoughfactorsalwayscomeinpairs,3364actuallyhasatotalofninefactors. Findwhattheyareandexplainwhythereisanoddnumberoffactors.Find another4-digitnumberwithanoddnumberoffactors.

Usefulskillsforthistopic

Vocabulary

Solvetheseproblems

1 Thereare7peopleintheDinglefamily.The Dinglesmakesuretheybuypacketsoffood theycanshareequallysothateachfamily membergetsthesameamountwithnoneleft over.CirclethefoodstheDingleswillbuy,then writethenumberofitemseachfamilymember willreceivefromeachpacketorcontainer.

2 Howcanyoubreakdown348tomakethecalculationof348 × 6easier?

3 Wouldyourmethodworkforanynumber?

4 Explainandprovideanotherexampletosupportyourreasoning.

Multiplicationand division Multiplicationand division Multiplicationand division division Multiplicationand Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division division Multiplicationand division Multiplicationand division Multiplicationand division division Multiplicationand division Multiplicationand division division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division Multiplicationand division

Multiplication isusedtocalculatetheproductoftwonumbers.Youcanthinkof multiplicationas‘lotsof’anumber.

Forexample,ifyouhave3nests,eachcontaining3Eastereggs,youhave3lots of3.Thisiswrittenas3 × 3andyouhaveatotalof9Eastereggs.

Ifyouhave142nestseachwith84Eastereggs,youhave142lotsof84eggs,or 142 × 84 = 11928Eastereggs.

Division,ontheotherhand,isawayofsplittinganumberintoequalparts.Ifwe haveeighttennisballsandsplitthemintofourgroups,wehavetwogroups.In otherwords,8 ÷ 4 = 2.

Inthischapter,welookattheconnectionbetweenmultiplicationanddivision.We willseehowmultiplicationcanbeusedtocheckadivisioncalculationandhow divisionistheinverseofmultiplication.

4A Mentalstrategiesfor multiplication

Themultiplicationtablesupto12 × 12formthebasisofmanyofthemultiplication mentalstrategiesthatwelookatinthissection.Itisimportanttobeabletorecallyour ‘tables’veryquickly.

4A Wholeclass LEARNINGTOGETHER

Discusseachstrategyasaclass,thencompletetherelatedactivities.

1 Multiplyingby4

Tomultiplyanumberby4,doubleit,thendoubletheresult.

Thisworksbecause4 = 2 × 2.

Forexample,tocalculate45 × 4mentally:

Double45toget90,thendouble90toget180.

45 × 4 = 180.

Multiplythesenumbersby4.

21 a 77 b 255 c 1026 d 1050 e 2222 f

2 Multiplyingby10,100and1000

Numbersthataremultiplesof10alwaysendinzero.Forexample: 11 × 10 = 11012 × 10 = 12013 × 10 = 130

Tomultiplyawholenumberby10,writeazeroattheendofthenumber.

23 × 10 = 2309898 × 10 = 98980123456 × 10 = 1234560

Tomultiplyawholenumberby100,writetwozeroesattheendofthenumber.

Thisworksbecause100 = 10 × 10.Forexample:

23 × 100 = 23009898 × 100 = 989800123456 × 100 = 12345600

Whenmultiplyingby1000,wewritethreezeroesattheend;for10000wewrite fourzeroes;andsoon.

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Multiplythesenumbersby10,thenby100andthenby1000.

934 a 1001 b 10101 c 15462 d 848084 e 295034957 f

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Multiplyingby8

Tomultiplyanumberby8,double,thendoubleagain,thendoubleforathird time.Thisworksbecause8 = 2 × 2 × 2.

Forexample,tocalculate15 × 8mentally:

Double15toget30,thendoubleagaintoget60,thendoubleagainto get120. 15 × 8 = 120

Usethe‘doublethreetimes’strategytomultiplythesenumbersby8.

Multiplyingby9

Ifwewanttoget9lotsofsomething,itiseasiertofind10lotsandthentake1lot away.Forexample: 9 × 17 = 10lotsof17takeaway1lotof17 = 170 17 = 153

Multiplyeachnumberby9.

Multiplyingby11

Ifwewanttoget11lotsofsomething,find10lotsandthenadd1lotmore. Multiplythesenumbersby11.

Multiplyingby5

Therearetwowaystomultiplyby5.

• Thefirstwayistomultiplyby10,thenhalvetheresult.Thisworksbecause 5 = 10 ÷ 2.

• Thesecondwayistohalvethenumber,thenmultiplytheresultby10. Usethesestrategiestomultiplytheseoddnumbersby5.

a 99

b 111

c 102

d 450

e Writefivenumbersofyourown.Multiplyeachnumberby5.

Multiplyingby20

Whenyouwanttomultiplyby20,doublethenumber,thenmultiplyitby10.This worksbecause20 = 2 × 10.

Completethesemultiplications.

× 20 a

× 20 c

× 20 b

× 20 d

8 Multiplyingby6

Ashortcuttomultiplyinganumberby6istofirstmultiplythenumberby3,then multiplyitby2(ortheotherwayaround).Thisworksbecause6 = 3 × 2.

Usethisstrategytomultiplythesenumbersby6.

Multiplyingby25

Youcanmultiplynumbersby25byskip-countingintwenty-fives.Useyourfingers tokeeptrackofhowmanyyouhavecounted;lateron,youcankeeptrackofthe skip-counting‘inyourhead’.

Forexample,for25 × 7:

255075100125150175

Skip-counttomultiplythesenumbersby25.

Nowthinkupavariationonthesamestrategytomultiplythesenumbersby25.

10 Usementalstrategiestosolvethesewordproblems.

a 25monkeyseachate11bananas.Howmanybananaswereeaten?

b AtthezooMaryfed11monkeys23peanutseach.Maryate2bagswith 19peanutsineachherself.Howmanypeanutswereeatenaltogether.

11a Makeupyourownstrategyformultiplyingby19.(Hint:Dosomethinglikethe ‘multiplyingby9’strategyandthe‘multiplyingby20’strategy.)

b Makeupyourownstrategyformultiplyingby21.(Hint:Dosomethinglikethe ‘multiplyingby11’strategyandthe‘multiplyingby20’strategy.)

c Writeastrategyformultiplyingby30.Testyourstrategyonfivedifferent numbers.

4B Breakinga multiplicationapart

Thisistherectangulararrayfor4 × 17.

Ifwecountallofthedotsinthearray,wewillfindthat4 × 17 = 68. Wecansplit17into1tenand7ones.

Thisbreaksthearrayaparttoshow multiplicationchunks.

Wecandothemultiplicationineachchunkfirst, thenaddtofindtheproductof4and17.

Insteadofdrawingarrays,youcandrawmultiplicationdiagramstohelpyou‘see’the multiplication.Thisdiagramusesthechunks4 × 10and4 × 7toshow4 × 17.

Youcanusemultiplicationdiagramstoexplainhowtomultiplylargenumbers.

Example1

Drawamultiplicationdiagramfor14 × 28,thenuseittocalculatetheanswer.

4B Individual APPLYYOURLEARNING

1 Usethemultiplicationdiagramandcompletethecalculationforeachofthese.

a 5 × 18

b 12 × 13

2 Completethesemultiplications.Trytosolvethem‘inyourhead’.Thefirst multiplicationhasbeendoneforyou.Remembertodoallthemultiplications beforetheadditions.

12 × 7 = 10 × 7 + 2 × 7 = 70 + 14

d Usetheabovestrategytosolvethesemultiplications.

3a Multiplythesenumbersby44. i 13 ii 43 iii 128

4a Multiplythesenumbersby18. i 14 ii 82 iii 107

4C Themultiplication algorithm

Multiplicationdiagramsareconnectedtothe algorithm formultiplication.An algorithmislikearecipethatgivesyoustepstofollow.Thediagramshelpexplainhow thealgorithmworks.

Thismultiplicationdiagramfor17 × 23givestheproductsforeachchunk.

Wecanfindtheproductof17and23bymultiplyingthechunksandaddingthem together.

Themultiplicationalgorithmisamoreefficientwaytofindtheproductof17and23. Thisishowthemultiplicationalgorithmworksforthemultiplication23 × 17.

Setoutthenumberssothedigitslineup accordingtotheirplacevalue.

Startwiththeones.Multiplytheonesdigit in17bytheonesdigitin23.

Say‘7times3is21’.Write‘1’intheones columnandcarry‘2’tothetenscolumn.

Nextworkwiththetens.Multiplytheones digitin17bythetensdigitin23.

Say‘7times2is14’.Addthe2tenscarried frombeforetomake16.Write‘6’inthe tenscolumnand‘1’inthehundreds column.

Nowmultiplythetensdigitin17bythe onesdigitin23.Thiswillgiveacertain numberoftens,sostartbywriting‘0’inthe onescolumn.

Say‘1times3is3’.Write‘3’inthetens column.

Next,multiplythetensdigitin17bythe tensdigitin23.Say‘1times2is2’. Write‘2’inthehundredscolumn.

Thefinalstepistoadd161to230. Theproductof23 × 17is391.

Example2

Calculate482 × 87usingthemultiplicationalgorithm. Explaineachstepinthealgorithmusingamultiplicationdiagram.

3374 = 2800 + 560 + 14isthesum38560 = 32000 + 6400 + 160isthe ofthechunksinthebottomrowsumofthechunksinthetoprow ofthemultiplicationdiagram.ofthemultiplicationdiagram.

4C Individual APPLYYOURLEARNING

1 Usethemultiplicationalgorithmtocalculatethese.

× 43 a

× 78 b

× 32 c 480 × 64 d

× 498 e

× 1002 f

2a Jesse’scarreleases119gramsofcarbonintotheatmosphereforeachkilometreit travels(aslongasJessedrivesataconstantspeed).Howmuchcarbonisreleased ifJessetravels:

12kilometres? a 46kilometres? b 108kilometres? c

3a InthesupermarketnearKatie’shousethereare23cartonsofeggson6shelves and14cartonsofeggson8shelves.Eachcartonholds12eggs.Howmany eggsarethereintotal?

b Aliismovinghouse.Threeoftheroomshave16boxeseachand6roomshave 19boxeseach.Howmanyboxesisthatintotal?

c Laurenhas6drawerswith32itemsofclothingineachofthem.Ineachofher 3cupboards,thereare83itemsofclothing.Howmanyitemsofclothingdoes Laurenhave?

4 Usethemultiplicationalgorithmtocalculate a to h

a 9 × 9 + 7

b 98 × 9 + 6

c 987 × 9 + 5

d 9876 × 9 + 4

e 98765 × 9 + 3

f 987654 × 9 + 2

g 9876543 × 9 + 1

h 98765432 × 9 + 0

i Whatdoyounotice?

5 Danieldrewamapofthehighwaythatrunspasthishomeinthecountry.

Beach Cousin’s house Pool Home School Railway station Movie theatre

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Schoolis13kilometresfromhome.Fromhome,thenearestrailwaystationis 2timesthisdistanceandthenearestmovietheatreis5timesthedistance.The nearestswimmingpoolis17kilometresfromhome.Fromhome,hiscousin’shouse is2timesthisdistanceandthebeachis4timesthedistance.

WhatisthedistanceDanieltravels:

a fromhometoschoolandback?

b fromhishometotherailwaystation?

c fromhishometovisithiscousin?

d fromtheswimmingpooltoseeamovie?

e fromhishometothebeachandbackhome?

f fromhishometothemovietheatreandthenreturninghomeviahis cousin’shouse?

4D Connectingmultiplication anddivision

Divisionisaboutsplittingorsharingquantities equally.

Divisionistheinverseoperationofmultiplication.Whenweknowonemultiplication fact,weknowtwodivisionfacts.

Wecanseethisonthemultiplicationtable. 0 12 3 4 56 7 8910 1112

0000000000000 1 0123456789101112 2 024681012141618202224 3 0369121518212427303336 4 04812162024283236424448

Thetableshowstwowaysof multiplyingtoget54: 9 × 6 = 54and6 × 9 = 54

Ifwereverseboth multiplications,wefindthat: 54 ÷ 6 = 9and54 ÷ 9 = 6

Thismeanswecanusethemultiplicationtableinreversetododivisioncalculations.

Example3

Copyandcompletethesesentencesbyfillinginthegaps.

a If5 × 9 = 45,then45 ÷ 9 = and45 ÷ 5 =

b If12 × 4 = 48,then48 ÷ = 4and48 ÷ = 12.

Solution

a If5 × 9 = 45,then45 ÷ 9 = 5and45 ÷ 5 = 9.

b If12 × 4 = 48,then48 ÷ 12 = 4and48 ÷ 4 = 12.

Wecanalsousetheword divisible: 24isdivisibleby8because24 ÷ 8 = 3 withnoneleftover.

Inthedivision24 ÷ 8 = 3,24isthe dividend,8isthe divisor and3is the quotient.

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Thedivisoristhenumberthatyou divideby.

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Divisionwithremainder

Sometimes,thedivisordoesnotdivideexactlyintothedividend,andthereisa remainder leftover.Thiscanbeseenbydrawinganarray.Forexample,thisisthe closestarraywecandrawifwetrytofindout25 ÷ 8.

Thearrayusesonly24oftheballoons,withoneballoonleftover. So,wecanmake3groupsof8with1remainder.

Wewritethisas:

25 ÷ 8 = 3remainder1

Wesaythedivisionis exact iftheremainderis0.

Sharingofteninvolvesremainders.Herearetwodivisionstoriestoshowdivisionwith theremainder.

Divisionstory1

Alexboughtaboxof30chocolatestosharewithher3friends.CanAlexandher friendssharethechocolatesequally?

Thereare4people,includingAlex.Themultiplesof4are4,8,12,16,20,24,28,32 4 × 7 = 28,so28isthelargestnumberofchocolatesthatcanbeeatenifeachperson getsthesamenumber.Therewillbe2chocolatesleftover.

Wewritethisas:

30 = 4 × 7 + 2or30 ÷ 4 = 7remainder2

Eachofthe4peoplewillget7chocolates.Therewillbe2chocolatesleftover.

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Divisionstory2

Nickhas22skateboardwheels.Hewantstousethemtobuildasmanyskateboardsas hecan.

HowmanyskateboarddeckswillNickneed?Howmanywheelswillhehaveleftover? Divide22by4.

Weget5lotsof4,with2leftover.

Wewritethisas:

5 × 4 + 2 = 22or22 ÷ 4 = 5remainder2

Nickcanbuild5skateboards.Hewillhave2wheelsleftover.

Inthisexample,22isthedividend,4isthedivisor(thenumberyoudivideby),5isthe quotientand2istheremainder.

Theremainderisalwayssmallerthanthedivisor.Wecanseethat22isnotdivisibleby 4becausewhenwetrytodivide,thereisaremainder. 4D Wholeclass LEARNINGTOGETHER

1 Numbersthataremultiplesof10endinzero.Divideeachofthefollowing numbersby10.

2 Mentallydivideeachnumberby100.Forexample400 ÷ 100 = 4

3 Mentallydivideeachnumberby1000.Forexample4000 ÷ 1000 = 4

4 Mentallydivideeachnumberby4byhalvingandhalvingagain.Checkyour answerbydoubling,thendoublingagain.

4D Individual APPLYYOURLEARNING

1 Copyandcomplete.Thefirstonehasbeendoneforyou.

a If4 × 10 = 40,then40 ÷ 10 = 4 and40 ÷ 4 = 10 .

b If3 × 9 = 27,then27 ÷ 9 = and27 ÷ 3 = .

c If6 × 7 = 42,then42 ÷ 6 = and42 ÷ 7 = .

d If12 × 8 = 96,then96 × 8 = and96 ÷ 12 = .

e If144 × 72 = 10368,then10368 ÷ 144 = and10368 ÷ 72 = .

2 Usethecorrespondingmultiplicationtocheckthateachcalculationiscorrect.

121 ÷ 11 = 11 a 162 ÷ 9 = 18 b

÷ 63 = 8 c 1170 ÷ 45 = 26 d

÷ 73 = 98 e

3 Whichofthesenumbersisdivisibleby8? 241836569496

÷ 57 = 102 f

4 Usethemultiplicationtableinreversetowritetwodivisionstatementsfor eachnumber.

5 Copyandcompletethefollowing.

a 49 = 7 × 7,so55 = 7 × 7 + and55 ÷ 7 = 7remainder

b 94 = 7 × 12 + ,so94 ÷ 12 = remainder

6 Tammyfillspaperbagsfromasackcontaining20kilogramsofsugar.Each fullbagofsugarweighs3kilograms.HowmanybagscanTammyfill?How muchsugarwillshehaveleftover?

7 Copytheseandfillinthemissingnumbers.Themissingnumberiseithera divisor,aquotientoraremainder.

a 15 = 4 × + 3

b 29 = 4 × 7 +

c 30 = 12 × +

d 157 = 15 × +

e 12 = 4 × +

f 14 ÷ 5 = 2remainder

g 26 ÷ 5 = remainder

h 191 ÷ 10 = remainder

i 192 ÷ 3 = remainder

3rd

4E Theshortdivision algorithm

Thedivisionalgorithmisunusual,asitstartsontheleftofthenumberandsharesout thebigpiecesfirst.Wecanseethisifwedivide488by3.

Thispictureshowsthenumber488.

Todivide488by3wetrytomake3equalgroups.Webeginwiththehundreds.There are4hundreds.Whenweshare4hundredsbetween3people,eachperson’sshareis 1hundredandthereis1hundredleftover.

Nowwedealwiththe1hundredandthe8tens.Convertthe1hundredinto10tens andtrytomake3equalgroups.

Thereare18tens.Whenweshare18tensbetween3people,eachperson’sshareis 6tens.

Thereare8onestoshare.Whenweshare8onesbetween3people,eachpersongets 2,with2leftover.

Werecordthisusingthealgorithm3) 488,i.e.488 ÷ 3. Youcanusetheshortdivisionalgorithmwhenyouaredividingbya1-digitnumber. Ifwewanttodivide392by8,wecanuseshortdivision.

8 49 ) 39 72 8into3hundreds. Wedonothaveenoughhundreds. Convert3hundredsand9tensto39tens. 8into39tensis4tenswith7tensleftover. Write‘4’(tens)abovethelineandcarrythe7. 7tensand2onesis72ones. Nowdivide8into72.

72 ÷ 8 = 9,sowrite‘9’(ones).

392 ÷ 8 = 49

Example4

Calculate228 ÷ 7.Usemultiplicationtocheckyourwork.

Solution Dothedivision.Checkbymultiplying.Thenaddtheremainder.

)

Factorsanddivision

Themultiplicationstatement: 4 × 3 = 12 isequivalenttothedivisionstatement: 12 ÷ 3 = 4

Whenwedivide12by3,wegettheanswer4,withzeroremainder. 4and3arefactorsof12becausetheydivideinto12exactly.

Wecanuseshortdivisiontotestwhetherwehaveafactorofalargernumber.For example,anumberisdivisibleby7ifitisamultipleof7.Soanumberisdivisibleby7 iftheremainderis0whenitisdividedby7.

Ifyouwanttofindsomefactorsofanumber,startbydividingthatnumberbysmall primenumbers.Forexample,divide441by3usingtheshortdivisionalgorithm.

3 147 ) 4 14 21

Theremainderis0andthedivisionisexact.

441 ÷ 3 = 147

Theequivalentmultiplicationstatementis: 147 × 3 = 441

Thistellsusthat3isafactorof441and147isalsoafactorof441.

Example5

a Divide322by7tofindoutwhether7isafactorof322.

b Divide274by7tofindoutwhether7isafactorof274.

Solution

Ifanumberisdivisibleby7,then7isafactorofthatnumber. Ifanumberis not divisibleby7,then7is not afactorofthatnumber.

7 46 ) 32 42

322 ÷ 7 = 46(remainder0) So,46 × 7 = 322 7isafactorof322.and322isdivisibleby7.

b 7 39 ) 27 64 r1

274 ÷ 7 = 39remainder1 274isnotamultipleof7. 7isafactorof274.7isnotafactorof274.

Individual APPLYYOURLEARNING

1 Usetheshortdivisionalgorithmtocalculatethese.Usemultiplicationtocheckyour answers.

848 ÷ 4 a 435 ÷ 5 b 912 ÷ 8 c

÷ 9 d

2 Copythesestatementsandfillintheblanks. Thefirsttwohavebeendoneforyou.

Youwillneedtodotheshortdivisionineachstatement.

a Idivided127by6andgotremainder 1 .

So6 isnot afactorof127.

b Idivided216by6andgotremainder 0 .

So6 is afactorof216.

c Idivided313by3andgotremainder

So3 afactorof313.

d Idivided414by3andgotremainder

So3 afactorof414.

e Idivided413by3andgotremainder

So3 afactorof413.

f Idivided414by9andgotremainder

So9 afactorof414.

3 Copyandcompletetheseshortdivisions

a 9) 585

b 6) 498

c 4) 308

d 8) 544

4 Copyandcompletetheseshortdivisions.

a 4) 327

b 8) 739

c 3) 814

d 7) 463

1a Multiplythesenumbersby20. 748170239

b Multiplythesenumbersby19. 636123340

c Multiplythesenumbersby21. 821382413

d Multiplythesenumbersby30. 943194715

e Multiplythesenumbersby6. 17130309600

f Multiplythesenumbersby25. 54090120

2 Completethismultiplicationtable.

3 Calculatethesemultiplications.

4 ThepopulationofWintonintheyear2000was2476people.Ifthepopulation increasesby37peopleeachyear,whatwillthepopulationbein:

a 2015?

b 2020?

c 2027?

5 Acarparkcanhold150cars.Ifeachdriverpays $6.00perdayandthecarparkis fulleveryday,howmuchmoneywillthecarparkownercollectin:

a 5days?

b 1week?

c 3weeks?

6 Dothecorrespondingmultiplicationtocheckifeachdivisioncalculationiscorrect. 276 ÷ 12 = 23 a

7 Callumismakingup17partybags.Findthenumberofeachitemperbagandthe remainderifthereare: 89snakes a 36jellybeans b 112sourglowworms c 224Smarties d

8 Copythesestatementsandfillinthemissingnumbers.(Themissingnumbersarea divisor,aquotientoraremainder.) 17 = 3 × + 2 a

9 Elsieteachesatotalof96recorderstudents. Shehas16groupsofstudentsaltogether.How manystudentsareineachgroup?

10 Laurenswims1500metreseverytimeshegoes toswimmingtraining.IfLaurenswamthe samedistanceeachdayfor6days,howfar wouldsheswim?

11 Usetheshortdivisionalgorithmtocalculate these.

a 628 ÷ 4 b 26019 ÷ 9

c 795 ÷ 6

d 26019 ÷ 8

12 Whichofthesenumbersintheboxbelowaredivisibleby:

13 Aschoolisorganisingatriptoascience museum.Thebusesavailablecancarryeach 45students.Theschooldecidestofilleachbus toitsmaximumcapacity.Additionally,each studentisgivenalunchboxthatcosts$6.The schoolhasatotalbudgetof$1200forthe lunchboxes.

a Howmanystudentscantheschoolaffordto takeonthetrip?

b Howmanybuseswillbeused?

c Howmanyspareseatswilltherebeonthebuses?

4G Challenge–Ready,set,explore!

Challenge1:43

Inthissection,youwillgetplentyofpracticeatdividingby 43.Youmustaddorsubtractnumberslike43, 2 x 43 = 86 and3 x 43 = 129.Therewillbesometrialanderror. Keepacarefulrecordofyourcalculationsandwhatyou havechecked.Manyofthestepsarerelatedtothesteps youwillhavealreadydone.

1 Hint:Ifyouknowthat19992isthelargestmultipleof17thatislessthan20000, thenthefirstmultipleover20000is19992 + 17 = 20009.Youcanobtainother multiplesof17byaddingorsubtracting17.Forexample,20009 + 17 = 20026 and19992 17 = 19975.

a Whatisthelargestnumberwith6digitsthatisamultipleof43?

b Whatisthelargestnumberwith6differentdigitsthatisamultipleof43?

c Whatisthelargestoddnumberwith6differentdigitsthatisamultipleof43?

d Whatisthelargest6-digitmultipleof43whosedigitsaddupto43?

e Whatisthesmallest6-digitmultipleof43whosedigitsaddupto43?

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Challenge2:OnesandTwos

2 Findthesmallestnumberdivisibleby83thatismadeupcompletelyofthedigits 1and2.Tohelpyoursearch,usesome(orall)oftheseideas.

• Multiplesof83thatendin1 must betheresultofmultiplyingbyanumber endingin7.

• Multiplesof83thatendin2 must betheresultofmultiplyingbyanumber endingin4.

• Ifyouknowanumberhas83asafactor,youcaneasilygenerateothermultiples of83byadding83toitrepeatedly.

• Theonly3-digitnumbersthatarecomposedentirelyofonesandtwosare between111and122,andbetween211and222.

3 Findthelargest5-digitnumberthatismadeupofonlyeightsandsevensandis divisibleby58.Usetheshortcutsyoulearnedinquestion 1 tosavetime.

CHAPTER

Usefulskillsforthistopic

• recallingquicklythemultiplicationfactsto12 × 12

• representingafractionaspartofacollection

• representingwholenumbersandfractionsonanumberline

• findingequivalentfractions

• comparingandorderingfractions

• understandingoffactorsandmultiples

Vocabulary

Theword‘fraction’comesfromtheLatinword frango,whichmeans‘Ibreak’.

Numerator

• Denominator

• Vinculum

• Unitfractions

• Properfractions

• Improperfractions

• Mixednumbers

• 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’s engage 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 Whichislarger?

• Equivalentfractions

Whichfractionislarger 2 5 or 3 10 ?

Showandexplainyour thinkingwithapartner.

Fractions Fractions FractionsFractions Fractions 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 FractionsFractions Fractions 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

Fractionsareusedtodescribepartsofacollectionorpartsofanobject. Weusefractionswhenwecook. Forexample,acakerecipemight contain 1 2 acupofsugarand11 2 cupsofflour.

Weusefractionswhenweshare things.Forexample,if8friends boughtapizzaandsharedit equally,eachpersonwouldget 1 8 ofthepizza.

Weusefractionswhenwecomparedistances.Forexample,Bunburyin WesternAustraliais 2 3 of thedistancealongthe SouthWesternHighway fromAlbanytoPerth.

Perth Bunbury

Albany

5A Namingandrepresenting fractions

Afractionhasanumberonthetopandanumberdown below.Thesenumbershavespecialnames.

Thetopnumberiscalledthe numerator.Thenumber downbelowiscalledthe denominator.Onewayto rememberwherethedenominatorgoesistosay‘Dfor denominator,Dfordown’.

Thelinebetweenthenumeratorandthedenominatoris calledthe vinculum.

5 8 13 numerator denominator

Weusefractionstodescribepartofawhole.Hereisarectanglethathasbeencutinto 5equalpieces.

Threeofthepiecesareshaded.Wesay 3 5 oftherectangleisshaded.

3 5 the number of shaded pieces the number of equal pieces

Thenumeratorrepresentsthenumberofpiecesshaded,andthedenominator representsthetotalnumberofpieces.

Tohelpyourememberthedifference,thinkofthemlikethis: N isfor numerator;itmeansthe number ofequalpartsthatweareinterestedin. D isfor denominator;itmeans divided intothismanyequalparts.

3rd

Wecanalsousefractionstodescribepartofacollection ofobjects.

Wecandraw13blocksandshowthefraction 8 13 .

Thecircledgroupofblocksis 8 13 ofthetotalcollection ofblocks.

Thenumeratoristhenumberofblockscircled.The denominatoristhetotalnumberofblocks.

Fractionsonthenumberline

Thefirstfourwholenumbersandzeroaremarkedonthisnumberline.Thewhole numbersareequallyspaced.

Fractionscanbemarkedonanumberline,too.

Halves

Thisishowtomarkthefractions

and 5 2 onanumberline.Firstmarkin 0and1.

Nowbreakthelinebetween0and1into2equalpieces.Eachpieceisone-half.

Copyhalvesacrossthenumberlineandlabelthemarkers:

Thenumbers

Wereadtheseas‘one-half’,‘two-halves’,‘three-halves’,‘four-halves’,andsoon.

Quarters

Drawanumberlinefrom1to3.Dividethenumberlinebetween0and1into4equal pieces.Eachpieceisone-quarter.

Copyquartersacrossthenumberlineandlabelthemarkers

andsoon.

Wecanseethat:

2 4 isthesameas 1 2 4 4 isthesameas1 6 4 isthesameas 3 2 8 4 isthesameas2 10 4 isthesameas 5 2 12 4 isthesameas3 Thenumbers

Wereadtheseas‘one-quarter’,‘two-quarters’,‘three-quarters’,andsoon. Example1

Areamodelforfractions

Rectangles

Thisisarectangle.Wethinkofitas‘thewhole’. Ithasthevalueof1.

Iftherectangleisthewhole,thenthisisone-half. 1 2 1 2 ofthewhole

Thisisone-quarter. 1 4 1 4 ofthewhole

Thisisone-third. 1 3 1 3 ofthewhole

Thetopnumber,ornumerator,tellsushowmanypartsoftherectangleareshaded.

Thebottomnumber,ordenominator,tellsushowmanyequalpartstherectangleis dividedinto.

Example2

Drawtworectangles.Shade 1 5 ofthefirstrectangle.Shade 3 5 ofthesecond rectangle.

Solution

Shading 1 5 and 3 5 ofarectanglecouldbedoneinseveraldifferentways.Hereis onewayforeachfraction.

Squares

Wecanalsousesquarestomakefractionpictures.

Thinkofthissquareas‘the whole’.

Thesesquareshavebeen colouredtoshow 1 2

Thissquarehasbeen colouredtoshow 1 3

Thesesquareshavebeen colouredtoshow 1 4 .

Example3

Drawasquare.Drawlinesonthesquaretoshoweighths,thenshade 3 8 .

Drawanothersquare.Drawlinestoshoweighthsinadifferentway,thenshade 3 8

Solution

Thesquarecouldbeshadedin otherwaystoshow 3 8 . Howelsecouldthesquaresbe shaded?

Circles

Wecanusecirclestomakefractionpictures,too.Alwaysmakesurethatthecircleis dividedintoanumberofequalpieces.Thebestwaytodothisistodrawlinesfromthe centreofthecircle.

Thesecircleshavebeenshadedtoshow 1 2 , 1 3 and 1 7

Thispizzahasbeencutinto8equalpieces. 5piecesofpizzahavepineappleonthem. 3piecesofpizzadonothaveanypineapple. Writethefractionforthepartofthepizzathathaspineappleonit.

Solution

5partsoutofatotalof8pieceshavepineappleonthem,so 5 8 ofthepizzahas pineappleonit.Theansweris 5 8 .

Example4

5A Wholeclass LEARNINGTOGETHER

1 Therewere3litresoforangejuiceinSusan’sfridge.Shedrank1litre.What fractionoftheorangejuicedidSusandrink?

2 Ahmedhas19greenmarblesand37redmarbles.Firstcalculatethetotal numbersofmarbles.Thenworkoutthefractionofmarblesthataregreen.

3 Veronica’skitchentabletophasanareaof300000mm2.Veronica’sschool diarycoversanareaof43000mm2.Veronicaputsherdiaryinthemiddleof thetable.Whatfractionofthetabledoesitcover?

4 Chalkontheplayground

Youwillneedchalk.Drawanewnumberlineontheplaygroundforeach question.

a Markthewholenumbers0, 1, 2, 3and4onyournumberline.Nowmark themultiplesof 1 3 between0and4.

b Markthewholenumbers0, 1, 2, 3and4onyournumberline.Nowmark themultiplesof 1 4 between0and4.

Individual APPLYYOURLEARNING

1 Thesearefootballscarves.Writethefractionthatcorrespondstothebluepartof eachscarf.

a b

2 Writethefractionshownbythestaroneachnumberline.

3 Drawanumberlinefrom0to1.Markthesefractionsonit.

1 2 , 1 3 , 1 4 , 1 5 and 1 6

Whatdoyounoticeaboutthefractions?

4 Makethreecopiesofthissquare.

Useshadingtorepresent 3 8 inthreedifferentways.

5 Someofthesejuicebottlesareempty.Writethenumberofbottlesthatareempty asafractionofeachgroup.

6 Vinceboughtarectangularchocolatebardividedinto16equalsquares.Heate7 squaresofchocolateatrecess.WhatfractionofthechocolatebarisleftforVince toeatatlunchtime?

Kerribroughtaboxof12doughnutstoshareformorningtea. 2 3 ofthedoughnuts werestrawberry-filled.Therestwerechocolate-iced.HowmanyofKerri’s doughnutswerechocolate-iced?

Drawanumberlinefrom0to1.Markthesefractionsonit.

1 2 , 1 4 , 1 8 and 1 16

Whatdoyounoticeaboutthefractions?

Drawanumberlinefrom0to1.Markthesefractionsonit.

1 3 , 1 6 and 1 12

Whatdoyounoticeaboutthefractions?

10 Drawthreenumberlines,oneundertheother,onanA3sheetofpaper.Makethe firstnumberline10cmlong,thesecondnumberline20cmlong,andthethird numberline30cmlong.

Markeachnumberline0atoneendand1attheother.

Markthesefractionsoneachnumberline: 1 2 , 1 3 and 1 5

Whatdoyounoticeaboutthedifferentlengthnumberlines?

11 Whatfractionofeachsquarehasbeenshaded?

12 Thisis 1 3 ofawhole.

Drawwhatthewholemightlooklike.

13 Showhow 1 8 couldberepresentedoneachdiagrambelow:

5B Typesoffractions

Properandimproperfractions

Wecallafractiona properfraction ifthenumeratorislessthanthedenominator.For example, 1 4 and 3 4 areproperfractions.

Ifthenumeratorisgreaterthanthedenominator,orequaltothedenominator,then thefractioniscalledan improperfraction

Forexample, 5 4 and 4 4 areimproperfractions.

Example5

Labeleachfractionas‘proper’or‘improper’.

Wholenumbersasfractions

Allwholenumberscanbewrittenasfractions.Forexample,1 = 4 4 and2 = 8 4

Ifthenumeratorandthedenominatorarethesamenumber,wegetafractionthatis equivalentto1.Forexample, 4 4 = 1and 100 100 = 1.

Ifthenumeratorisamultipleofthedenominator,thefractionisequivalenttoawhole number.

Forexample, 12 4 = 3and 49 7 = 7.

Everywholenumberisalsoafraction.

Forexample,3 = 3 1 and227 = 227 1

Example6

Writethewholenumberequivalenttoeachimproperfraction.

Mixednumbers

A mixednumber isawholenumberplusafractionsmallerthan1.Forexample,11 6 isa mixednumber.Itmeans1plus 1 6 more.

Improperfractionsareeitherwholenumbersorcanbewrittenasmixednumbers. Ifwedividearectangleinto4equalpieces,eachpieceis 1 4 ofthewhole.

orfour-quartersisthesameas1.

Nowweextendthedrawingbyaddingonpiecesofsize 1 4 .

6 4 isthesameas 12 4 or11 2 .

9 4 isthesameas21 4

Wecanseethesameresultusinganumberline.Thisnumberlineismarkedusing quartersandmixednumbers.

Example7

Writethesefractionsasmixednumbersinsimplestform.Makethewholenumber partaslargeaspossibleandwritethefractionpartinsimplestform.

5B Wholeclass LEARNINGTOGETHER

1 Sorteachofthesefractionsintooneofthefollowingcategories:‘proper fraction’,‘improperfraction’or‘mixednumber’.

Addthreemoreexamplestoeachcategory.

5B Individual APPLYYOURLEARNING

1 Writethesefractionsaswholenumbersormixednumbers.

2 Convertthesemixednumberstoimproperfractions.

3 Conisplanningaparty.Heisallowing 1 2 abottleofsoftdrinkand 3 8 ofa pizzaforeachpersonwhocomestohisparty.

a If13peopleattendCon’sparty,exactlyhowmuchsoftdrinkwillheneed?

b If13peopleattendCon’sparty,exactlyhowmanypizzaswillheneed?

c If21peopleattendCon’sparty,howmuchsoftdrinkandpizza willheneed?

4 a Howmanypiecesofropeoflength 1 8 metrecanbecutfromapieceof ropeoflength61 2 metres?

b Howmanylengthsof 1 6 metrecanbecutfromarope81 3 metreslong?

c Howmanylengthsof 3 4 metrecanbecutfromarope7 3 12 metreslong?

5C Equivalenceand simplifyingfractions

Equivalentfractions

Twofractionscanmarkthesamepointonthenumberline.Forexample,wecanshow 1 2 and 2 4 onnumberlinesasfollows.

• Drawanumberline.Label0and1asshown.

• Dividethelinebetween0and1into4equalpiecestogetquarters.Eachlengthis 1 4

• Drawasecondnumberlinedirectlybelowthefirstnumberline.

• Dividethesecondnumberlineinto8equalpiecestogeteighths.Eachlengthis 1 8 .

Noticethattwolengthsof 1 8 isthesameasonelengthof 1 4 .Thismeansthat 2 8 and 1 4 markthesamepointonthenumberline.

Fractionsthatmarkthesamepointonthenumberline,suchas 2 8 and 1 4,arecalled equivalentfractions.Ifyoulookatbothnumberlines,youwillbeabletofindother pairsofequivalentfractions,suchas 3 4 and 6 8 , 2 4 and 4 8 , 4 4 and 8 8 . Wecanfindequivalentfractionswithoutdrawingtwonumberlines.

Thinkaboutwhathappenedtothenumeratorandthedenominatorof 2 4 . 2 4 4 8 × 2 × 2

Thenumerator (2) andthedenominator (4) wereboth multipliedbythesamewholenumber (2)

Wecangetanequivalentfractionifwemultiplythenumeratorandthe denominatorbythesamewholenumber.

Thisalsoworksinreverse. 4 8 2 4

Ifwedividethenumeratorandthedenominatorin 4 8 by2,weget 2 4

Wecangetanequivalentfractionifwedividethenumeratorandthe denominatorbythesamewholenumber.

Example8

a Useanumberlinetoshowthatthefractions 2 3 and 4 6 areequivalent.

b Whatwholenumberwerethenumeratoranddenominatorof 2 3 multipliedby togettheequivalentfraction 4 6 ?

c Giveanotherequivalentfractionfor 2 3

Solution

a Markanumberlineinthirds,andlabelthem.Thencuteachthirdintotwo equalpiecestogetsixths.

2 3 and 4 6 markthesamepointonthenumberline.

b Thenumeratoranddenominatorwerebothmultipliedby2.

c Multiplythenumeratoranddenominatorof 2 3 by5.

Thisgives 10 15,whichisequivalentto 2 3

Youcanalsomultiplybyanyothernumber.Forexample,youcanmultiplythe numeratoranddenominatorof 2 3 by6toget 12 18

Simplestform

Afractionisinsimplestformiftheonlycommonfactorofthenumeratorandthe denominatoris1.

Fortheequivalentfractions 1 2 , 2 4 , 3 6 , 4 8 and 50 100,thesimplestformis 1 2 .

1 2 isinsimplestformbecause1istheonlynumberthatisafactorofboth thenumeratoranddenominator.

3 17 isinsimplestformbecause1istheonlynumberthatisafactorofboth thenumeratoranddenominator.

4 6 isnotinsimplestform,as2isafactorofboththenumeratoranddenominator.

Dividingthenumeratorandthedenominatorby2,wefindthesimplestformof 4 6, whichis 2 3

Example9

Simplifyeachfraction.

Sometimesyoumayneedtodothedivisionsintwoormoresteps.

Reducethefraction 84 126 toitssimplestform.

Simplestformandequivalence

Thefractions 16 24 and 14 21 bothhave 2 3 astheirsimplestform.

Theybothmarkthesamepointonthenumberline.Theyareequivalentfractions.

Butwecannotgofrom 16 24 to 14 21 bymultiplyingordividingbythesamewholenumber.

Totestwhethertwofractionsareequivalent,wechecktoseeiftheyhave thesamesimplestform.

Example11

Showthat 40 24 isequivalentto 15 9 byreducingeachtoitssimplestform.

Bothfractions 40 24 and 15 9 simplifyto 5 3,sotheyareequivalenttoeachother.

5C Wholeclass LEARNINGTOGETHER

1 WhoamI?

Someonereadseach‘whoamI’totheclass,oneclueatatime.

Iamequivalentto 4 5 . WhoamI? a

Mynumeratoris16. Iamcloserto1thanto0.

Threeofmeisequivalentto 12 16

Iamequivalentto 25 100 .

Mydenominatoris4.WhoamI? b

2 Seta3-minutelimit.Yourchallengeisto:

a usemultiplicationtofindasmanyfractionsasyoucanthatare equivalentto 3 4

b usedivisiontofindasmanyfractionsasyoucanthatare equivalentto 180 700

3 Drawatablewiththreecolumnsonthewhiteboardandlabelthem‘Smaller than 3 5’,‘Equivalentto 3 5’,and‘Largerthan 3 5 ’.Sorteachfractionintothe correctcolumn.

Individual APPLYYOURLEARNING

1 Useanumberlinetoshowthatthetwofractionsareequivalent.

and

2 Writethreeequivalentfractionsforeachofthese.Describetheprocessyouused tocreatethesefractions.

3 Copythesefractions,thenfillinthemissingnumeratorsanddenominatorsto makeequivalentfractions.

4 Copyeachofthese.Fillintheboxestoshowwhichnumberthenumeratorand denominatorwere multiplied bytoarriveattheequivalentfraction.

Thefirstonehasbeendoneforyou.

5 Copyeachofthese.Fillintheboxestoshowwhichnumberthenumeratorand denominatorwere divided bytoarriveattheequivalentfraction.

Thefirstonehasbeendoneforyou.

6 a Howmanythirdsareequivalentto 6 9 ?

b Howmanyfifteenthsareequivalentto 2 3 ?

c Howmanyquartersareequivalentto 15 12 ?

d Howmanytwelfthsareequivalentto 1 3 ?

3rd

7 Simplifyeachfractionbydividingthenumeratorandthedenominatorbythesame wholenumber.

8 Areciperequires 2 5 ofacupofsugar.Ifyouwanttodoubletherecipe,howmuch sugardoyouneed?

9 Simplifythefraction 12 16 toitssimplestform.Isitequivalenttoanyotherfractions?

5D Comparingfractions

Whenthedenominatorsoftwofractionsarethesame,theonewiththe larger numeratoristhelargerfraction.

Thesetworectanglesshow 7 8 and 5 8 .

Youcanseethat 7 8 islargerthan 5 8

Itisnotassimpletocomparefractionswhenthedenominatorsaredifferent.

Thesetworectanglesshow 3 5 and 4 7 .

Youcanseethat 3 5 islargerthan 4 7

Butsometimesitisdifficulttodrawdiagramsthatshowclearlywhichofapairof fractionsislarger.

Thebestwaytocomparefractionsistofindanequivalentfractionwith thesamedenominatorforeachfraction.

Is 3 4 largerorsmallerthan 13 16 ?

Convert 3 4 intosixteenths. 3 4 = 3 × 4 4 × 4 = 12 16 12 16 issmallerthan 13 16,so 3 4 issmallerthan 13 16 .

Example13

Is 3 2 largerorsmallerthan 4 3 ? Solution

6isacommonmultipleofbothdenominators,soconvert 3 2 and 4 3 intosixths.

Larger,smallerorequivalent

Whenwecompareanytwofractions,wecanusesymbolsbetweenthem:

• ifthefirstfractionis smallerthan thesecondfraction,thesymbol<wouldbeused

• ifthefirstfractionis equivalentto thesecondfraction,thesymbol=wouldbeused

• ifthefirstfractionis largerthan thesecondfraction,thesymbol>wouldbeused.

Anumberlinecanbeusedtocomparetwofractions.Forexample,tocompare 2 3 and 3 4 welocatethemonthenumberline.Weknowthatnumberstotheleftonthe numberlinearesmaller,so 2 3 < 3 4 .

Ifwewanttocompare 8 20 and 12 30 wecanmarkthemonthenumberline.

Weseethattheymarkthesameplaceonthenumberline,sotheyareequivalent.

Comparingunitfractions

Fractionsthathave1asthenumeratorarecalled unitfractions

Forexample, 1 2 ,

and 1

Herearesomerectanglesthatshowunitfractions.

areunitfractions.

Youcanseethatthemorepartsthewholeisdividedinto,the smaller eachpartis. Thismeansthat1partoutof4islargerthan1partoutof20.So 1 4 islargerthan 1 20 .

Whenwecompareunitfractions,theonewiththelargerdenominatoris the smaller fraction.

5D Wholeclass LEARNINGTOGETHER

1 Setupapieceofstringasanumberline.Makethesefractioncards.

Eachstudentpegstheirfractiononthenumberlineinorderfromsmallestto largest.Discusswherecardswereplacedandthestrategyused.

HINT

-Whichfractionscouldbeusedasabenchmarkforplacingtheothers?

2 Repeattheactivityaboveusingthesefractioncards.

3 Adominostandingonitsendcanbereadasafraction.For example,thisdominocouldbereadas 3 5 or 5 3 .

Youwillneedaclasssetofdominoes(afterremovingthe dominoeswithblankspaces).

Workinsmallgroups.Eachgroupneedsahandfulofdominoes. Orderyour‘dominofractions’fromsmallesttolargest.The fractionswillvarydependingonwhichwayyoustandthemup.

5D Individual APPLYYOURLEARNING

1 Whichislarger?Recordusingsymbols=,<,or>.

2 Orderthesefractionsfromsmallesttolargest.

3 Copyandcompletethetablebychoosingthecorrectsymbol(‘issmallerthan<’, ‘isequivalentto=’or‘islargerthan>’)thatmakesthestatementtrue.Thefirst onehasbeendoneforyou.

4 Nameafractionthatis: between 7 10 and 1 4 a greaterthan 4 5 butlessthan1 b lessthan 2 3 butgreaterthan 4 12 c near0 d almost1 e lessthan 1 900 f

5 Amilkshakerecipeneeds 7 8 ofacupofmilk. Athickshakerecipeneeds 3 4 ofacupofmilk. Whichshakeneedsmoremilk?

6 SwanseaSoccerClubhastwoteams:theAteamandtheBteam.Bothteamshave thesamenumberofplayers.TheAteamhad 3 4 ofitsplayersturnuptotraining. TheBteamhad 7 12 ofitsplayersturnup. Whichteamhadmoreplayersturnuptotraining?

3rd

1 Writeeachfractioninwords.

2 Writethefractionthatmatchestheshadedpartofeachfootballscarf.

3 Drawafootballscarftoshoweachfraction.

4 Jimhasstartedastampcollection.Hehas12stampsfromHongKong,23stamps fromtheUnitedStatesand5fromtheUnitedKingdom.WhatfractionofJim’s stampcollectionis:

a fromHongKong?

b fromtheUnitedStates?

c fromtheUnitedKingdom?

5 TheBetterBakerymakesamixeddozenselectionofbreadrolls.Ofthe12rolls, 3arepoppyseed,4arewholemeal,2arewhite,1ismultigrainand2are sunflowerseed.

a Whatfractionoftherollsarewhite?

b Whatfractionoftherollsarewholemeal?

c Whatfractionoftherollsare not multigrainorsunflowerseed?

6 Drawanumberlinestartingat0andendingat1,andmarkthesefractionsonit.

1 2 , 1 4 , 1 7 , 1 8 and 1 9

Whatdoyounotice?

7 Drawanumberlinestartingat0andendingat1,andmarkthesefractionsonit.

1 3 , 1 6 , 1 9 and 1 18

Whatdoyounotice?

8 Copyandshade 1 4 ofeachsquare.

9 Makefourcopiesofthissquare.Shadeeachsquaretoshow 5 8 in fourdifferentways.

10 Writethreeequivalentfractionsforeachofthese.

11 Copythesefractionsandwritethemissingnumeratorsanddenominators.

12 Copythesediagrams.Fillintheboxestoshowwhatthenumeratorand denominatorwere multiplied bytoarriveattheequivalentfraction.

13 Copythesediagrams.Fillintheboxestoshowwhatthenumeratorand denominatorwere divided bytoarriveattheequivalentfraction.

14a Howmanytenthsareequivalentto 6 20 ?

b Howmanyquartersareequivalentto 6 8 ?

c Howmanythirdsareequivalentto 12 9 ?

d Howmanysixteenthsareequivalentto 3 12 ?

15 Reducethefractionsineachpairtotheirsimplestformtoshowthattheyare equivalent.

16 Writethesefractionsinorder,smallesttolargest.

17 Nameafractionthatis: between 5 9 and 1 5 a greaterthan 9 10 butlessthan1 b lessthan 3 4 butgreaterthan 8 16 c near0 d

18 BerniceboughttwoidenticalboxesoforangesandlabelledthemBoxAandBoxB. Lastweek,sheate 1 3 oftheorangesfromBoxAand 4 9 oftheorangesfromBoxB. Whichboxhasmoreorangesleftinit?

19 Acakerecipeneeds 5 8 ofakilogramofflour.Asconerecipeneeds 10 12 ofa kilogramofflour.Whichrecipeneedsmoreflour?

20 Copythesefractionsandlabelthem‘properfraction’,‘improperfraction’or‘mixed number’. 3

21 Convertthesefractionstomixednumbersorwholenumbers.Writethefraction partinitssimplestform.

22 Convertthesemixednumberstoimproperfractions.

5F Challenge–Ready,set,explore!

Plainsplitnumbers

Aplainsplitnumberisamultipleof3withsomeveryspecialfeatures.Whenyousplita plainsplitnumberintoone-thirdofitselfplustwo-thirdsofitself,youdonotneedto useanydigitmorethanonce.

Forexample,lookatthenumber138.Itisdivisibleby3: 1 3 of138is46.

Soifwedivide138intoone-thirdandtwo-thirdsweget:138 = 46 + 92(one-thirdof 138plustwo-thirdsof138)

Whatisunusualaboutthesplitisthatweusedonlysevendigits(1, 2, 3, 4, 6, 8, 9)in thesumwithoutanyrepetition.

Thismeansthat138isaplainsplitnumber,asitusessevendifferentdigitsinitssplit withoutrepeatinganyofthem.

Challengequestions

1 Showthatanyplainsplitnumberthatusessevendifferentdigitsinitssplitislarger than99andsmallerthan150.

2 Findalloftheplainsplitnumbersthatusesevendifferentdigitsintheirsplit.

3 Therearetwoplainsplitnumbersthatuseeightdifferentdigitsintheirsplits.One ofthemis207because207 = 69 + 138.Findtheotherplainsplitnumberthatuses eightdifferentdigits.

4 Therearefiveplainsplitnumbersthatuseninedifferentdigitsintheirsplits.Oneof themis801because801 = 267 + 534.Findtheotherfourplainsplitnumbersthat useninedifferentdigits.

5 Ifaplainsplitnumberusesalldigitsinitssum,itcannotbelargerthan1500but mustbelargerthan999.Explainwhy.

6 Findanumberthatisamultipleof4andusesninedifferentdigitswhenitissplit intoone-quarterplusthree-quarters.

Usefulskillsforthistopic

• representingfractionsonanumberline

• anabilitytoworkcomfortablywithmultiplesandfactors

• comparingandorderingfractions

Vocabulary

• Numerator

Equivalent

• Denominator

• Multiple

• Factor

• 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 Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Whattwofractionscouldbeaddedtogive 2 3 ? Findasmanysetsoffractionsaspossible.

Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic Fractionarithmetic

Fractionscanbetreatedlikeordinarynumbers.Wecanaddonefractionto another,ormultiplytwofractionstogether. Ifyoulookaroundyourhome,youmightseesituationswhereyouaddfractions.

Forinstance,thinkofaloafofbreadbeingcutintoslices.Eachsliceisafractionof thewholeloaf.Ifyoucuttheloafintotenequalslices,theneachsliceis 1

ofthe loaf.Whenyouusetwoslicestomakeasandwich,youhave

of theloaf.Ifyoumakeanothersandwichforafriend,you’veused2 ×

oftheloaf.

6A Addingfractions

Addingfractionswiththesamedenominator

Addingfractionswiththesamedenominatoristhesameasanyotheraddition.

Ifwewanttoadd 1 5 and 2 5,wecandrawa diagramlikethis.

Wecountthetotalnumberofpiecesofsize 1 5 andwrite: 1 5

Wecanaddtwofractionsonthenumberline.Forexample,

First,dividethenumberlinefrom0to1intosixths.

Thenshow 2 6 asjumpsonthenumberline.

Add 3 6 bymaking3morejumpsof 1 6 .

Wecanseethat: 2 6 + 3 6 = 5 6

Addingfractionswithdifferentdenominators

Weuseequivalentfractionstoaddfractionswith differentdenominators.

Wecanshow 1 2 + 1 4 usingarectangle.

Wecandrawthesamediagramdifferentlyto showthatone-halfplusone-quarterequals three-quarters.

Thisworksbecause 1 2 isequivalentto 2 4 .

Wecanalsoseethisusingthenumberlinebelow:

Solution

Drawarectangleanddivideitinto5equal pieces.Shade 1 5

Ifwecut 1 5 intotwoequalpiecesweget two-tenths.Cuteachfifthintotwo.Nowthe rectangleisdividedintotenths.

Shadeanother 3 10 toshowtheaddition.

1 5 + 3

Wecanuseequivalentfractionstomakethedenominatorsofbothfractionsbeing addedthesame.Thenaddingthefractionsisstraightforward.

Example3

Add 1 3 and 1 6

Solution

Thesefractionsdonothavethesamedenominator.Wecanchange 1 3 toan equivalentfractionwithadenominatorof6. 1 3 = 1 × 2 3 × 2 = 2 6 (Multiplythenumeratorandthedenominatorby2.)

Lowestcommonmultiple

Sometimesyouwillneedtoconvert both fractionstoequivalentfractionsthathave thesamedenominator.

Forexample,ifwewanttoworkout 1 2 + 1 3,weneedtofindequivalentfractionsfor both 1 2 and 1 3

Wedothisbyfindingthelowestcommonmultipleofbothdenominators.

Thelowestcommonmultipleof2and3is6.

Nowwefindequivalentfractionsfor 1 2 and 1 3 thathave6asthedenominator.

Usingequivalentfractionswiththesame denominator,theadditionbecomesas shownontheright.

Example4

Add 3 4 and 1 3

Solution

Thefractionsdonothavethesamedenominator.

Thedenominatorsofthetwofractionsare3and4.Thelowestcommonmultiple of3and4is12.Sonowwechange 3 4 intoafractionthathas12asdenominator.

Wealsochange 1 3 intoafractionthathas12asdenominator.

(continuedonnextpage)

Wecannowadd 9 12 and 4 12 becausetheyhavethesame denominator.

Weconvert 13 12 toamixednumber.

Thismeansthatouradditionhasbecome: 3

6A Wholeclass LEARNINGTOGETHER

1 Drawnumberlinestoshowtheseadditions.Writetheanswers.

2 UsethefractionwallinBLM16tohelpyoucalculateeachaddition.

3 Copytheseadditionsandfillintheblanks:

4 Drawcirclesorrectanglescutintohalves,thirdsandsixthstoshoweach additionanditssolution.

1 Calculate:

2 Drawanumberlinetocalculateeachaddition.

2 7 +

3 Murrayrecordedhowmuchbreadhisfamilyateeachdayfor1week.

a OnwhichdaysdidMurray’sfamilyeatmorethan1loafofbread?

b HowmuchbreaddidMurray’sfamilyeatfromMondaytoFriday?

c HowmuchbreaddidMurray’sfamilyeatontheweekend?

d HowmuchbreaddidMurray’sfamilyeatforthewholeweek?

4 Copythese,writethemissingnumeratorsanddenominators,andthensolve.

5 Addthesefractions.Youmightliketouseanumberlinetohelpyoufigureoutthe answers.

6 Copyandcompleteeachaddition.

3rd

7 Janineate 1 8 kilogramofgrapes.HersisterYvonneate 1 4 kilogramofgrapes.What fractionofakilogramofgrapesdidtheyeataltogether?

8 Writetheadditionforeachpicture.Usethepicturetohelpyoucalculate eachaddition.

9 Addthesefractions,thenwriteeachanswerasamixednumber.Rememberanumber linemaybeuseful.

10 Acakereciperequires 1 2 ofacupofoilforthemixture.Anextra 1 5 ofacupis neededtogreasethepan.Whatfractionofoilisneededaltogethertobake thiscake?

11 Convertthefractionsineachadditiontoequivalentfractionswiththesamedenominator, thenaddthem.Recordtheanswerinitssimplestform.

12 Atthefruitshop,theSmithfamilybought 1 3 kgofapples, 1 2 kgoforangesand 1 4 kgofgrapes.Whatwastheweightofthebagcontainingallthefruitto carryhome?

13 Inanefforttogetfit,MrWalshwalked21 2 kmonMonday,then33 4 kmon

Tuesday.OnWednesdayhewalked41 5 kmandonThursdayhewalkedafurther 5 3 10 km.Hethenrestedfortheremainderoftheweek.Whatdistancedid MrWalshwalkthatweek?

14 ThepetroltankofTom’scarshowsthereis 1 8 ofatankofpetrolremaining. Tomstopsatthepetrolstationandaddspetroltothecarsoitis 3 4 full. WhatfractionofpetroldidTomaddtothetank?

6B

Subtractionoffractions

Fractionswiththesamedenominator

Thiscakehasbeendividedinto6equalpieces. Theshadedpartofthecakehasicingonit,andthespotted parthaschocolatesprinkles.Theicedpartofthecakemakes up 3 6 ofthetotalcake.

Ifsomeoneeatsanicedpieceofcake,itisthesameas subtracting 1 6 ofthecake. Only2piecesofcakewithicingareleft.

Wecanwritethisasasubtraction: 3 6 1 6 = 2 6

Usingrectangles

Wecanuserectanglescutintoequalpiecestohelp explainsubtraction.Thesediagramsshow 2 10 taken awayfrom 7 10 .

Taking 2 10 awayleaves 5

Subtractiononthenumberline

Wecanalsouseanumberlinetoshowthesubtraction 7 10

10 .

Subtracting

Subtractingfractionswithdifferentdenominators

Wecanusenumberlinesordiagramstosubtractfractionsthathavedifferent denominators.

Forexample,ifwewanttotake 1 10 from 4 5,wecandrawadiagramtoshowwhat happens.

Thisdiagramhasbeencutinto5equalpieces.Then4ofthepieceshavebeenshaded toshowthefraction 4

Westartbycuttingeach 1 5 pieceintotwoequalpiecestomaketenths.Eachsmall pieceis 1 10

Take 1 10 away.

So, 4 5 1 10 = 7 10

Weuse equivalentfractions whenwesubtractfractionsthathavedifferent denominators.

Example5

Take 1 4 from 3 8 . Solution Write 1 4 asanequivalentfractionwithadenominatorof8.

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Ifdenominatorsarethesame → subtractnumerators.

Ifdenominatorsaredifferent → makeequivalentfractions,then subtract.

1 AtaprimaryschoolinGeelong,theYear3classmanagedtopickup41 2 kg ofrubbishattheirlocalbeachforeshore.TheYear4classcollected53 4 kgof trash.Therubbishwascombinedandsortedandthenthelocalcounciltook away41 8 kgforrecycling.Whatweightofrubbishwaslefttobethrownin thebin? Howcouldweuseanumberlinetohelpsolvethisproblem?Workwitha partnerthensharepicturesandsolutionswiththeclass.

3rd

6B Individual APPLYYOURLEARNING

1 Workoutthesesubtractions.Drawadiagramornumberlinetoshowyourworking.

2 Workouteachsubtraction,thensorttheanswersontoatablewiththreeheadings:

Maramadeacakeandcutitinto8equalpieces.Herbrotherate1piece.Mara tooktheremainingcaketoaparty,whereherfriendsate5pieces.Shetookhome whatwasleft.WhatfractionofthewholecakedidMarabringhome?

4 Matthewisanapprenticeelectrician.Hestartedthedaywith 7 8 ofawholerollof cable.Heused 1 3 ofawholerollinthemorning,and 1 4 ofawholerollinthe afternoon.HowmuchcabledidMatthewhaveleftattheendoftheday?

5 Therearetwostrategiesyoucanusewhensubtractingmixednumbers:

• youcansubtractthewholenumberfirst,thendealwiththefractions,or

• youcanconvertbothfractionstoimproperfractions,thendealwiththemas usual.

Usethemostsuitablestrategytosubtractthesemixednumbers.

6 Janeran101 4 kmforafundraisingfunrun.Benran105 6 km.Whatfractionofa kmdidBenrunfurtherthanJane?

7 Apizzahasbeencutinto8slices.Johneats 1 4 ofthepizzaandJennyeatsone slice.Howmuchofthepizzaisleft?

8 Astudentscored 18 20 onaspellingtest.Onfurtherevaluationtheteacherfound thestudenthadactuallyonlycorrectlyspelled 3 4 ofthequestions.Howmany questionshadtheteachermarkedincorrectlythefirsttime?

9 MrsMarshhad5kgofflourinhercupboard.Sheused21 5 kgforacake,then somemoreforsomecookies.Shewasleftwith1 6 10 kgflour.Howmuchflourdid MrsMarshuseforthecookies?

6C Findingafractionof anumber

Garyhas9apples.Hewantstotake 1 3 oftheapplestoschool.Howmanyapples isthat?

Wecanfindoutbydividingthewholegroupinto3equalgroupsandtaking1ofthose groups.

Garytakes3applestoschool.

Wecanwritethisas:9 ÷ 3 = 3,or 1 3 of9apples = 3apples

Whatistwo-thirdsof9apples?

Ifone-thirdoftheapplesis3apples,thentwo-thirdsistwiceasmany.

Two-thirdsoftheapples = 2 × 3apples = 6apples

Wecanusethemultiplicationsign × insteadoftheword‘of’.Wewrite9as 9 1 . 2 3 of9 = 2 3 × 9 1

Two-thirdsof9applesis6apples. Example7

a Whatisone-fifthof 20chocolatebars?

SAMPLEPAGES

b Whatisthree-fifthsof20chocolatebars?

a One-fifthof20chocolatebars

= 1 5 × 20 1 = 1 × 20 5 × 1 = 20 5 = 4chocolatebars

b Three-fifthsof20chocolatebars = 3 5 × 20 1

= 3 × 20 5 × 1 = 60 5 = 12chocolatebars

6C Wholeclass LEARNINGTOGETHER

1 Usecounterstofindthefractionofeachcollection.

2 Attheshoppingmall,Gregfoundapairofrunnersonspecial.Theyusually cost $85butwerereducedtosellfor 1 4 offthefullprice.Howmuchcould Gregbuytherunnersforatthesale?Whatmethodwouldyouusetofigure thisout?Shareyourthinking.

6C

Individual APPLYYOURLEARNING

1 Calculateeachofthese.Drawapicturetohelp.

SAMPLEPAGES

2 Thereare5peopleintheGreenfamily:Mum,Dadandthreechildren.Dadbought 15potatoesfordinner.Theseweretobesharedequally,butthenMumandoneof thechildrenwentouttohavedinnerwithafriend.

a Howmanypeoplearehomefordinner?

b Copythesestatementsandfillintheblanks. □ □ ofthefamilyishomefordinner.

SoMrGreenshouldcook □ □ ofthe15potatoes.

c Howmanyofthepotatoesshouldhecook?

3 Agroupof4friendsgotoalocalcafeforbreakfast.Thebillcomesto $34and theydecidetosplititevenly.Howmuchdoeseachpersonneedtopay?

4 TheO’Brienfamilyhas2parentsand3childrenunder12.Recentlytheyallcame downwithacoughandcold.Thelabelonabottleofcoughsyrupsuggestedthe recommendeddoseforanadultwas20ml.Therecommendeddoseforachild under12was 1 4 ofthisamount.Howmuchisneededtogiveeveryoneinthe O’Brienfamilyonedoseofthemedicine?

5 Anewmoviereleaseisshowingatthelocalcinema.Themoviestartsat12noon andrunsfor1hourand45minutes.Onethirdofthewaythroughsome refreshmentswillbebroughttoyourseat.Whattimewilltherefreshmentsarrive?

1 Writetheanswerstotheseadditionsasproperfractionsormixednumbers.

2 Drawanumberlinetocalculateeachaddition.

Addthefractions,thenwritetheanswerasaproperfractionormixednumber.

4 Copyandcompletetheseadditions. 3

5 Convertthefractionsineachsettoequivalentfractionswiththesame denominator,thenaddthem.

6 Attheendoftheirholiday,theRussellfamilydrovebackhomefromFingalBayto Brunswick.Onthefirstday,MrsRusselldrove 1 4 ofthedistance.Onthesecond day,MrRusselldrove 3 8 ofthedistanceandMrsRusselldrove 5 16 ofthedistance. HowmuchofthedistancehomehadtheRussellsdriveninthosetwodays?

7 Workoutthesesubtractions.

8 SharkBaySchoolbought15kilogramsofpottingmixtomakeavegetablegarden. Theyused2kilogramsforcherrytomatopots,3kilogramsforcucumberpotsand 4kilogramsforthecapsicumpots.Whatfractionofthepottingmixwasleftfor thegarden?

9 Calculate 5 6 of18. a Calculate 2 3 of9. b Calculate 2 5 of9. c

10 Jason’sSeafoodreceivedadeliveryof325kilogramsoffreshprawns.Jasonwants toselltheprawnsinbulklotsof6kilograms.Howmany6-kilogrambulklotscan Jasonsell?

11 Amrita’shomeworkshouldtakeher90minutestocomplete.Shedoeshomework forthesameamountoftimeeachday,andshehas6daystocompletethe homework.HowmanyhourswillAmritaneedtoworkonherhomeworkeachday untilitisfinished?

12 A3kgbagoforangescontains12oranges.Howmanyorangesin 2 3 ofthebag?

13 AnewfenceistobeplacedacrossthefrontoftheJones’homeproperty.The fencecomesin2 5mpiecesandeachpiecewillcover 1 8 ofthetotallength.How longwillthefencebeintotal?

6E Challenge–

Sylvester’suniqueunitfractions

SylvesterisreadingabookaboutEgyptianfractions.ThemethodstheEgyptiansused around1650BCwereverycomplicated.Thoughnotverymuchsurvivesofwhatthey did,weknowthattheyusedtablestoassistthemintheircomputations.Oneofthe interestingthingstheEgyptiansdidinvolvedtheuseofunitfractions.Unitfractions have1asthenumerator.

Forexample, 1 2 , 1 15 and 1 237 areunitfractions.

TheEgyptianshadspecialtablesforwritingafractionthatisnotaunitfractionasa sumofunitfractions.Theyperformedquitecomplexcomputationsusingthisidea.

Sylvesterusedonlyunitfractions,justliketheEgyptians.Forexample,hewrote 7 12 as thesumoftwounitfractions.Hediditintwodifferentways:

ThereisanotherwayofgettingananswerlikeSylvester’sfirstone.Youcarefullyselect aunitfractiontosubtract. Example8 Write 7 12 asthesumofunitfractions.

Solution

Weneedtochooseaunitfractiontosubtract.Wechoosethisfractionsothatitis lessthan 7 12 butasclosetoitaspossible.Wedothisbyfindingthefirstmultiple of7thatislargerthan12.Weknow1 × 7issmallerthan12.

Let’stry2.Now,2 × 7 = 14isgreaterthan12,therefore 1 2 < 7 12 andsowe subtract 1 2 7 12 1 2 = 7 12 6 12 = 1 12

Soweget: 7 12 = 1 2 + 1 12

Example9

Write 9 10 asthesumof3unitfractions.

Solution

Weneedtochooseaunitfractiontosubtractandthenworkwithwhatisleftover.

Wechoosethisfractionsothatitisascloseto 9 10 aspossiblebutlessthanit. Weobtainthisbyfindingthefirstmultipleof9thatislargerthan10.

Let’stry2.Now,2 × 9 = 18isjustgreaterthan10sowesubtract 1 2 9 10 1 2 = 2 5 Repeatfor 2 5 1 × 2istoosmall.Sois2 × 2 = 4.

Let’stry3.Now,3 × 2 = 6isgreaterthan5sowesubtract 1 3 . 2 5 1 3 = 1 15

Soweget: 9 10 = 1 2 + 1 3 + 1 15

Challengequestions

1 Writethesefractionsasthesumoftwounitfractions:

2 Whenyouadd 1 3 and 1 4,youget 1 3 + 1 4 = 4 + 3 3 × 4 Nowaddthesefractionsinasimilarway: 1 5 + 1 6

3 Usethisideatowritethesefractionsasthesumoftwodifferentunitfractions.

4 Writethesefractionsasthesumofthreedifferentunitfractions.

5 Whatisthelargestansweryoucangetbyaddingtwodifferentunitfractions?

6 Findafractionsmallerthan 9 10,butlargerthanyouranswertoquestion4,that cannotbewrittenasasumoftwounitfractions.Howmanycanyoufind?

Usefulskillsforthistopic

• anunderstandingofthebase-tensystemofhundreds,tensandones

• knowledgeoffractionsbeingpartofawhole

• usingnumberlinestorepresentnumbers,includingfractionsandmixednumbers

• anunderstandingofdecimalnumberstohundredths

Vocabulary

• tenths

• hundredths

• thousandths

• decimalpoint

• decimalnotation

• Theword‘decimal’comesfromtheLatinword decem,meaningten 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 Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage

Whichofthefollowingistheoddoneout? • 0.40

2 5

4 10

0 25

Decimals Decimals DecimalsDecimals Decimals Decimals DecimalsDecimals Decimals Decimals DecimalsDecimals 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

Decimalnumbersarebuiltupfromwholenumbersandfractionssuchas 1 10, 1 100, 1 1000,andsoon.

Decimalnumbersareusedformeasurementsinthemetricsystem,suchas measuringheight.Forexample,herearetheheightsoftheManeyfamily (inmetres).

Wealsousedecimalnumbersinourcurrencysystem.Forexample,185centsis writtenas$1.85.

7A Decimalnumbers

Theword‘decimal’comesfromtheLatinword decem,whichmeans‘ten’.Our numbersystemisbasedon‘lotsof’ten.

Decimalnumbersusethefractions 1 10, 1 100, 1 1000,andsoon,aswellaswhole numbers.Thedecimalpointisusedtoindicatewherethefractionpartsstart.

Forexample,thenumber3946 572means: 3thousands + 9hundreds + 4tens + 6ones + 5tenths +

So3946.572 = 3 × 1000 +

Tenths

Decimalnumberscanbeshownonanumberline.Thisnumberlineshows0,1and2.

Cutthelinebetween0and1into10equalpieces.Eachpiecehasalengthof

10 .

Labelthefirstmarker 1 10,thencontinuetolabelacrossthenumberline.Remember that 10 10 isthesameas1.Afterthis,thenumberlineshowswholenumbersandtenths: 1 1 10,1 2 10,andsoon.

Thenumberlineshowsfractionsanddecimalsthatareequivalent.

Betweeneverywholenumberandthenext,thenumberlinecanbemarkedintenths. Hereisanumberlinemarkedintenthsbetween20and21.

Wecanseethat:

20 1 = 20 1 10

20.8 = 20 8 10,andsoon

Hundredths

Ifwecutthenumberlinebetween0and 1 10 into10equalpieces,wegethundredths.

Eachpiecehaslength 1 100.Usingdecimalsthatbecomes: 1 100 = 0 01

Wecanmarkacrossthenumberlineinhundredths,startingat0.

Theseconddecimalplaceisforhundredths.Wewrite:

Whenwegetto1,wecontinueinstepsofone-hundredth.

Thisnumberlineshowshundredthsbetween2.9and3.1.

Lookatthenumberline.Youcanseethat:

Example1

a Writethedecimalnumber87.46onaplace-valuechart.

b Writethedecimalnumber87.46asasumoftens,ones,tenthsandhundredths.

c Mark87,87 46and88onanumberline.

Solution

Ifwecutthepartofthenumberlinebetween0and 1 100 into10equalpieces,we getthousandths.

Thethirddecimalplaceisusedforthousandths,so4thousandthsiswrittenas0.004. Whenwemagnifythenumberlinebetween0and0.01,wecanseemoreclearly where0 004isplaced.

Thefollowingnumberlineshowsthousandthsfurtheralongthenumberline, between19.09and19.1.

Writethenumber29

Decimalsasmixednumbersorfractions

Ifwewanttowrite2.45asamixednumber,wedoitlikethis.

2 45 = 2 + 4 10 + 5 100 = 2 + 40 100 + 5 100 = 2 45 100 = 2 9 20

However,thereisaquickerwaytowritefractionsasmixednumbers.Aswe’veseenin thepreviousexample,2 45goesasfarasthehundredthsplace.Sowewrite100asthe denominatorand45asthenumerator.

2 45 = 2 45 100

Thesameshortcutworksforthousandths.Thedecimalnumber12.765goesasfaras thousandthsandithas765afterthedecimalpoint,so:

12 765 = 12 765 1000 = 12153 200

Thedecimal0.08islessthan1.Wecanwriteitasafraction.

0 08 = 8 100 = 2 25

Comparingdecimalnumbers

0 76has7tenthsand0 392has3tenths. Wecanplacethesenumbersonanumberline.

UNCORRECTED

0.76istotherightof0.392onthenumberline,so0.76islargerthan0.392.

Uncorrected 3rd sample pages

Example3

Putthesenumbersinorder,smallesttolargest. 7.4267.8457.417

Solution

Lineupthenumbersusingthedecimalpointasamarker,sothatdigits withthesameplacevaluearelinedupundereachother.Allofthe numbershavethesamewholenumberpart,sowecomparethetenths.

Twoofthenumbershave4tenthsandonenumberhas8tenths. 8tenthsislargerthan4tenths,so7.845isthelargestnumber.Adjust theordertoshowthis.

Becausetwonumbershave4tenths,weneedtocomparethehundredths. 2hundredthsislargerthan1hundredth,so7 426islargerthan7 417. Theorder,smallesttolargest,is: 7 417

1 a Drawanumberlinebetween13and14andshowwherethesedecimal numbersarelocated.

2 Decimallines

a Drawontheboardahorizontallineabout50centimetreslong.Callits length1.Howlongdoyouthink1tenthmightbe?

Wecanusepicturesoflineslikethesetomodeldecimalnumbers.

b Whichdecimalnumbersareshownbythesepictures?

3 Convertthesedecimalsintofractionsormixednumberswithdenominatorsof100.

4 Matchthedecimalnumbertotheclues DecimalNumber

0 553

0 5

UNCORRECTEDSAMPLEPAGES

0 539

0 055

0 95

Thisdecimalnumberistheclosesttoonewhole

Thisdecimalnumberhasthesamenumberoftenthsand hundredths

Thisdecimalnumberisthesmallest

Thisdecimalnumberisequalto 1 2

Thisdecimalnumberhas9thousandths

7A Individual APPLYYOURLEARNING

1 Copythesestatementsandfillintheblankswithdigits0to9.Thefirstonehas beendoneforyou.

2 Copyandcompletethesestatements.

3 Markeachnumberonanumberline,thencopythestatementandfillinthe blanks.Thefirstonehasbeendoneforyou.

a 5.48isbetween5. 4 and5. 5

b 9 37isbetween9 and9

c 29.91isbetween29. and30.

d 19.98isbetween19. and20.

e 19 99isbetween19 9and

3rd sample

4 Convertthesenumbersintofractionsormixednumbersthelongway.Thendoit theshortway.Thefirstonehasbeendoneforyou.

a 296.74

Longway:

296.74 = 296 + 7 10 + 4 100 = 296 + 70 100 + 4 100 = 296 74 100

Shortway:296.74goesasfarashundredths. Wehave74afterthedecimalpoint.

So:296.74 = 296 74 100

5 Writethesefractionsandmixednumbersasdecimals.

6 Convertthesemixednumbersintodecimals.

7 Writetheseasdecimals. 7 +

8 Writethesedecimalsasfractions.

9 Converteachdecimalintoamixednumberorfraction.

10 Drawthenumberlineandmarkeachpairofnumbersonit. Thencirclethelargernumberineachpair.

a 4.1and3.9

b 2.24and2.42

11 a Writeanequivalentfractionwithadenominatorof10foreachfraction.

1 2 1 5 3 5 2 5

b Nowwriteeachfractioninpart a asadecimal.

12 a Writeeachfractionasanequivalentfractionwithadenominatorof100.

3 4 7 25 1 20 13 20

b Nowwriteeachfractioninpart a asadecimal.

13 Writethesemixednumbersasdecimals. 21 4 81 5 63 5 2 7 50

14 Writeeachsetofnumbersinorder,smallesttolargest.

a 6.14.6101.66.40.610.16.0

b 0.30.40.340.430.0340.0430.4030.004

c 0.0001960.20.99340.0360.820.4009

d 1.111.110.1110.010111.0111.0111

15 Writethenextfournumbersinthesecountingsequences.

a 10 810 911 0

b 26 726 826 9

c 82 4682 4782 48

16 Matchthedecimalnumbertotheclues

008 Thisdecimalnumberisthelargest

.84

.084

Thisdecimalnumberroundsto0.5

Thisdecimalnumberhas4thousandths

Thisdecimalnumberisequivalentto 8 10

Thisdecimalnumberisthesmallest

7B Roundingdecimals

Ifwewanttomakeanestimateofthesumoftwonumbers,sometimesitisusefulto roundthenumberstothenearesttenorhundred.Forexample,supposewecalculate 31 + 68 = 99andwanttocheckthatouranswerisaboutright.Werounddown31to 30androundup68to70.

Sowefindthat31 + 68isapproximatelyequalto30 + 70 = 100.

Thisisausefultechniquetocheckthatouranswerisreasonableor‘withinthe ballpark’.

Roundingcanalsobeusedwithdecimalnumbers.

Inthenumber12 561820754thereareninedigitsafterthedecimalpoint,buttomake iteasiertousewecanroundittotwoplaces.Thiswouldgiveusanapproximatevalue ofthenumber.

Onanumberline,12.561820754wouldbeabouthere:

Wecanseethat12 561820754islargerthan12 56andsmallerthan12 57.

Becauseitiscloserto12.56wewouldround12.561820754to12.56.

Whatifwewanttoround12.561820754toonedecimalplace?Thatis,whatdowe doifweneedtoknowwhether12 561820754iscloserto12 5or12 6?

Therearesomerulesthatcanhelpusroundwholenumbersanddecimalnumbersthat donotrequiretheuseofanumberline.

Ifwewanttoroundtoonedecimalplace,thefirstdigitafterthedecimalpointisour roundingdigit.Thenwefollowthesesteps.

Lookattheverynextdigittotherightoftheroundingdigit.

• Ifthenextdigitis0, 1, 2, 3or4,thentheroundingdigitstaysthesame.

• Ifthenextdigitis5, 6, 7, 8or9,thentheroundingdigitgetslargerby1.

In12 561820754,ourroundingdigitis‘5’andthedigitafteritisa6,sotherounding digitgetslargerby1.Ourroundingdigitbecomes‘6’. Wethendiscardalltheotherdigitsaftertheroundingdigit.

So12 561820754roundedtoonedecimalplaceis12 6.

3rd

Example4

a Round235tothenearestten.

b Round428tothenearesthundred.

c Round9.87654321tothenearesthundredth.

Solution

a Whenweround235tothenearestten,ourroundingdigitis3.

235

Thenumbertotherightofthe3is5,sotheroundingdigitgetslargerby1. Theotherdigitsarediscardedandreplacedbyazerotoholdtheplacevalue.

240

Rounding235tothenearesttengives240.

b Whenweround428tothenearesthundred,ourroundingdigitis4.

428

Thedigittotherightofthe4is2,sotheroundingdigitstaysthesame. Thedigitsafteritarediscardedandreplacedbyzeroestoholdtheplacevalue.

400

Rounding428tothenearesthundredgives400.

c Whenweround9 87654321tothenearesthundredth,ourroundingdigitis7.

9.87654321

Thedigittotherightofthe7is6,sotheroundingdigitincreasesby1andthe digitsafteritarediscarded.

9.88

Rounding9.87654321tothenearesthundredthgives9.88.

Roundthesenumberstothenearesttenth.

Roundthesenumberstothenearestwholenumber.

Roundthesenumberstothenearesthundredth.

4 Fillintheblankspacestomakeeachofthestatementsbelowtrue.Youcan useeachofthedigits1,3,5,7and9onlyonce:

a 85roundedtothenearestwholenumberis4

b 7 ___9roundedtothenearesttenthis7 2

c 4 2___4roundedtothenearesthundredththis4 29

d 5 098roundedtothenearestwholenumberis___

e 9 68roundedtothenearesttenthis9

7C Addingdecimals

Wecanadd0.2to0.6asfollows.

0.2 + 0.6 = 2

Wecanfollowthesamestepsasanadditionalgorithmforwholenumbers.Startby liningupthenumberswiththedecimalpointsandtheplacesaligned.

0 . 2 + 0 6

0 . 8

Say‘2tenthsplus6tenthsis8tenths’. Write‘8’inthetenthscolumn.

Example5

Completetheseadditions.

UNCORRECTEDSAMPLEPAGES

Wecanusetheadditionalgorithmtoaddtwo,threeormoredecimalnumbers.

Firstdealwiththethousandths. 5thousandthsplus7thousandthsmakes 12thousandths,plus4thousandthsmakes 16thousandths.

Write‘6’inthethousandthscolumn,and carry1intothehundredths.

Nowworkwiththehundredths.

3plus6makes9hundredths,plus8plus 1makes18hundredths.

Write‘8’hundredthsandcarry1intothe tenths.

Nowaddthetenths,theonesandfinally thetens.

Example6

Flynnwenttothesupermarket.Heboughtapacketofcornchipsfor $3.85,an energydrinkfor $1.90andabagofavocadosfor $4.35.HowmuchdidFlynn spend?

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Whendoingawrittencalculation,itisimportanttolineupthe numbersaccordingtotheirplacevalue,andhavethedecimalpoints aligned.

7C Wholeclass LEARNINGTOGETHER

Delightfuldecimalnumbers

1a Followthesestepstofindoutyourowndelightfuldecimalnumber:

• Thetensdigitisthenumberofchildreninthefamilyofapersonsitting nearyou.

• Theonesdigitisthesumofthedigitsofyourhousenumber.Ifyougeta sumgreaterthan9,addthedigitsagain.Forexample,237wouldbe 2 + 3 + 7 = 12,then1 + 2 = 3,sotheonesdigitwouldbe3.

• Thetenthsdigitisthenumberofpetsyouhave.(Usethesumofthedigitsif youhavemorethan9pets.Forexample,11gives1 + 1 = 2.)

• Thehundredthsdigitisthenumberoflettersinyourfirstname.(Usethesum ofthedigitsifyouhavemorethan9lettersinyourfirstname.)

• Thethousandthsdigitisthenumberofpeoplewholiveinyourhome. (Usethesumofthedigitsifmorethan9peopleliveinyourhome.)

Thenumberyounowhaveisyourdelightfuldecimalnumber.

b ThisishowPetearrivedathisdelightfuldecimalnumber.

• PetesitsnexttoLuna,whoisoneoffivebrothersandsisters,soPete’stens digitis6.

• Pete’shousenumberis27.Thedigitsin27sumto2 + 7 = 9,sohisones digitis9.

• Petehasnopets,sohistenthsdigitis0.

• Petehas4lettersinhisname,sohishundredsdigitis4.

• 5peopleliveathisplace,givinghisthousandthsdigit. Pete’sdelightfuldecimalnumberis69.045. AddyourdelightfuldecimalnumbertoPete’s.

c Nowaddyourdelightfuldecimalnumbertotwoofyourclassmates.

1 Usetheadditionalgorithmtocalculatethese.

2 Nancywenttothepaintstoreandboughtatinofpaintfor $32 85,apaintbrush for $12 75andsometilesfor $86 05.Howmuchdidshespendintotal?

3 Tanihasfivedifferentbankaccounts.Herearetheirbalances.

a HowmuchmoneydoesTanihaveintotal?

b IfTaniputthebalanceofaccountAandaccountEintoaccountD,howmuch wouldbeinaccountD?

c Tanioweshermum $42.00.Fromwhichtwoaccountsshouldshewithdrawall themoneytogetclosestto $42.00?

4 Richardmeasuredthelengthofthreeshelvesinmetres.

Shelf Blue Black White

Length 1.986m 2.012m 1.884m

a Whatisthelengthoftheblueandthewhiteshelvestogether?

b Whatisthelengthoftheblackandthewhiteshelvestogether?

c Whatisthetotallengthofthethreeshelves?

d Whichtwoshelveswouldfitbestina4-metrespace,leavingthesmallestgap?

5 Addthesedecimalnumbers.Remembertolineupthedecimalpoints.

a 12 5 + 1 2 + 21 1

b 114 577 + 5 472 + 77 2

c 1 11 + 7 74 + 77 7

d 53 02 + 190 7 + 7 702

e 7 245 + 0 702 + 37 74 + 588 3

f 1.0025 + 104.7442 + 1.27 + 0.2

6 Patwalked81 5 kilometresandBillwalked2.4kilometres. Whatisthetotaldistancethattheywalked?

7 Maryhadtwomatsinherhallway.Onewas7 6 10 metreslongandtheotherwas 3.25metres.Whatwastheirtotallength?

8 ToddandCarmelwereplayinggolf.Todds’schipshotwent143 4 metresand Carmel’swent2.75metresfurther.WhatwasthedistanceofCarmel’sshot?

9 Annebought11 2 kgofsugar.Anne’smumhadalreadybought2 5kgofsugar. Howmuchsugardidtheyhaveintotal?

10 Tomused4.3litresofpaintwhenhepaintedhisbedroom.HisbrotherJakeused 41 4 litrestopainthisbedroom.HowmuchpaintdidTomandJakeuseintotal?

3rd

7D Subtractingdecimals

Weusethesubtractionalgorithmtosubtractdecimals.Besuretoalignthedecimal pointsandtheplaces.

Example7

Calculatethesesubtractions.

a 2.4 1.3

b 12.26 9.74

Whenthedecimalnumbersaredifferentlengths,alignthedecimalpointandthe places.Wewriteazeroontheendoftheshorternumbertoshowthatplaceisempty.

Example8

Calculate12 583 3 9.

Calculatethesesubtractions.

Solution

a Writeazeroattheendof6 8tomatchthe4inthehundredthsplaceof3 94.

b Writetwozerosattheendof19.4tomatchthe6inthehundredthsplaceand the7inthethousandthsplaceof9.567.

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Whendoingawrittencalculation,itisimportanttolineupthe numbersaccordingtotheirplacevalue,andhavethedecimalpoints aligned.

1 Useoneofthesubtractionalgorithmstocalculatethese.

2 Subtract4.5from:

.8 a

.214 c

3 Subtract122.095from:

.898 a

.06

.678 b 4306.03 c

4 Moirabought0.85mofcolouredribbon.Shecutoff0.29m.Howmuchribbon wasleft?

5 TwotrucksdeliveredgraveltoGrumpy’sGravelYard.Altogethertherewas8 441 tonnesofgraveldelivered.Thesecondloadwas4 963tonnes.Howmuchgravel wasdeliveredinthefirstload?

6 Feliciawenttothelocalsupermarketandboughtthreeitemscosting $17.45 altogether.Oneitemcost $5 65andanothercost $8 85.Howmuchdidthethird itemcost?

7 Arectangularpaddockhadaperimeterof1200metres.Thewidthofthepaddock was220 85metres.Whatwasthelengthofthepaddock?

Multiplyingdecimalsby10

Whathappenswhenwemultiply0.2by10? 0 2 = 2 10 Ifwemultiply 2 10 by10weget2.

Wecanseethisusingdecimalstickpictures.

Thisis1: Andthisis0.2:

Thisiswhat10lotsof0.2lookslike.

Itisthesameas2ones.

0.2 × 10 = 2

0 2 × 10 = 2 10 × 10 = 20 10 = 2

Let’strymultiplying0.12by10.

0.12 = 1 10 + 2 100

0 12 × 10 = ( 1 10 × 10) + ( 2 100 × 10)

= 10 10 + 20 100 = 1 + 2 10 (10tenths = 120hundredths = 2 10 ) = 1.2

Nowlet’stryamultiplicationthatinvolvesthousandths. 0.349 = 3 10 + 4 100 + 9 1000

0.349 × 10 = ( 3 10 × 10) + ( 4 100 × 10) + ( 9 1000 × 10) = 30 10 + 40 100 + 90 1000 = 3 + 4 10 + 9 100 = 3.49

Haveyounoticedapatternwhenmultiplyingby10?

Whenwemultiplyby10,eachdigitmovesoneplacetotheleftintheplace-value chart.

Uncorrected 3rd sample

Multiplyingdecimalsby100

Multiplyingdecimalsby100isthesameasmultiplyingby10andthenmultiplyingby 10again.

Whenwemultiplyby100,eachdigitmovestwoplacestotheleftintheplace-value chart.Wemayhavetoinsertzeroesifnecessarytofillinanyblankspaces.

Let’sseewhythisworks.

1.2isthesameas1 2 10 1.2 × 100 =(1 × 100)+ ( 2 10 × 100) = 100 + 200 10 =

Wecanalsousetheplacevaluecharttodemonstrate:

Weaddedazerotofillthegapintheonescolumn.

Example10

Multiply2 347by100. a Multiply19 4by1000. b

Thedecimalpointdoesnotmove.

Whenmultiplyingadecimalnumberby10,eachdigitmovesone placetotheleft.

Whenmultiplyingadecimalnumberby100,eachdigitmovestwo placestotheleft.

Whenmultiplyingadecimalnumberby1000,eachdigitmovesthree placestotheleft.

7E Individual APPLYYOURLEARNING

1 Copytheplace-valuechartforeachnumberintoyourworkbook.Showeach numberonthechart.Multiplyeachnumberby10andshowitontheplace-value chart.Thencompleteeachmultiplication.(Thefirstonehasbeendoneforyou.)

a 34.25 × 10

.25 × 10 = 342.5

6.83 × 10

.83 × 10 =

80 09 × 10

.09 × 10 =

2 Dessiemeasuredhershoe.Itwas21.7cmlong.IfDessieplaced10ofhershoes endtoend,howlongwouldthelinebe?

3 Stephenmade2 5Lofcordialandpouredout10glassesforhisfriends.Eachglass contained0.125Lofcordial.Howmuchcordialwasleftinthejug?

4 Hamishmeasuredthelengthofoneofhispencils,asshown.Ifhelaid10ofthese pencilsendtoend,howlongwouldthelineofpencilsbe?

5 a Chloeis1.67mtall.Howmanycentimetrestallisshe?(Thereare100 centimetresin1metre.)

b Oscarweighs32.45kg.Howmanygramsisthat?(Thereare1000gramsin 1kilogram.)

6 Copythistable.Completeitbymultiplyingeachnumberby10,100and1000.

a 0.492

b 83.06

c 507.08

d 9 23

e 99 999

7 Converttheseweightsandmeasures.

5.62kg = _____g

.745t = ____kg

8 Copythistable.Converteachfractionormixednumbertoadecimal,then multiplyby10,100and1000.Writeyouranswerasadecimal.

9 Alocalbakerysellscupcakesfor $3.75each.Tencustomerscomeinandorderten cupcakeseach.Howmanycupcakeswillthebakeryneedtosupplyandwhatwill thetotalcostbe?

10 Sophierunsasmallonlinestorewhereshesellscustom-madeT-shirts.Each T-shirtcosts $12.95.Shereceivesalargeorderof150T-shirtsfromacorporate client.Whatwouldthecostbetotheclient?

Whenwedivide 1 10 by10weget 1 100 .Thisisbecause10hundredthsmake 1 10

Wecanwritethisusingdecimals.

0 1 ÷ 10 = 0 01

Whenwedivideby10,eachdigitmovesoneplacetotherightintheplace-value chart.Weaddazeroinanymissingspacesasaplaceholder.

Wecanwritethenumber12.4likethis:

Dividingby10gives:

Divide12 3by10,thenby100,thenby1000.

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Thedecimalpointdoesnotmove.

Whendividingadecimalnumberby10,eachdigitmovesoneplaceto theright.

Whendividingadecimalnumberby100,eachdigitmovestwoplaces totheright.

Whendividingadecimalnumberby1000,eachdigitmovesthree placestotheright.

2 Thistableshowsthenutritioninformation,inmilligrams,fromapacketoflolly snakes.Copythetableandconverttheamountstograms.(Remember: 1g = 1000mg.)Recordyouranswersinthethirdcolumn.

7F Individual APPLYYOURLEARNING

1 Divideeachnumberby10.

2 Maryandher9friendslovelicorice.Marybought2.5moflicoriceandcut itintoequalpiecesforherselfandher9friends.Whatlengthoflicoricedid theyeachget?

3 Steveplacedtenblocksofwoodthesamelengthendtoendandmeasured thetotallength.Thelengthwas1.4m.Howlongwaseachblock?

4 a Suedividedanumberby10andgot22.3.Whatwasthenumbershe startedwith?

b Zoedividedanumberby10andgot0 065.Whatwasthestarting number?

c Minhdividedanumberby20andgot0 03.Whatwasthestarting number?

5 Bottlesofsparklingwateraresoldinpacksof10atalocalsuperpmarket.The regularpricefortheentirepackis $18 50.Whatwouldbethepriceofone bottlewhensoldindividually?

Converttheseweightsandmeasures.

7 Janboughtapacketof10CrunchyBiscuits.Thebiscuitsweighed0.6kg altogether.Jan’sbrotherateoneofthebiscuits.Whatwastheweightofthe remainingbiscuits?

Reviewquestions–

Demonstrateyourmastery

1 Writethesefractionsandmixednumbersasdecimals.

2 Writethesedecimalsasmixednumbersandfractions.

3 Thetableshowsthelengthoffourlines.Eachoftheselinesneedstobemade longer.Copyandcompletethetable.

4 Writeeachsetofnumbersinorderfromsmallesttolargest.

5 Converteachdecimalintoamixednumber.

Convertthesemixednumbersintodecimals.

7 Usetheadditionalgorithmtocalculate:

Powersof10

Ournumbersystemisbasedonpowersof10.Inthissection,wewillinvestigatea methodofwritingnumberssuchas50000usingmultiplesofpowersof10.

Toillustratehowthishappens,let’shavealookathowthemanagerofahaberdashery shopmightbuybuttons.

Onepacketholds10buttons.Wewritethisas:1 × 10 = 10buttons.

Onecardholds10packetsof10buttons.Thismakesasquarewith10rowsof10.

Wesay10times10is‘10squared’,or‘10tothepowerof2’.

Wewritethisas:10 × 10 = 102 = 100

Oneboxholds10cards,eachwith10packetsof10buttons.Thismakesacubewith 10layersof10rowsof10.

Wesay10times10times10is’10cubed’or’10tothepowerof3’.

Wewrite:10 × 10 × 10 = 103 = 1000

Ifwemultiply4factorsof10togetherweget10 × 10 × 10 × 10.Thisiscalled‘10to thepowerof4’.

Wewrite:10 × 10 × 10 × 10 = 104 = 10000.

Ifwemultiply5factorsof10togetherweget10 × 10 × 10 × 10 × 10.Thisiscalled ‘10tothepowerof5’.

Wewrite:10 × 10 × 10 × 10 × 10 = 105 = 100000. Youmighthavenoticedashortcut.

• When102 iswritteninexpandedform,ithastwozeroesintotal:100.

• When103 iswritteninexpandedform,ithasthreezeroesintotal:1000.

• When104 iswritteninexpandedform,ithasfourzeroesintotal:10000.

3rd

Example12

Write1000000asapowerof10.

Solution

1000000has6zeroes,soitbecomes:

1000000 = 106

So1millionis106

Wewillnowextendthisideatoincludenumbersthatarenot1, 10, 100or1000. 4000is4 × 10 × 10 × 10,soitcanbewrittenas4 × 103 .

Example13

Write4000000asamultipleofapowerof10.

Solution

4000000 = 4 × 1000000 = 4 × 106

So1millionis106

Challengequestions

1 Writetheseproductsaspowersof10.

a 10 × 10 × 10 × 10 × 10 × 10 × 10

b 10 × 10 × 10 × 10

c 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

d 10 × 1

2 Writeeachproductasapowerof10,thenwriteitasanumber.

a 10 × 10 × 10 × 10 × 10

b 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

c 10 × 10 × 10

d 10 × 10 × 10 × 10 × 10 × 10

Writeeachnumberasamultipleofapowerof10. Onethousand a Onemillion b Tenthousand c Eighteenmillion d

4 Writeeachpowerof10asaproductof10s,thenwritethenumber. 107 a 104 b 1012 c 101 d

5 Writeeachnumberasaproductof10s,thenasapowerof10. 100000 a 100000000 b 10000 c 100000000000000 d

6 Writeeachnumberasamultipleofapowerof10. 5hundred a 23thousand b 16million c 7billion d

7 Writeeachnumberasamultipleofapowerof10.

g 430000000 h

8 Someworldpopulationfiguresfor2005areshowninthetablebelow.Writethe populationandtheareaforeachcountryasmultiplesofpowersof10.

9 Writethesemultiplesofpowersof10aswholenumbers.

SAMPLEPAGES

10 Harveyhas10paddocks.Eachpaddockisdividedinto10sections,andeach sectionhasenoughpasturefor6cows.Whatisthegreatestnumberofcowsthat Harveycanhaveonhisfarm?

Writeyouranswerasamultipleofapowerof10,andasanumber.

11 Harveyisoneof10neighbourswhoareallfarmers.Eachfarmerhassetuptheir farminthesamewayasHarvey.Whatisthemaximumnumberofcowsthatcould grazeinHarvey’sdistrict?

Writeyouranswerasamultipleofapowerof10,andasanumber.

UNCORRECTEDSAMPLEPAGES

Usefulskillsforthistopic

• understandingoffractionsanddecimals

• theabilitytoconvertfractionsandmixednumberstodecimals,andtoconvert decimalstofractionsormixednumbers.

Vocabulary

‘Percent’comesfromtheLatinwords percentum,whichmean‘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 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

What’sinaname?

Writeyourfullnameonapieceofpaper.

1 Whatfractionofthelettersinyournameareconsonants?

2 Canyouwritethisfractionasadecimal?

3 Whatpercentageofthelettersinyournamearevowels?

4 Whichwastheeasiesttocalculate?Why?

5 Whatstrategiesdoyouusetoconvertbetweenfractions,decimalsand percentages?

Number Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages PercentagesPercentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages PercentagesPercentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages PercentagesPercentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages Percentages PercentagesPercentages Percentages Percentages Percentages

UNCORRECTEDSAMPLEPAGES

Percentagesareanotherwayofwritingfractionordecimalquantities.

Thereare100flowersinBree’sgarden.Fiveoftheflowersarered.

Wecansaythat‘5percent’oftheflowersarered.‘5percent’isaquickwayof saying‘5outofonehundred’.

Thesymbolforpercentis %,so5percentiswrittenas5%.InBree’sgarden5% of theflowersarered.

8A Fractions,decimalsand percentages

Fractionsandpercentages

Apercentageisanotherwayofwritingafractionwithadenominatorof100.

Example1

TheRosebudCinemahas100seats.OnSaturdaynight,75ofthe100seatswere filled.Writethisasapercentageofthenumberofcinemaseats,thenasafraction ofthenumberofcinemaseats.

Solution

75outof100 = 75% So75% ofthecinemaseatswerefilled. 75outof100 =

= 3 4 So 3 4 ofthecinemaseatswerefilled.

Wecanalsowritefractionsthatdonothave100asadenominatoraspercentages. Firstconvertthefractiontoanequivalentfractionwithadenominatorof100,then writethefractionasapercentage.Forexample:

Youcanalsomultiplythefractionby100

Wholenumberscanbeconvertedtopercentages.

Toconvertapercentagetoanequivalentfractionormixednumber,firstwriteitasa fractionwithadenominatorof100,thensimplifyit.Forexample:

Weoftenuse100% todescribe‘all’ofsomething.Forexample,‘100% ofthestudents finishedtheirswimminglessons’meansthatallofthestudentstooktheswimming classes.

Percentageslessthan100% describesomethinglessthanthewhole.Forexample, ‘only72% ofstudentsarrivedontime’meansnoteveryonewasprompt.

Percentagesgreaterthan100% indicatethatsomethingwasmorethanexpected.For example,100studentswereexpectedtoattendthedance,but150studentsshowed up.Theattendanceismorethan100%

Decimalsandpercentages

Adecimalcanalsobewrittenasapercentage.Toconvertadecimaltoapercentage, firstwriteitasafractionwithadenominatorof100.

Adecimalthathashundredthsasthelastplaceconvertseasilytoapercentage.

Example2

Likewise,apercentagecanbeconvertedtoadecimalbyfirstwritingitasafraction withadenominatorof100,thenconvertingittoadecimal.Forexample:

Example3

Writethesepercentagesasdecimals.

Wholeclass LEARNINGTOGETHER

Ordereachgroupfromsmallesttolargest.Explainhowyoucancomparethe numbersintheirdifferentforms. 1

8A Individual APPLYYOURLEARNING

1 Writeeachoftheseasapercentage.

2 Writethepercentageforeachsituation.

a 18outof25studentsintheclasslikepizzamorethanpasta.

b 4outof5animalsinthenatureparkarenative.

3 Writethesepercentagesasfractionsintheirsimplestform.

4 Writethesepercentagesasdecimals.

5 Accordingtothebureauofstatistics,inVictoria,29.9% ofthepopulationwere bornoverseasin2021.Howmanypeoplewouldthisbeinagroupof1000?

6 50childrenspentthedayatLunaPark.27ofthemwereboys.Whatpercentage ofthechildrenweregirls?

7 Inaclassof25students,23ofthechildrenwerepresentforrollcall.What percentagewereabsent?

8 Ernestrequires 2 5 ofacupofsugarforacakeheisbaking.Whatisthisasa percentage?

9 Therearesomepigsandducksonafarm.25% oftheanimalsonthefarmhave fourlegs.Whatfractionarebirds?

10 Inasoccermatch,Josiescored18outofthe25ofthegoalsforthematch.What percentageisthis?

11 Achocolatebarweighs50grams.60% ofthebarisdarkchocolate.Howmany gramsofdarkchocolateareinthebar?

12 Inafundraisingevent,440ofthe500ticketsweresold.Whatpercentageofthe ticketswerenotsold?

13 Thereare36Year6studentsattheSwanRiverPrimarySchool. 3 4 ofthestudents submittedtheirassignmentontime.Howmanystudentssubmittedtheir assignmentlate?

8B Percentage‘of’aquantity

Weoftenneedtocalculateapercentageofaquantity.

Forexample:

Tocalculateapercentageofanothernumber,convertthepercentagetoafraction, thenmultiply.Theword‘of’tellsusthatweneedtomultiply.

Example4

Calculate20% ofeachnumber.

% of80 = 20 100 × 80 1 = 1 5 × 80 1 = 80 5 = 16

Unitarymethod

Thereissometimesaneasierwaytocalculatepercentages.Itiscalledthe unitary method.Firstworkout10% bydividingthenumberby10% inyourhead,thenworkin multiplesorfractionsof10%.

Example5

% of80 = 80 ÷ 10 = 8

% of80 = 2 × 8 = 16

Percentageincrease

Weusepercentageincreasewhenwetalkaboutanincreaseinwages,whenafeeis addedtoabillandwheninterestispaidonabankaccount.Tocalculateapercentage increase,firstcalculatethepercentage‘of’thenumber.Togetthenewnumber,add thepercentageincreasetotheoriginalnumber.

Example6

Simonleavesa5% tipforgoodtableservicewhenhegoesouttoeatinarestaurant. Headdsthispercentageontoeachfoodbill.Calculatethetotalcostofamealfor Simon,includingthe5% tip,ifthefoodbillcomestoatotalof:

$100 a

$60 b

$300 c

Solution

5% of $100 = 5 100 × 100 = 1 20 of100 =$5

Therestaurantbill,plus

Percentagedecrease

Percentagedecreaseisusedtodescribeasituationsuchasadropinthenumberof peopleattendingfootballmatchesthisyearcomparedtolastyear,orwhenanitemis discountedbecauseitisonsale.Tocalculateapercentagedecrease,calculatethe percentage‘of’thenumber.Togetthenewnumber, subtract thepercentagedecrease fromtheoriginalnumber.

Example7

Katrinawenttoasaleandboughtajumperthatusuallysellsfor $100at40% discount.WhatpricedidKatrinapayforthejumper?

Solution

40% of $100 = 40 100 × 100 =$40

Originalprice discount = saleprice $100 −$40 =$60

Katrinapaid $60forhernewjumper.

8B Individual APPLYYOURLEARNING

1 Calculatethesepercentages.

2 25% ofthewaterusedinAustralianhouseholdsisusedforwashingclothes. Whatistheamountofwateronehouseholdwoulduseinonedayforwashing clothesifitsdailywaterusageis:

100litres? a 150litres? b 300litres? c

3 Calculatethetotalamountafterthepercentageincreaseforeachofthese examples.

a Hamishearned5% interestonabankbalanceof $20.

b Lastyear,300childrenattendedbasketballclinicsatPortHaven.Thisyear, thenumberofchildrenattendingbasketballclinicshasrisenby10%

c Stella’sbeanplantwas30centimetrestallatthestartofJanuary.Bythe startofFebruary,herbeanplanthadgrown20% taller.

JakeandThomasSmithearned5% onthebalanceoftheirbankaccountlast year.Jakesaved $340andThomassaved $715.Howmuchmoneydideach brotherhaveinthebankattheendoftheyear,includinginterest?

5 Calculatethetotalamountafterthepercentagedecreaseforeachofthese.

a Tonilost8% ofherweightof50kilograms.

b 12% ofthe150treesinTrevor’sgardendiedbecauseofthedrought.

c SalesatBrightBooksaredown50% onlastweek’sfigureof250books.

6 Stephaniesaves25% ofherpocketmoneyeachweek.Howmuchdoesshe havelefttospendifshereceives: $10? a

12? b

7? c

7 Afootballstadiumhas25000seats.Foraparticularmatch,85% oftheseats werefilled.Howmanypeopleattendedthematch?If 1 5 ofthepeoplewho attendedwerechildren,howmanychildrenattendedthematch?

1 Writeeachoftheseasapercentage.

2 Writethemissingfractions(ormixednumbers),decimalsandpercentages.

3 Calculatethesepercentages.

20% of100 a 50% of120 b

80% of50 c 10% of900 d

5% of20 e 8% of200 f

100% of99 g 1% of1000 h

4a Writedownanumberthatismorethan30% of50.

b Writedownanumberthatis30% morethan50.

5 Foreachofthese,calculatethetotalamountafterthepercentage increase

a Abankbalanceof $130earned8% interest.

b Lastyear,200childrenattendedtheDaptohearingclinic.Thisyearthenumber ofchildrenattendingtheclinichasrisenby18%

c Francescawas130centimetrestallatthestartofJanuary.Bytheendof Decembershehadgrown20% taller.

6 Foreachofthese,calculatethetotalamountafterthepercentage decrease

a Cillahad180treesinhergarden.Tenpercentofthemdiedbecauseofthe drought.

b Chocolatesalesare50% downonlastyear’sfigureof1000.

8D Challenge–

Canetoads

In1935,102canetoadswereintroducedintoAustraliatohelpcontrolbeetleslivingin thesugarcanefieldsofQueensland.Theyquicklybecameapestandhavenowspread intoNewSouthWales,WesternAustraliaandtheNorthernTerritory.

SAMPLEPAGES

Challengequestions

1 Inthefirst6monthsthecanetoadpopulationhadincreasedto589timesthe originalnumberreleased.Howmanytoadsisthis?

2 Ifacanetoadtravels40kilometresperyear,howfarwouldittravelin: 2years? a 5years? b 15years? c 8years3months? d

3 Theaveragecanetoadweighs1.85kilograms.TheNorthernTerritoryFrogWatch programcaught29264canetoadsin1year.

a Whatwasthetotalmassofthecanetoadscaught?

b Howmanygramsisthis?

4 Thelandnowoccupiedbycanetoadsis700000squarekilometres.

a Ifthereare5canetoadsperhectare,howmanycanetoadsisthis?

b Ifthereare15toadsperhectare,howmanycanetoadsarenowinAustralia?

5 5% ofcanetoadtadpolesgrowintoanadult.Calculatethenumberof survivorsfrom:

100tadpoles a 1000tadpoles b 5000tadpoles c 35000tadpoles d 2400tadpoles e 403680tadpoles f

6 Everyyeareachfemalecanetoadproduces2clutchesofbetween8000and 35000eggs.If0.5% ofcanetoadeggssurvivetoadulthood,calculatethenumber thatwouldsurvivefrom:

200eggs a 1000eggs b 5000eggs c 8000eggs d 10000eggs e 13000eggs f 35000eggs g 180000400eggs h

7 0.5% ofcanetoadeggssurvivetoadulthood.

a If40femalecanetoadseachproduce16000eggsin1year,howmanyadult canetoadswillthisbe?

b If85femalecanetoadseachproduce9000eggsin1year,howmanyadult canetoadswillthisbe?

c If1000000femalecanetoadseachproduce32000eggsin1year,howmany adultcanetoadswillthisbe?

d If795883femalecanetoadseachproduce70000eggsin1year,howmany adultcanetoadswillthisbe?

8 Anewmethodofcontrollingcanetoadsisintroducedthatcauseshalfthe populationtodieeachyear.Howmanyyearswillitbebeforethecanetoad populationisbacktolessthantheoriginal102toadsifthepopulationstartsat: 1000? a

Usefulskillsforthistopic

• asolidgraspofarithmetic

• anunderstandingofinverseoperations.

Vocabulary

Operations

• Input • Output

• Indices •

• Algorithm • Brackets

Pets

DanielandMarkhavesomepets.

• Theyeachhavethesamenumberofpets.

• Thetwoboysown9parrotsbetweenthem.Therestoftheiranimalsarelizards.

• Eachparrothas2legs.Eachlizardhas4legs.

• Daniel’sanimalshave2morelegsintotalthanMark’s.

• Danielownsthesamenumberofparrotsaslizards.

• IfMarkgaveDanielalizard,Daniel’sanimalswouldhave28legsinall.

• Howmanyparrotsandhowmanylizardsdoeseachboyhave?

• Whatstrategydidyouusetofindyouranswer?Didyoudrawapictureorwrite equationsorsomethingelse?

Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra Algebra Algebra Algebra AlgebraAlgebra Algebra Algebra Algebra Algebra

Inthischapterwelookatanimportantpartofmathematicscalled algebra Algebraisawayofsolvingproblemswhenpartoftheinformationisunknown.In apuzzletherearepartsweknow,forexampleinthe’Pets’puzzleontheprevious pageweknewwhattypesofpetstherewereandhowmanylegseachpethad. Tosolvethepuzzle,weneedtofindtheunknown.Inthe’Pets’puzzlewehadto findhowmanyofeachpetMarkandDanielowned.

Algebraisatoolthatenablesustodealwiththeunknowninapuzzle. Manyoftheideasinalgebra arebasedonwhatyoualready knowaboutnumbers.The wordalgebracomesfroma bookwrittenaround830AD byamathematiciannamed Al-Khwarizmi.Hisideas helpedspreadalgebrato Europe.

Theword‘algebra’comesfromthetitleofhisbook.Wegettheword‘algorithm’ fromhisname.

9A Findinganunknown

Inalgebra,weoftenuseasymbollike □ oraletter(suchas x)tostandforan unknownnumber.

Whenwearetoldthat □ + 6 = 18,weusuallyreadthisas‘Somethingplus6is18.’ The □ standsforanumber,butwhatisit?Youcanprobablysolvethisinyourheadby nowbecauseyouknowthat12 + 6 = 18.

Oryoucouldstartwithsomethingthatyouaresureof,suchas10 + 6 = 16,thengo up2more:12 + 6 = 18.Youcanprobablythinkofotherwaysoffindingoutwhatthe □ represents.

Example1

□ 2 = 14Whatis □?

Solution

Takingaway2from □ gives14.Thismeansthat □ mustbe2morethan14. So □ = 16.

Wecanoftenusemathematicalsymbolstowritewhatwehavebeentold.Thisisan importantskill.

Forexample:Martinesaid,‘Ifyouadd3tomyage,youget12’.Whatis Martine’sage?

Youcanprobablysolvethisinyourhead,butwewillpractiseusing □ torepresent Martine’sageinstead.

Martinehasalreadysaidthatadding3to □ gives12.Ifweusethe+sign, thisbecomes:

3 + □ is12

Nowreplace‘is’withthe = sign.Youget:

3 + □ = 12

Fromthisyoucanworkoutthat □ = 9.

Usinginverseoperationshelpsustochecktheaccuracyofour calculation.

Additionistheinverseofsubtraction,soyoucanusesubtractionto solveadditionproblemslikethisone.

Example2

Iamanumber.Fourtimesmeis28.WhatamI? Solution

Youcandothisinyourheadbyrememberingyourtables: 4 × 7 = 28.Sothenumberis7.

Wecanalsowritethisinformationinanotherway,using □ torepresentthe unknownnumber:

4 × □ = 28Whatnumberis □?

Wecanthenanswerthisbyusingourtables: 4 × 7 = 28,so □ = 7. Anotherwayistousedivision:

Thenumberis7.

9A Wholeclass LEARNINGTOGETHER

1 Findthenumberthat □ representsineachquestion.Thendiscussthedifferent methodspeopleusedtoworkout □

+ 6 =

2 Use □ torepresentthenumberyouneedtofind.Use +, , ×,or ÷ and the = signtowritedownwhatyouaretoldabout □,thenfindoutwhat □ represents.

a Ian’sageplus4is14.

b Laura’ssister’sageplus6is12.

c 3addedtoDavid’sweightinkilogramsis30.

d Celiahasaboxwithanumberofchocolatesinit.HerfriendTammyate 2chocolatesandtherewere10left.

9A Individual APPLYYOURLEARNING

1 Findthenumberthat □ representsineachquestion.

2 Writethesestatementsusingalgebra.Use +, –, × or ÷ andthe = sign,and usea □ toreplacetheword‘something’. Somethingplus4is11. a Somethingtakeaway5is9. b Twolotsofsomethingis12. c Somethingsharedbetweentwois4. d

3 AtThomasStreetPrimarySchoolthereare4classes.Eachclasshasthesame numberofchildreninit.Thereare6staffmemberswhoworkattheschool. Thetotalnumberofchildrenandadultsintheschoolis106.Howmany childrenareineachclass?

4 Agroupofchildrenwasonabus.Threechildrengotoffthebusatthefirst stop.Atthenextstop,12childrengotonthebus.Therearenowtwiceas manychildrenonthebusastherewereatthestart.Howmanychildrenare onthebusnow?

3rd

9B Orderofoperations

Lookattheequationbelow: 5 + 7 × 3 = □

Tomsayswesimplyworklefttoright,sowehave5 + 7 = 12,thenmultiplythisby 3toget36.

Jane,however,suggestswesolvethemultiplicationfirst,7 × 3 = 21,andaddthisto5. Thentheanswerwouldbe26.

Whoiscorrect?

Weneedaconsistentapproachtosolvingaproblemsothateveryonecanreadand solveaprobleminthesameway.Everyonecanusethisagreedconventionoforderof operationsandobtainthesameanswer.Theagreedorderisasfollows:

• Calculateanythinginsidethebracketsfirst.

• Computeanyfractionsorindicesnext,forexample,32 = 9.

• Nextcomesmultiplicationanddivision.Thesearetreatedequallybecausetheyare inverseoperations.Assuchjustworkfromlefttorightastheyappear.

• Finallywesolveanyadditionandsubtractionworkingfromlefttorightasthey appear.Additionandsubtractionaretreatedequallybecausetheyareinverse operations.

Evaluate50 15 × 3. Istheanswer35 × 3 = 105or50 45 = 5?

Solution

Theagreedorderofoperationstellsusanymultiplicationanddivisionmustbe solvedbeforeanyadditionandsubtraction.Inthiscasewewouldneedtosolve theequationasfollows:

50 15 × 3(calculatethemultiplication,15 × 3first)

= 50 45(calculatethesubtractionnext) = 5

Example3

9B Wholeclass LEARNINGTOGETHER

1 Solvethefollowingequations:

2 Canyouuseuptosix4’stocreateequationswherethesolutionis0through to10?Youcanuseanyoperationsaslongasyoufollowtherulesforthe orderofoperations.Forexample,4

3 TheMitchellfamilyuserainwaterfromtheirtanktowatertheirgarden.At thestartofJune,theirtankhad2800litresinit.After13days,thetankheld 6700litres.Duringthistime,itrainedregularlyandtheMitchellsdidnotneed towaterthegarden.Eachday,thetankcollectedthesameamountofwater. Howmuchwaterwascollectedeachday?

9B Individual APPLYYOURLEARNING

1 Solvethefollowingequations:

2 Canyouuseonly3’sandcreateequationsthatfollowtheorderofoperations andequalallthewholenumbersfrom0to20?

3 Cherylneedstosave $50over12months.Shehasalreadysaved $14.

a Howmuchdoessheneedtosaveeachmonth?

b Writeanequationwiththeorderofoperationsforthissituation.

4 Scottsavedthesameamountofmoneyeachweekfor6weeks.Thenhis grandfathergavehim $34.NowScotthas $100.

a HowmuchmoneydidScottsaveeachweek?

b Writeanequationwiththeorderofoperationsforthissituation.

9C Numberpatterns

Patternsoftenoccurinmathematicsandtheycanbeusedtofindthenextnumberina sequenceoramissingnumberinasequence.

2, 4, 6, 8, 10,…

Inthiscountingsequenceweadd2tofindthenextnumberandcontinuethepattern, sothenextthreenumberswouldbe12, 14, 16.

Adifferentsequencecouldbeformedbydoublingeachnumber,forexample:

2, 4, 8, 16,

Anothersequencecouldbemadeusingmorethanonestep,forexample,doublethe number,thenadd1.

Startingat2thenextnumberwouldbe5(double2is4andadd1toget5),then11 (double5is10andadd1toget11).

Thenumberpatternwouldbe2, 5, 11, 23, 47, 95,...

Example4

Sanjitmakesapatternbymultiplyingbyanumberandadding. ThefirstnumberinSanjit’spatternis3,thesecondnumberis8,andthethird numberis18.

Canyoufindtheruleandcompletethetable?

1stnumber 3

2ndnumber 8

3rdnumber 18

4thnumber 38

5thnumber

6thnumber

7thnumber 318

8thnumber

Solution

The1stnumberis3

The2ndnumberis2 × 3 + 2 = 8

The3rdnumberis2 × 8 + 2 = 18

Sothe5thnumberis2 × 38 + 2 = 78

The6thnumberis2 × 78 + 2 = 158

The8thnumberis2 × 318 + 2 = 638

Sotherulethatletsyouworkoutthe nextnumberinSanjit’spatternis multiplybytwoandthenaddtwo.

Number

1stnumber 3

2ndnumber 8

3rdnumber 18

4thnumber 38

5thnumber 78

6thnumber 158

7thnumber 318

8thnumber 638

9C

Wholeclass LEARNINGTOGETHER

1 Kaitlyn’sfavouritebiscuitsaresoldinpacketsof4.

a Copyandcompletethetableontheright.

b IfKaitlynbuys2packetsofbiscuits,how manybiscuitsisthisintotal?

c IfKaitlynbuys3packetsofbiscuits,how manybiscuitsisthisintotal?

d IfKaitlynbuys8packetsofbiscuits,how manybiscuitsisthisintotal?

e IfKaitlynbuys30packetsofbiscuits,how manybiscuitsisthisintotal?

f IfKaitlynbuys200packetsofbiscuits, howmanybiscuitsisthisintotal?

2 Sonjais9andherbrotherJaredis16.Sonjadrewupatabletocomparetheirages astheygetolder.

a WhenSonjais10,howoldwillJaredbe?

b WhenSonjais13,howoldwillJaredbe?

c WhenSonjais14,howoldwillJaredbe?

d WhenSonjais18,howoldwillJaredbe?

e WhenJaredis24,howoldwillSonjabe?

f Challenge:WhenJaredwastwiceSonja’s age,howoldwasSonja?

Individual APPLYYOURLEARNING

1 Writethenextfivenumbersineachsequence.

2 Ateachercreatesapatternforherclass.Thepatternlookslikethis: 1, 5, 9, 13, □, 21,... Whichnumberismissing?

3 Hereisanumberpattern:1, 3, 5, □, 9, □, 13,... Whatisthesumofthefirst8numbersinthispattern?

4 Apatternbegins:1, 2, 5, 8, 11, 14, 17. Whichnumberisthenextinthepattern?

14 A 20 B 14 C 20 D

5 Dannyisbuildingpyramidsfromblocks.Sofarhehasbuilt3pyramids.

Tobuildthepyramidthatis3blockshigh(P3),hefirstmadea3 × 3squareof9 blocksonthetable.Thenheputa2 × 2squareof4blocksontopofthe3 × 3 square.Finally,heputa1 × 1squareof1blockonthetop. DannywantstobuildpyramidP4.Hehadonly50blockswhenhestartedandhe can’tusetheblocksfrompyramidsP1, P2andP3becausehegluedthem together.Whichofthefollowingiscorrect?

A DannydoesnothaveenoughblockslefttobuildP4.

B DannyhasexactlyenoughblockslefttobuildP4,withnoblocksleftover.

C DannyhasenoughblockslefttobuildP4,with1blockleftover.

D DannyhasexactlyenoughblockslefttobuildP4,with20blocksleftover.

6 Antheaisbuildingpyramids,usingthesamemethodasDannyinquestion 5.So far,shehasbuilt4pyramids(seebelow).Shehasexactlytherightnumberof blockslefttobuildanotherP4.HowmanyextrablockswillAntheaneedtoborrow ifshewantstobuildP5insteadofanotherP4?

9blocks A 16blocks B 25blocks C 30blocks D

9D Reviewquestions–Demonstrateyourmastery

1 Findthenumberthat □ representsineachquestion.Thendiscussthedifferent methodspeopleusedtoworkout □

a □ + 8 = 19

b □ –2 = 8

c 2 × □ = 10

d □ ÷ 2 = 7

e □ + 12 = 52

f □ × 2 + 18 = 68

2 Writethesestatementsusingalgebra.Use +, –, × or ÷ andthe = sign,anduse

a □ toreplacetheword‘something’.Findthenumberthat □ representsineach question.

a Somethingplus12is25.

b Somethingtakeaway9is27.

c Twolotsofsomethingis66.

d Somethingsharedbetweentwois36.

3 Solvethefollowingequations:

a (56 + 43)÷ 11 = □

b 1 4 of80 3 × 6 = □

c 9 × 5 +(12 + 43)= □

d 56 ÷ 8 × 3 = □

e 100 62 × 2 = □

f 66 ÷ 11 × 9 ÷ 6 = □

4 Writethenextfivenumbersineachsequence.

a 10, 20, 30, 40, 50, …

b 2, 4, 8, 16, 32,

c 3200, 1600, 800, 400, 200, …

d 3, 7, 15, 31, 63, …

9E Challenge–

Emma’sEs

Emmaismakingpatternswithtiles.Shecallshershapes E1,E2 and E3,dependingon thenumberofhorizontaltilesineach E

Challengequestions

1 Sheused8tilesfor E1 and11tilesfor E2.

a Howmanytileswillsheusefor E5?

b Howmanytileswillsheusefor E7?

c Howmanytileswillsheusefor E10?

d Whatdoyounoticeaboutthepattern?

e Emmahasexactly100tiles.Howmany Escouldshemakebeforerunningout oftiles?

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2 Emmadecidedtobuildupher Esinanewway.Ittookalotmoretiles.Shecalled thesenewshapes WideE1, WideE2 and WideE3.

a Howmanytileswillsheusefor WideE6 inthisnewpattern?

b Thelastdigitsofthenumberoftilesusedtomakethe WideEsfollowanew pattern.Writeasentencethatdescribesthepattern.

c Howmanytileswillsheneedaltogethertocreateallthe WideEsfrom WideE1 to WideE10?

d Emmamade WideE8 and WideE9.Thenshebrokethemintopiecestomake thebiggestnumber WideE shecould.Whatnumber WideE wasit?

CHAPTER

Usefulskillsforthistopic

• quickrecallofmultiplicationfactsto12 × 12

• theabilitytomeasurelengthanddistancesinmillimetres,centimetresandmetres usingrulersandtapemeasures

• theabilitytoconvertameasurementincentimetrestometres,ortometresand centimetres

• theabilitytocalculatetheperimeterandareaofarectangularshape.

Vocabulary

centimetre

• millimetre • metre • kilometre

• squarecentimetre • squaremetre • squarekilometre

• 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 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

Trueorfalse?

Canyoudecideifthesestatementsaretrueorfalse?Explainwhy.

1 Ifyouknowtheperimeterofarectangularyardis50metres,youcanassumeall foursidesareequalinlength.

2 Atypicalsmartphoneisusuallylessthan20cminlength.

3 Theareaofarectangleiscalculatedbyaddingthelengthandwidthtogether.

4 Ifyouwanttobuildarectangulargardenthathasaperimeterof40metres,you needtomeasureeachsideseparately.

5 Arectangulargardenmeasuring4metresby6metreshasenoughspacefor 24floweringplantsifeachneeds1squaremetre.

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 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 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 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 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

Ifwewanttoknowhowlongorwideorhighsomethingis, wecanmeasureitwithameasurementtool.

Weusearulertomeasureshortlengths,suchasthewidth ofyourhand.

Atapemeasurecanbeusedtomeasurelongerlengths, suchasyourheight.

Surveyorsuseatoolcalledatheodolitetomeasurelong distances,likethelengthofaroad.

Nomatterwhattoolweuse,wecanfindoutsomething’ssizeandthenusemaths toworkoutotherinformationaboutit.

10A Length

InAustraliaweusethe metricsystem ofmeasurement. Theunitsoflengthinthemetricsystemarebasedonthemetre.

Youwilloftenseetheword‘metre’abbreviatedastheletter m.Aswellasthemetre, wealsousethemillimetre(mm),thecentimetre(cm)andthekilometre(km)to measurelength.Theprefixbefore‘metre’tellsyouaboutthesizeoftheunit beingused.

milli means 1 1000 so1millimetreis 1 1000 ofametre.

Whenusingdecimals:1mm = 0.001m

centi means 1 100 so1centimetreis 1 100 ofametre.

Whenusingdecimals:1cm = 0 01m kilo means1000so1kilometreis1000metres.

Choosingunits

Therearetwoimportantthingstorememberwhenmeasuringlength.First,youneed toselectthemostsuitableunitofmeasurementfortheobjectyouwanttomeasure. Second,youneedtoselecttherightmeasuringinstrument.

Forexample,ifyouwanttoknitapairofgloves,itwouldnotbeverysensibleto measurethewidthofyourhandinwholemetres.Itwouldbemoreappropriatetouse centimetresormillimetres.

Builders,furnituremakers,architectsandelectriciansnearlyalwaysusemillimetres, evenforverylargemeasurements.Mosttapemeasuresaremarkedoutinmillimetres.

Example1

Measurethelengthofthispencilinmillimetres.

Solution

Thepencilis97millimetreslong.

Whenmakinganymeasurement,wecanonlybeasaccurateasourmeasuring instrumentallows.Thepencilin Example1 mightactuallybe96.76938mmlong. Butifthesmallestunitontherulerweareusingismillimetres,thenwemustmake ourmeasurementtothenearestmillimetre.

Example2

Measurethelengthofthispenciltothenearestcentimetre.

Solution

Thepencilis10centimetreslong. Noticethatthemeasurementhasbeenroundeduptowholecentimetresbecause itiscloserto10cmthan9cm.

Whenyouaremeasuringandwanttocomparemeasurementsoraddorsubtract them,alwaysusethesameunits.

10A Wholeclass LEARNINGTOGETHER

1 Estimatethelengthofeachitem,thenmeasureitasaccuratelyasyoucan.

a Theheightofthisbookinmillimetres

b Thewidthofyourclassroominmetres

2 Withoutusingaruler,estimateandcutpiecesofpaperstreamertothese lengths.Writeyourestimateoneachstreamer.

a 30mm

b 50mm

c 20cm

d 60cm

e 1m

f Nowaskafriendtousearulerortapemeasuretocheckyourestimates, andtowritethenewmeasurementoneachstreamer.

g Howclosewereyourestimates?

10A Individual APPLYYOURLEARNING

1 Writetheunityouwouldusetomeasureeachoftheseitems.Thenwritethe nameofthemeasuringinstrumentyouwouldusetomakethemeasurement.

a Thelengthoftheplayground

b Thelengthofyourshoe

c ThedistancefromAustraliatoNewZealand

d Thelengthofagrainofrice

2 Listthreeobjectsinyourclassroomthathavealengthbetween:

a 1metreand2.5metres

b 25centimetresand100centimetres

c 75millimetresand150millimetres

d 3.5metresand4metres

3 Writethelengthofeachpencilinmillimetres.

4 a Jennycuta70-centimetrepieceofstringfromalengthof10metres.How muchstringwasleft?

b ThenAlicutapieceofstring125centimetreslongfromthestringleftover afterJennycutherpiece.Howmuchoftheoriginalstringwasleft?

5 Tinaismakingaframeforaportraitshepaintedatschool.Sheneeds2piecesof timber240mminlengthand2piecesoftimber180mminlength.

a WhatisthetotallengthoftimberframeTinaneedsinmillimetres?

b IfTinacutthepiecesfroma1-metrelengthofframe,howmuchwouldbe leftover?

Weusedifferentunitstomeasurethelengthofdifferentkindsofobjects.Inmany cases,wecouldusetwodifferentunitstomeasurethesameobject,soitisimportant tobeabletochangefromoneunittoanother.

Metresandcentimetres

Weconvert metrestocentimetres bychangingeachmetreinto100centimetres.This isthesameasmultiplyingby100.

Solution

6 75m =(6 75 × 100) cm = 675cm

Weknowthat1metreisequalto100centimetres,sotoconvertfrom centimetresto metres wemake‘lots’of100centimetres.Forexample:

456cm = 400cm + 56cm = 4lotsof100cm + 56cm = 4m56cm = 4.56m

Thisisthesameasdividingby100.

456cm = 456 100 m = 4.56m

Example4

Convert675centimetrestometres.

Solution

675cm = 675 100 m = 6 75m

Metres,centimetresandmillimetres

Thereare100centimetresin1metreand1000millimetresin1metre,soweknow thatthereare10millimetresin1centimetre.

Toconvert fromcentimetrestomillimetres,wemake‘lots’of10millimetres. Thisisthesameasdividingby10.Theamountleftoveriswrittenasadecimalpart ofacentimetre.

Toconvertfrom centimetres to millimetres wemultiplyby10.

Example5

a Convert75millimetrestocentimetres.

b Convert105centimetrestomillimetres.

Solution

a 75mm = 75 10 cm = 7 5cm

b 105cm =(105 × 10) mm = 1050mm

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Toconvertfrom millimetrestometres,wemake‘lots’of1000millimetres.Thisisthe sameasdividingby1000.Theamountleftoveriswrittenasadecimalpartofametre.

Example6

Convert2350millimetrestometres.

Solution

2350mm = 2350 1000 m = 2.350mor2.35m

Metresandkilometres

Toconvert frommetrestokilometres,wemake‘lots’of1000metres.Thisisthesame asdividingby1000.Theamountleftoveriswrittenasadecimalpartofakilometre.

Example7

Convert6850metrestokilometres.

Solution

6850m = 6850 1000 km = 6.850kmor6.85km

Toconvert kilometres tometres,multiplyby1000.

Example8

Convert1.2kilometrestometres.

Solution

1.2km =(1.2 × 1000) m = 1200m

Toconvertfromoneunittoanotherwemultiplyordivideby10,100 or1000:

10B Wholeclass LEARNINGTOGETHER

1 a Useatapemeasuretomeasuretheheightofyourclassroomdoor incentimetres.

b Convertyourmeasurementtometres.Usethetapemeasuretocheck youranswer.

2 Measurethewidthofyourtableinmillimetres.Convertyourmeasurement tocentimetres.

3 Sylvia’sclassroomhas10tables.Eachtablemeasures125centimetresin width.ThetablesinSylvia’sclassroomareplacedsidebyside.Calculatetheir totalwidthin: millimetres a

10B Individual APPLYYOURLEARNING

1 Converteachmeasurementtometres.

Expressyouranswerasadecimalifitisnotawholenumber.

Rankthelengthsfromsmallesttolargest.

2 Converteachmeasurementtocentimetres.

Expressyouranswerasadecimalifitisnotawholenumber.

Rankthelengthsfromlargesttosmallest.

3 Converteachmeasurementtokilometres.

Expressyouranswerasadecimalifitisnotawholenumber.

Converteachmeasurementtomillimetres.

11cm a

3m b

0.2cm c

4.3m d

0.8cm e

0.2m f

Rankthelengthsfromsmallesttolargest.

5 Martinboughttwopiecesofwoodfromthetimberyard.Onepiecewas3m 65cmlong.Theotherpiecewas4m45cmlong.Ifbothpiecesofwoodwere placedendtoend,whatwouldtheirtotallengthbe?

6 Fredais1m47cmtall.Billis129cmtall.Findtheirheightdifferencein:

a metres

b centimetres

c millimetres

7 Kimboughtacablethatwas4.25metreslong.Shecut6piecesofcable 40centimetres inlengthfromit.Whatlengthofcablewasleft?

8 ThetopofHarry’shouseis11 45metreshigh.Thejacarandatreeinhisfrontyard is4m 70cmhigherthanthetopofhishouse.Howhighisthetree?

9 JennyiswalkingfromFishvilletoCodtown,adistanceof6.75kilometres.Shestill has1320metreslefttowalk.Howfarhasshewalkedalready?Giveyouranswer inkilometres.

10 Evancut3lengthsofdeckingmeasuring2.5metres,1.65metresand 895millimetres. Whatwasthetotallengthinmetres?

11 Winston’spaceis65centimetresinlength.Howmanymetreswouldhetravelin 15paces?

12 Lulucut20piecesofstringfromaballofstring.Eachpieceshecutwas 1 55metresinlength.Therearestill19metresofstringleftontheballofstring, sohowmuchdidLuluhavetobeginwith?

13 Brittanyplaced25pencilsinarow,endtoend.Eachpencilwas145millimetresin length.

a Whatisthelengthoftherowofpencilsinmillimetres?

b Howmuchshortof5metresisthis?

10C Perimeter

Theword‘perimeter’comesfromtwoGreekwords: peri,meaning‘around’and metron,meaning‘measure’.So‘perimeter’meansthemeasureordistancearound something.Itisthetotallengtharoundtheedge.

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Imaginewalkingaroundtheedgeofasoccerpitch.Youwouldwalk91metres, 46metres,91metresthen46metresagain.(Thesemeasurementsaretheminimum sizeforaninternationalsoccerpitch.)

Theperimeterofthesoccerpitchisthesumoftheselengths.

Perimeter = 91 + 46 + 91 + 46 = 274metres

Theperimeterofanystraight-sidedshapeisthesumofthelengthsofitssides.Shapes withthreeormorestraightsides,suchassquares,trianglesandhexagons,arealso knownas polygons

Theperimeterofthistriangleiscalculatedasfollows.

Perimeter = 7 + 9 + 11

27centimetres

Example9

Calculatetheperimeterofthisrectangle.

Solution

Theperimeteroftherectangleisthesumofthelengthofitssides.

Perimeter = 13 + 2 + 13 + 2 = 30m

Theperimeteroftherectangleis30metres.

Example10

Calculatetheperimeterofthisquadrilateral.

Solution

Theperimeterofthequadrilateralisthesumofthelengthsofitssides.

Perimeter = 14 + 11 + 8 + 9 = 42cm

Theperimeterofthequadrilateralis42centimetres.

Calculatetheperimeterofthispentagon.

8 mm

These marks are used with two-dimensional shapes to indicate sides of equal length.

Solution

Theperimeterofapentagonisthesumofthelengthsofitssides.

Perimeter = 8 + 8 + 8 + 8 + 8

= 5 × 8 = 40mm

Theperimeterofthepentagonis40millimetres.

Wholeclass LEARNINGTOGETHER

1 Workinpairstoestimatethenmeasuretheperimeterof:

a thecoverofthisbook

b thetopofyourdesk

c thedoorofaclassroomcupboard

d theclassroomdoor

Checkeachother’smeasurements.

10C Individual APPLYYOURLEARNING

1 Thesepolygonsaredrawnon 1-centimetregridpaper. Calculatetheperimeterof eachshape.

2 Calculatetheperimeterofeachpolygon.Allmeasurementsareincentimetres,so remembertoputcmaftereachanswer.Thepolygonsarenotdrawntoscale.

3 Youcancalculatetheperimeterofarectanglebydoublingthetwoside measurementsandaddingthemtogether.Calculatetheperimeterof theserectangles.

4 The and = marksontherectanglesbelowindicatewhichsidesareequalin length.Calculatetheperimeterofeachrectanglebyaddingthetwo measurementsgiven,thendoublingtheresult.

5 Copyandcompletethistable.Usethelengthandwidthofeachrectangleto calculateitsperimeter.

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6 Onlysomeofthemeasurementsaregivenforthesepolygons.Usethegiven measurementstoworkoutthemeasurementsyoudonotknow.Thencalculate theperimeterofeachpolygon.Allmeasurementsareinmetres.

10D Area

Theareaofarectangletellsus‘howlarge’theinsideofthatrectangleis.Itisthe amountofmaterialneededto‘cover’therectanglecompletely,withoutany gapsoroverlaps.

Areaismeasuredin squareunits:squaremillimetres,squarecentimetres,square metresandsquarekilometres.

A squaremillimetre isanareaequaltoasquarewithsidelength1millimetre.

1 mm

Thesymbolforsquaremillimetresismm2.(Thesmall‘2’means‘squared’.)

A squarecentimetre isanareaequaltoasquarewithsidelength1centimetre.

1 cm

Thesymboliscm2 .

A squaremetre isanareaequaltoasquarewithsidelength1metre.

Thesymbolism2 .

Australiahasthesixthlargestareaintheworld,afterRussia,Canada,China, USAandBrazil.

Australiahasanareaof7692024km2 .

Theformulaforcalculatingareaofarectangle

Tocalculatetheareaofapostagestamp,we couldcoveritwithasquaremillimetregrid andcounteachsquaremillimetre.Thiswould takeawhile,butwewouldeventually discoverthattheareaofthestampis 750mm2 .

Thereisaquickerwayoffindingtheareaof arectanglethancountinglittlesquares.We canfindtheareaofarectanglebyfindingthe productofitslengthandwidth.

Thisistheformulaforcalculatingtheareaofarectangle.Itworksforallrectangles.

Area = length × width

Thelengthandwidthmustbeinthesameunitofmeasurement.

Ifweusetheformula,theareaofthestampis:

Area = 25mm × 30mm = 750mm2

Example12

Calculatetheareaofarectanglethatmeasures16cmby8cm.

Solution

Area = length × width = 16cm × 8cm = 128cm2

Asquareisaspecialtypeofrectangle.Itswidthanditslengthareequal.So,thearea ofasquarecanbecalculatedbymodifyingtheformulafortheareaofarectangle.

Area = length × length = length2

Wereadthisas‘lengthsquared’.

Example13

Calculatetheareaofasquarewithsidelength9cm.

Solution

Area = length2 =(9cm)2 = 9cm × 9cm = 81cm2

10D Wholeclass LEARNINGTOGETHER

1 Workingroups.Youwillneed1-centimetregridpaper.Drawfourpairsof rectanglesthathavethesameareabutdifferentperimetres.Discussyour resultswiththeclass.

10D Individual APPLYYOURLEARNING

1 Theserectanglesaredrawnoncentimetregridpaper. Calculatetheareaofeachrectangle.

2 Findtheareaofasquarethathasasidelengthof:

3 Copythistable,thencalculatetheareaandperimeterofeachrectangle.

4 Calculatetheareaofeachrectangle.(Therectanglesarenotdrawntoscale.)

5 Whichisthecorrectareaforeachrectangle?

6 Findtheunknownside,thencalculatetheareaofeachrectangle.

10E Areaofcompositeshapes

Sometimesweneedtocalculatetheareaofashapethatis not arectangleorasquare. Ashapemadeupfromtwo(ormore)shapesiscalleda compositeshape

Tofindtheareaofacompositeshape,wecansplittheshapeintopieces,findthearea ofeachpiece,andthenaddtheareastogether.

Area = AreaA + AreaB

= 3cm × 4cm + 2cm × 3cm

= 12cm2 + 6cm2

= 18cm2

Anotherwayistofindtheareaofalargerrectangle,then‘takeaway’thebitthatis notincluded.

Area = AreaC AreaD

= 7cm × 3cm 3cm × 1cm = 21cm2 3cm2 = 18cm2

Example14

Acompositeshapehasbeensplitintwodifferentways.Calculatetheareaforeach.

Solution

a Area = AreaA + AreaB + AreaC

= 2cm × 1cm + 4cm × 1cm + 7cm × 2cm

= 2cm2 + 4cm2 + 14cm2

= 20cm2

b Area = AreaD + AreaE + AreaF

= 4cm × 2cm + 3cm × 2cm + 2cm × 3cm

= 8cm2 + 6cm2 + 6cm2

= 20cm2

10E Individual APPLYYOURLEARNING

1 Calculatetheareaofthesecompositeshapes.

2 a Calculatetheareaofthe blueregion.

b Tim’sloungeroommeasures 17m × 8m.Heboughta rectangularcarpetandplaced itonthemiddleofthelounge floor,leavinga1-metremargin allaround.

Calculatetheperimeterand areaofTim’snewcarpet.

2 Askippingropeis5.83metreslong.Howmanyskippingropescanbemadefroma ropeof length55metres?Howmuchisleftover?

3 Calculatetheperimeterofeachpolygon.

4 Copyandcompletethistable.Usethelengthsandwidthstocalculatethe perimeterandareaforeachrectangle.

5 Calculatetheareaandperimeterofeachshape.

10G Challenge–Ready,set,explore!

Victoria’snativetrees

BelowyouwillfindalistoftreeswhicharenativetoVictoriaalongwithsome importantinformationabouteachone:

Challengequestions

1 Listthetreesinorderfromsmallesttolargest.

2 Whatistherangeofheightsamongthetreeslisted?Calculatethedifference betweenthetallestandshortesttreeinmetres.

3 WhatisthedifferencebetweenyourheightandtheheightofaShiningGum?

4 Howmighttheheightsofthesetreesaffecttheirecosystem?Considerfactorslike sunlight,shade,andhabitatforwildlife.Whichtreesmightbemorebeneficialfor certainanimals?

5 Canyoufindsomethingintherealworldthatisapproximatelythesameheightas eachtree?Itmightbeabuilding,monument,orsomethingelse.Becreative!For example,aBlackwoodreaching25misapproximatelythesameheightasthe ShrineofRemembranceinMelbourne,Victoria,or5to6giraffesstandingontop ofeachother!

Usefulskillsforthistopic

• experienceinreadingscalesortapemeasures

• experiencewithmeasuringcontainers

• previousexperienceincomparingmassesofdifferentobjects

• knowledgeoftherelationshipbetweentonnes,kilograms,gramsandmilligrams

• theabilitytomultiplyanddividedecimalsby10,100and1000

Vocabulary

Cubiccentimetre

• Cubicmetre • Graduatedscales • Litres

• Millilitres • Kilolitre • Megalitre

• Tonnes •

• Grams • Milligrams • Kilograms

1 Drawanitemthathasavolumethatwouldbemeasuredin:

a millilitres

b litres

c kilolitres

2 Drawanitemthathasamassthatwouldbemeasuredin:

a grams

b kilograms c tonnes

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Inthischapter,wearegoingtoinvestigatemeasuringvolume,massandcapacity. Learningaboutthesemeasurementsisimportantas ithelpsusunderstandandmanageeveryday activitiessuchasbaking,shopping,science experimentsorpackingforatrip.

Volumeistheamountofspacesomethingtakesup.For example,whenyoupourjuiceintoaglass,theamountof juiceismeasuredbyvolume.Knowingthevolumehelps youunderstandhowmuchjuicethereistodrink. Knowingthevolumehelpsyouunderstandhowmuch liquidtheglasscanholdwithoutspilling.

Capacityissimilartovolumebutisoftenusedtodescribehowmuchacontainer canhold.Forexample,knowingthecapacityofaglasstellsyouwhetheritcan holdagivenamountofjuiceorifthejuicewillspill.

Massistheamountofmatterinanobject.Forexample, whenyousteponascale,itmeasuresyourmass.

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11A Volume

Volumeisameasurementoftheamountofspacesomethingtakesup. Itdoesnotmatterifyouaremeasuringthevolumeofasolidoraliquid,asboth measurementsusethesameideas.

Themainunitsformeasuringvolumeare:

• the cubiccentimetre (cm3)forsmallobjects

• the cubicmetre (m3)forlargeobjects.

Tocalculatethevolumeofarectangularprism,wemeasureitslength,width andheight.

Thisrectangularprismhaslength3cm,width5cmandheight2cm.Itdoesnotmatter whichmeasurementswecallthelength,widthandheight.Ifweturntheprism around,itsmeasurementsarethesame.

Thevolumeoftherectangularprismincubic centimetresisthenumberofcentimetrecubes requiredtomakeit.Abase-tenoneanda centicubearebothgoodexamplesofa centimetrecube.

Thisdiagramshowstherectangularprism(fromabove)madeupfromcentimetre cubes.Ithastwolayers.Eachlayerisshowninadifferentcolour.

Wecancountthecubestofinditsvolume.Eachlayercontains3cubesinitswidth and5cubesinitslength,making15cubes.Thereare2layersofcubes,sothevolume oftherectangularboxis30cm3 .

Sincetheareaofeachlayerislength × widthandthenumberoflayersistheheight, wehavecalculatedthevolumeaslength × width × height.

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Thevolumeofarectangularprismisgivenbytheformula:

Volume = length × width × height

Usingtheformula,youcanworkoutthevolumeoftherectangularprismlikethis.

Volume = length × width × height

= 3cm × 5cm × 2cm

= 30cm3

Weusethesymbolcm3 toshowthatvolumeiscalculatedbymultiplyingthethree dimensionsoflength,widthandheight.

Example1

Findthevolumeofarectangularprism8cmlong,3cmwideand2cmhigh.

Solution

Volume = length × width × height

= 8cm × 3cm × 2cm

= 48cm3

Thevolumeoftheprismis48cm3 .

Example2

Calculatethevolumeofthiscube.

Solution

Becauseallthedimensionsofacubearethesame,theformulabecomes:

Volume = length × length × length (orlength3)

= 6cm × 6cm × 6cm = 216cm3

Thevolumeofthecubeis216cm3 .

Wecanalsofindthevolumeofobjectsthatarenotprisms.

Example3

a Howmanycubesof1cm3 wereusedtomakethis staircase?

b Whatisthevolumeofthestaircase?

c Howmanymorecentimetrecubeswouldyouneedto makea4cm × 4cm × 4cmcube?

Solution

a Numberofcubesinlayer1 ∶ 4 × 4 = 16

layer2 ∶ 4 × 3 = 12

layer3 ∶ 4 × 2 = 8Totalnumberofcubes ∶ layer4 ∶ 4 × 1 = 416 + 12 + 8 + 4 = 40

b Thereare40cubes,sothevolumeis40cm3

c Tomakea4cm × 4cm × 4cmcube,youneed64centimetrecubes. Thenumberofcubesneededis64–40 = 24cubes.

Ifyouaregiventhevolumeofaprismandthemeasurementoftwoofitssides,you canworkoutthelengthofthethirdside.

Uncorrected 3rd sample

Example4

Thisrectangularprismhasavolumeof120cm3 . Finditswidth.

Solution

Remember:itdoesn’tmatterwhichofthemeasurementsyoucallthelength,width orheight.Inthiscase,thewidthistheunknownedge. Tofindthewidth,youneedtodividethevolume(120cm3)bythelengthandthe heightoftheprism.Theformulalookslikethis.

Width = volume length × height

So,forthisrectangularprism:

Width =

Theprismhasawidthof4cm.

1 Usecentimetrecubestobuildthisrectangularprism.

a Findthevolumeoftheprismbycountingthe numberofcubes.

b Usetheformulatocalculatethevolumeof theprism.

c Ifthewholeprismwaspaintedred,howmany 1cm × 1cmfaceswouldbered?

2 Usecentimetrecubestobuildthisrectangularprism.

a Finditsvolumebycountingthenumberof cubesused.

b Usetheformulatocalculatethevolumeof theprism.

c Ifthewholeprismwaspaintedblue,how many1cm × 1cmfaceswouldbeblue?

11A Individual APPLYYOURLEARNING

1 a Usecentimetrecubestobuildtheserectangularprisms.

b Foreachprism,countthenumberofblocksyouusedineachlayer,thenadd themtogethertofindthevolumeoftheprism.

c Listtheprismsinorderofvolume,smallesttolargest.

d Usetheformulatocalculatethevolumeofeachprism.

2 Findthevolumeofarectangularprismthathasthedimensions: 5cm × 4cm × 2cm a 4m × 2m × 3m b 3m × 4m × 3m c 10cm × 6cm × 6cm d 8m × 2 5m × 5m e 10m × 5m × 2 8m f

3 Julietcollectssmallrectangularboxesthatmeasure4cm × 3cm × 2cm.

Howmanyofthesesmallboxeswouldfitinsidealargerboxthatmeasures 60cm × 30cm × 24cm?

3rd

Calculatethevolumeofacubethathassidelength: 3cm a 10cm b 12cm c 17cm d 25cm e

5 Copythistable,thencalculatethevolumeofeachrectangularprism.

6 Frankiewantstodigaholesothathecaninstallaswimmingpool.Ifthepoolis 7metreslong,4metreswideand1.5metresdeep,whatvolumeofsoildoes Frankieneedtoremove?

7 Bahirawantstoconcreteherdriveway.Herdrivewayis12metreslongand 3metreswide.Iftheconcreteneedstobe0.2metresdeep,whatvolumeofsoil doesBahiraneed toremove?

8 Findthemissinglength,widthorheightoftheseprisms.

a V = 60m3 L = 5m, H = 4m, W =?

b V = 36cm3 L = 6cm, W = 3cm, H =?

c V = 36m3 W = 3m, H = 3m, L =?

9 TheTerrificTeaCompanyimportedaboxofteathatwas12cmlongand8cm wide.Afterdrinkingone-thirdofthetea,therewasstill576cm3 oftealeftinthe box.Calculatetheheightofthebox.

10 Tarabuiltacubewithsidelength7cm.Howmany more centimetrecubesdoes sheneedtomakeacubewithsidelength: 8cm? a 9cm? b 20cm? c 100cm? d

11 a Howmanycentimetrecubeshavebeenusedtomake thismodel?

b Whatisthevolumeofthemodel?

c Howmany more cubesareneededtomakeacubeof sidelength3cm?

12 a Howmanycentimetrecubeshavebeenusedtomake thismodel?

b Howmany more cubesareneededtomakea 4cm × 3cm × 2cmprism?

13 a Howmanycentimetrecubeshavebeenusedtomake thismodel?

b Howmany more cubesareneededtomakea 5cm × 4cm × 3cmprism?

14 Joeismakingacake.Hiscaketinis12cmlongand6cmwide.Whenitisfull,his caketincontains648cm3 ofcakemix.FindtheheightofJoe’scaketin.

11B Capacity

Wecanmeasureliquidsusingmeasuresofvolume,suchascubic centimetres.Thereisalsoanothermeasurecalled capacity that weonlyuseforliquidsandgases.

Theword‘capacity’isusedtodescribehowmuch liquidacontainercanhold.Ajugthatholds1litre hasacapacityof1litre,evenifitdoesnotactually haveanyliquidinit.Measuringjugsandcontainers havescalesontheirsidesthataremarkedwith lines.Theselinesarecalled calibrations or graduatedscales,andtheyallowustomeasure liquidsaccurately.

Millilitresandlitres

Weusemillilitres(mL)orlitres(L)tomeasurethevolumeofaliquid.

1000millilitres = 1litre

1millilitreofwaterfillsaspaceequaltoonecentimetrecubeandhasamassof1gram.

1millilitre = 1cm3 and1000cm3 = 1litre

Toconvert millilitrestolitres,wemake‘lots’of1000millilitres.Thisisthesameas dividingby1000.

3rd

Example5

Convert17000millilitrestolitres.

Solution

17000mL = 17000 1000 L = 17L

Toconvert litrestomillilitres,wemultiplyby1000.

Example6

Convert8litrestomillilitres.

Solution

8L =(8 × 1000) mL = 8000mL

Kilolitresandmegalitres

Onecubicmetreofwaterisknownasa kilolitre andisequivalentto1000litres.

1000litres = 1kilolitreand1kilolitre = 1m3

Thereareapproximately750kilolitresofwaterina50-metreswimmingpool.In2005, theaverageAustralianused154kilolitresofwater.

Toconvertlitres tokilolitres,wemake‘lots’of1000litres.Thisisthesameasdividing by1000.

Example7

The25-metrepoolatHendersonSecondaryCollegeholds375000litresofwater. Whatisthisinkilolitres?

Toconvertfrom kilolitrestolitres,wemultiplyby1000.

Example8

Simiused982kilolitresofwaterlastyear.Convertthistolitres.

Solution

982kL =(982 × 1000) L = 982000L

Aunitformeasuringextremelylargequantitiesofliquidsisthe megalitre.Amegalitre is1000000litres.Oftenyouwillseethecapacityofwaterstoragedamsmeasuredin megalitres.

11B

Wholeclass LEARNINGTOGETHER

1 Fillameasuringjugtothe100mLmark.Putin25centicubes(orplastic base-tenones).Whatisthewaterlevelnow?

Nowaddanother25centicubestothejug.Howmuchdidthewaterrise?

Discussyouranswerwiththeclassandseeifyoucanexplainwhathappened.

2 Estimatethecapacityinmillilitresof:

a ateacup

b ayoghurtcontainer

c asmallfoodcontainer

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Useameasuringjugtocheckyourestimate.Howaccuratewereyou?

11B Individual APPLYYOURLEARNING

1 Convertthesemeasurementsinlitrestomillilitresbymultiplyingby1000. 7litres a 12litres b 342litres c 1000litres d

2 Convertthesemeasurementsinmillilitrestolitresbydividingby1000.

a 1000millilitres

b 13000millilitres

c 3420millilitres

3 Convertthesemeasurementsinlitrestokilolitresbydividingby1000. 4000litres a 18000litres b 39870litres c

4 Convertthesemeasurementsinkilolitrestomillilitresbymultiplyingby 1000000.(Thatis,multiplyby1000andthenmultiplyby1000again.)

a 4kilolitres

b 23kilolitres

c 815kilolitres

5 Ordertheamountsbelowfromsmallesttolargest.

a 3000milliltres

b 2kilolitres

c 1500litres

6 Rosie’sshedmeasures3m × 8m.Whenitrains,4mmofrainfallsontheshed roofeachhour.Whatisthetotalamountofwatercollected: in3hours? a in12hours? b

7 Sharonboughtarectangularwatertankmeasuring8m × 9m × 300mm.

a HowmanylitresofwatercanSharon’stankhold?

b Sharoncollectswaterfromaroofmeasuring6m × 2m.Ifrainfallsat5mm perhour,howlongwillittaketofillhertank?

8 TheMcGovernfamilyhasaspapoolthatcanhold1kilolitreofwater.They alsohavearainwatertankthatholds750litresandabucketthatholds 10000millilitres.

a Howmanybucketsofwaterwouldyouneedtofillthespapool?

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b Howmanytankswouldyouneedtofillthespapool?

c Explainhowyouconvertedtheunitstofindyouranswers.

11C Mass

Theunitsofmeasurementweuseformeasuringmassare milligrams, grams, kilograms and tonnes.Thebasicunitformeasuringmassisthekilogram,whichis abbreviated kg.

Eachunitformassisrelatedtotheotherunits.Asthisdiagramshows,wecanconvert fromoneunittoanotherbymultiplyingordividing.

Theprefix‘kilo’means1000.Ifyoucombine‘kilo’and‘gram’ittellsyouthatthereare 1000gramsin1kilogram.Theletter g isusedasanabbreviationforgrams.

1000grams (g)= 1kilogram (kg)

Ifwehave 4kilograms,wecanworkouthowmanygramswehave.

1kg = 1000g

So4kg = 4 × 1000g = 4000g

Thereare1000kilogramsin1tonne.Theletter t isusedasanabbreviationfortonnes. Ifwehave3000kilograms,wecanworkouthowmanytonneswehave.

1000kilogramsisthesameas1tonne.

So3000kg = (3000 ÷ 1000) t = 3t

Example9

Abagofpotatoesweighs3 5kg.Howmanygramsisthat?

Solution

Weneedto multiply thekilogramsby1000tofindthenumberofgrams.

3.5 × 1000 = 3500

Thereare3500gramsofpotatoesin3.5kilograms.

Example10

Patsybought250gramsofbeads.Howmanykilogramsofbeadsdidshebuy?

Solution

Weneedto divide thenumberofgramsby1000toconvertittokilograms.

250 1000 = 0.25

Patsybought0.25kgofbeads.

Example11

John’scarhasamassof1.35t.Howmanykilogramsisthat?

Solution

Toconverttonnestokilograms,weneedto multiply by1000:

1t = 1000kg

So, 1 35t = 1 35 × 1000kg = 1350kg

John’scarhasamassof1350kilograms.

Weneedasmallerunitthangramstomeasurethemassofverysmallobjects.

Aletterinanenvelopeweighsabout5grams.Ifwewanttomeasurethemassofthe postagestampontheenvelope,weneedtouseasmallerunitcalledmilligrams.

Milligramsareusedformeasuringthemassofobjectsthatareverylight,suchasa stamp,arosepetaloraleaf.

Theprefix‘milli’meansone-thousandth.Thereare1000milligramsin1gram. Milligramsareabbreviated mg.

1000milligrams (mg)= 1gram (g)

1milligram (mg)= 1 1000 ofagram(g)

Youhaveprobablyheardofsmallanimalscalledmillipedes.Despitetheirname, millipedeshaveabout60legs,not1000legs.Longago,peoplethoughttheselittle animalslookedasiftheyhadatleast1000legs,andthatishowtheygottheirname.

Example12

Acanoftomatoescontains40mgofsalt.Convertthismeasurementtograms.

Solution

Toconvertmilligramstograms,weneedto divide by1000:

1000mg = 1g

So, 40mg =(40 ÷ 1000) g = 0.04g

Example13

Apharmacistweighsout1 125gofpowderforeachcapsule.Howmanymilligrams ofpowderdoes heuse?

Solution

Toconvertgramstomilligrams,weneedto multiply by1000.

1g = 1000mg

So, 1.125g = 1.125 × 1000mg = 1125mg

11C Wholeclass LEARNINGTOGETHER

1 Youwillneedbathroomscalesorkitchenscales.Selectfivedifferentobjects fromaroundtheroom,andusethescalestomeasurethemassofeachobject. Thendrawupatableandconvertthemeasurementofeachobjectinto milligrams,grams,kilogramsandtonnes.

2 Astandardchickeneggweighs70grams.Calculatethe massofthesecharactersineggs.

a Billythebabybilbyweighs280grams.

b Wallytheweightlifterweighs175kilograms.

c Carltheclownfishweighs35000milligrams.

11C Individual APPLYYOURLEARNING

1a Jessicabought350gofgrapes.Howmanykilogramsofgrapesdid shebuy?

b Jamesbought4kilogramsofapples.Howmanygramsofapplesdid hebuy?

c Jordanbought1 5tonnesofgrapes.Howmanykilogramsofgrapes didhebuy?

2 a Howmanytonnesaretherein3875kg?

b Howmanykilogramsaretherein1.074t?

c Howmanykilograms aretherein2855g?

d Howmanygramsaretherein0.045kg?

e Howmanygrams aretherein4455mg?

f Howmanymilligramsaretherein4 072g?

3 Converteachmassfromtonnestokilograms.

4 Converteachmassfromgramstokilograms.

5 Converteachmassfromgramstomilligrams.

6 a Carly’sgrocerybagweighs5.75kg.How manygramsisthat?

b Paulloadedhistruckwith3.04tof furniture.Howmanykilogramsdid heload?

c Annabellebought805gofapples.Write thisinkilograms.

d WinnieWitchused4.25gofpowderedbarkinherpotion.Howmany milligramsofpowderedbarkdidsheuse?

e Rebeccaneeds1 35kgofflourtomakesomecakes.Howmanygramsof flourdoesshe need?

f Apacketofpotatochipscontained1256mgofsalt.Howmanygramsof saltwasthat?

11D Calculatingdifferent masses

Whenwewanttofindthetotalmassofanumberofobjects,weneedtoconvertthem alltothesameunitbeforewecanaddtofindthetotal.

Also,ifweneedtofindthedifferencebetweentwomassesthataremeasuredin differentunits,weneedtoconvertthemassestothesameunitbeforewecansubtract.

Example14

Whatisthetotalmassinkilogramsof3 5kgofpotatoesand875goftomatoes?

Solution

Convertthemassofthetomatoestokilograms,thenaddthemassestogether.

Potatoes3 5kg = 3 5kg

Tomatoes875g = 0.875kg

Total = 4 375kg

11D Wholeclass LEARNINGTOGETHER

1 Lookatsupermarketcataloguestofinditemsthathavetheirmassinkilograms anditemsthathavetheirmassingrams.Selectthreedifferentitemsandfind thetotalmass.Convertitemstothesameunitofmassbeforeadding.

2 Labelsonpackagedfoodsshowthemassoftheproductinside.Thisis calledthe netweight.Ifweaddthemassofthepackagingwehavethe grossweight.Youcanfindoutthemassofthepackagingbysubtracting thenetweightfromthegrossweight.Usethisinformationtosolvethe problembelow.

3rd

Grandmamade12identicaljarsofjamandpostedthemtoNorahforher birthday.Thegrossweightoftheparcelwas4.8kilograms.Eachjarandits lidhadamassof150grams.Grandmaused180gramsofcardboard,bubble wrapandstickytapeforpackaging.Calculatethenetweightingramsofthe jamineachjar.

Individual APPLYYOURLEARNING

1 Herearesomegroceryitemswiththeirweights.

Whatisthetotalmassof:

a theboxoforangesandthetinoffruit?

b thepacketofchipsandtheboxofcereal?

c thecarrotsandthewatermelon?

d theboxoforanges,thepacketofchipsandthecarrots?

e alloftheitems?

2 Copythistableandwritethemissingvalues.

3 Onesmalltinofspaghettihasanetweightof125g.Howmanysmalltinsshould Stellabuyifsheneedsatotalmassof:

a 500gofspaghetti?

b 1.25kgofspaghetti?

4 Alargetincontains425goftuna.Helenneeds1.7kgoftuna.Howmanylarge tinsoftuna doessheneedtobuy?

5 Onetable-tennisballweighs450mg.Whatwouldbethemassingramsofsix table-tennisballs?

6 Thenetweightofabagofpolystyreneballsis5g.Ifeachballweighs500mg, howmanyballsareinthebag?

1 Copythistable.Calculatethevolumeofeachrectangularprism.

2 Findthemissinglength,widthorheightoftheseprisms.

a V = 35m3 , L = 5m, H = 1m, W =? cm

b V = 48cm3 , L = 2cm, W = 6cm, H =? cm

c V = 108m3 , W = 4m, H = 9m, L =? m

3 Convertthesemeasurementsinlitrestomillilitresbymultiplyingby1000.

a 4litres

b 19litres

c 603litres

d 4294litres

4 Convertthesemeasurementsinmillilitrestolitresbydividingby1000.

a 5000millilitres

b 78000millilitres

c 1003millilitres

d 789millilitres

5 Convertthesemeasurementsinlitrestokilolitresbydividingby1000.

a 6000litres

b 46000litres

c 88000litres

d 1111001litres

6a Howmanykilogramsaretherein3395g?

b Convert6.09345gtokilograms.

c Howmanygrams aretherein3.02kg?

d Howmanygrams aretherein0 003kg?

e Howmanymilligrams aretherein32.09g?

f Howmanygrams aretherein12005mg?

7 Convertthesemeasurementstokilograms.

a 2.29t

b 4000000g

c 80000mg

8 Tinabought3.5kgofapples,2.08kgofpears,200gofspicesand850gofcoffee fromher localshop.WhatwasthetotalweightofTina’spurchasesinkilograms?

9 Findthemissinglength,widthorheightoftheseprisms.

a V = 100cm3 L = 8cm, H = 2.5cm, W =?

b V = 84m3 L = 12m, W = 3.5m, H =?

c V = 72cm3 W = 8cm, H = 6cm, L =?

11F Challenge–Ready,set,explore!

Measure,pour,stack,solve

1 ThevillageofSeccois sufferingabaddrought.Each personisallowedtotake home4litresofwaterperday fromthevillagewell.Guido hasonlytwobuckets,a3litre oneanda5litreone.Hetakes themtothewell.

Howcanhemeasureout exactly4litresofwaterusing histwobuckets?

2 TheLemonGroveSchoolholdsafete.Attheendofthefete,thereare24litresof lemonadeleftoverinacontainer.Threefamilieswinaprize:theycantakehome 8litresoflemonadeeach.Butthereareonlythreebucketsonhandtomeasureit out:13litre,5litreand11litrebuckets.Thelemonadecanonlybetakenhomein thebucketsandtheoriginalcontainer.

Howcantheytakehome8litreseach?

3 BabyMelissahastwowoodenblocks,eachmeasuring12cmx6cmx3cm.She canstandthemonanyside,andshecanplaceablocknicelybalancedontopof another.

a Howmanydifferentheights canshemakeusingoneor bothofherblocks?

b Howmanydifferentheights canMelissamakeifshehas threeofthewoodenblocks?

c Howmanydifferentheights canMelissamakeifshehas fourofthewoodenblocks?

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4 Pippahasboughtalargefishtank.Sheisconsideringhowtoestimateitscapacity tohelpensuretherightenvironmentforherfish.CanyoucreateawayforPippa toestimatethecapacityofherfishtank?

CHAPTER

Usefulskillsforthistopic

• readingandrecordingtimeusingbothdigitalandanalogueclocks

• convertingtimebetweena.m.andp.m.and24-hourtime

• understandingtimeduration

Vocabulary

Antemeridiem

• Postmeridiem • Elapsedtime • Itinerary • 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 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

Howlongdoesittakeyoutogettoschooleachday?Writethetimeyouusuallyleave homeandthetimeyouusuallyarriveatschool.Countontofindtheelapsedtime. Calculatehowmuchtimeyouspendtravellingtoandfromschooleachday. Howlongwouldthatbeover1week?

Ifyouareatschoolfor40weeksoftheyear,approximatelyhowmuchtimedoyou spendtravellingtoandfromschoolinoneyear?

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Inthischapter,wearegoingtoinvestigatereadingtime,recordingtimeand planningforevents.

Beingabletoreadandwritetimeisimportantformanagingourdailyactivities. Timehelpsusorganisewhatwedo.Whetheritiswakingupinthemorning, goingtoschool,meetingafriendorplanningafunactivity,knowingthetime helpsustostayontrack.

Whenwelearntoreadtimeindifferentformats,wecanbetterhandledifferent activities,suchascatchingabus,goingtothemoviesorarrivingatschoolatthe correcttime.

12A Readingand recordingtime

Thebasicunitofmeasurementfortimeisthe second

• Thereare60secondsin1minute.

• Thereare60minutesin1hour.

• Thereare24hoursin1day.

• Thereare7daysin1week.

Therearetwowaysofrecordingthetimeofday.Youcanuseeithera12-hourclockor a24-hourclock.

Usinga12-hourclock

Whenweusea12-hourclock,thedayisbrokenupintotwoblocksof12hours. Atimesuchas11:30a.m.meanshalf-pasteleveninthemorning(eleven-thirty),and 11:30p.m.meanshalf-pastelevenintheevening.

Weusetheabbreviation a.m. toshowthatwemeanthemorning.Theletters‘am’ comefromtheLatin antemeridiem,whichmeans‘beforenoon’.

Weusetheabbreviation p.m. toshowthatwemeantheafternoonorevening. Theletters‘pm’comefromtheLatin postmeridiem,whichmeans‘afternoon’.

Onadigitalclock,midnightisshownas12:00a.m.andmidday(12noon)isshown as12:00a.m.

Usinga24-hourclock

Whenweusea24hourclock,thedayismeasuredinoneblockof24hours.Alltimes aremeasuredfrommidnightononedayuntilmidnightthenextday.So11:30a.m.is writtenas1130and11:30p.m.iswrittenas2330.

Itisincorrecttousea.m.orp.m.with24-hourtimes.Midnightiswrittenas0000 (orsometimes2400)andmiddayiswrittenas1200.

Addingtime

JimwentaroundtohisfriendAman’splaceafterschool.IttookJim35minutestoride hisbiketoAman’s.JimandAmanwatchedamoviefor1hourand30minutes.Tofind outhowlongJimhasbeenawayfromhome,weneedtoaddthetwotimeperiods.

Ridinghisbike 35minutes

+Watchingamovie1hour30minutes

Totaltimeaway1hour65minutesor2hoursand5minutes

Thereare60minutesinonehour,soweconvert65minutesto1hour5minutes. ThetotalamountoftimethatJimhasbeenawayfromhomeis2hoursand5minutes.

Example1

Justinewasdoingadailytimedmathstest.OnMondayittookher2minutesand 50secondstocompletethetest.OnTuesdayittookher2minutesand40seconds. HowlongdidJustinespendonthetestsoverthetwodays?

Solution

Monday2minutes50seconds

Tuesday2minutes40seconds 4minutes90seconds

Totaltime = 4minutes + 1minute + 30seconds = 5minutes + 30seconds

12A Wholeclass LEARNINGTOGETHER

1 Drawa24-hourtimelineshowingtheamountoftimeyouspendsleeping, working,playing,eatingandtravellinginanormalschoolday.

a Howmuchtimedoyouspendeatingandsleepingeachday?

b Howmuchtimedoyouspendtravellingandworkingeachday?

c Howmuchtimedoyouspendeating,playingandtravellingeachday?

12A Individual APPLYYOURLEARNING

1 Caitlinistrainingforatriathlon.OnMondayshetrainedfor1hourand 25minutes.OnTuesdayshetrainedfor2hoursand10minutes.OnWednesday shetrainedfor1hourand40minutes.OnThursdayshetrainedfor2hoursand 5minutes.OnFridayshetrainedfor1hourand55minutes.Whatisthetotal amountoftimeCaitlinspenttraining?

2 Lachlanpractiseshisguitareverymorningandeveryevening.Eachdayherecords howlonghepractised.CalculatethetotaltimethatLachlanpractisedeachday (a to g).

a Monday:2minutes30seconds + 1minute15seconds

b Tuesday:4minutes45seconds + 3minutes25seconds

c Wednesday:5minutes27seconds + 2minutes45seconds

d Thursday:2hours25minutes + 3hours10minutes

e Friday:1hour30minutes + 2hours55minutes

f Saturday:4hours23minutes + 2hours55minutes

g Sunday:5hours8minutes + 3hours27minutes

3 Londonis11hoursbehindMelbourneduringdaylightsaving.Calculatethetime inLondonwhenitisthefollowingdaylightsavingtimesinMelbourne.

a 9p.m.Monday

b 5:45p.m.Tuesday

c 1109Saturday

d 7:24a.m.Sunday

12B

Elapsedtime

Timeplanning

Sometimesweneedtocalculatehowlongitisbetweentwotimes.Thisiscalled calculating elapsedtime.Theword‘elapsed’means‘goneby’,soelapsedtimemeans theamountoftimethathasgoneby,orpassed.

Wecalculateelapsedtimebybuildinguptowholeminutes,hoursordays,keeping trackoftheamountsoftimeaswego.Forexample,ifHelen’strainleftMaryborough at2:55p.m.andarrivedatDaisyHillat3:45p.m.,howlongdidittake?Wecanwork outtheelapsedtimebybuildingupfrom2:55to3:00.Thenthereare45minutes more.Helen’straintriptook50minutes.

Example2

Ranistartedherhomeworkat4∶45p.m.andfinisheditat6∶10p.m.Howlongdid shespenddoingherhomework?

Multi-eventplanning

Whenplanningforthetimeelapsedbetweentwoevents,therecanbeanumberof factorstoconsider.Forexample,whenplanningatripyoumayhavetonavigate multiplemodesoftransport.Itisimportanttoknoweachofthetimetablesandensure thereissufficienttimebetweenthemodesoftransporttomakesureallconnections aremet.

Sometimesyoumayneedtoworkforwards,andothertimesitmaybenecessaryto workbackwards.Forexample,ifyouwanttofindoutwhattimeyouwillarriveafter leavinghomeinthemorning,youwouldplanforwards.Alternately,ifyouneedtobe somewhereatacertaintime,itisbettertoworkbackwardsfromthattimetodecide whentosetoff.

Example3

Lilyisdrivingtothebeach.Sheleaveshomeat9:00a.m.Thedrivewilltake2hours and30minutes,andshewantstostopfora20minutetoiletbreakalongtheway. WhattimewillLilytellherfriendtomeetheratthebeach?

Solution

Lilyisleavinghomeat9:00a.m.soitisbesttostartbyworkingforwards fromhere.

9:00a.m.plus2hoursand30minutesdrivetime = 11:30a.m. 11:30a.m.plus20minutebreak = 11:50a.m.

Lilyshouldtellherfriendtomeetheraround11:50a.m.

Example4

Liamhassoccerpracticeafterschool,whichstartsat4:15p.m.Heneeds30minutes togetchangedandpreparedbeforepracticestarts.Ittakeshim15minutestowalk tothetraininggroundfromschool.WhattimeshouldLiamleaveschooltobeready forpracticeontime?

Solution

Liamneedstobereadytostartpracticeat4:15p.m.soitisbesttostartbyworking backwardsfromhere.

4:15p.m.subtract30minuteschangetime = 3:45p.m. 3:45p.m.subtract15minuteswalktime = 3:30p.m. Liamshouldleaveschoolat3:30p.m.inordertobereadyforpracticeontime.

12B Wholeclass LEARNINGTOGETHER

1 Thefollowingtimesareallonthesameday.Workwithapartnertocalculate howmuchtimehaselapsedbetween:

a 11∶33a.m.and3∶45p.m.?

b 1535and1755?

2 Thefollowingtimesareondifferentdays.Workwithapartnertocalculate howmuchtimehaselapsedbetween:

a 6∶35a.m.Mondayand11∶25a.m.Thursday?

b 1047Wednesdayand1326Saturday?

3 Davidiscyclingtothepark.Heleaveshishouseat2:00p.m.Thecyclingtrip takes1hourand15minutes,andheplanstotakea10-minuteresthalfway through.WhattimewillDavidreachthepark?

4 Danielisgoingtowatchamoviethatstartsat7:45p.m.Heneeds20minutes tobuysnacksandfindhisseat.Ittakeshim15minutestodrivetothecinema. WhattimeshouldDanielleavehishousetomakesurehe’sreadytowatch themovieontime?

12B Individual APPLYYOURLEARNING

1 TheTerm1timetableforMsChantry’sYear6classisshownbelow.

a Atwhattimedothestudentsstartschool?

b Howlongistheschoolday?

c HowmanyhoursperweekarespentdoingMathematics?

d Whichtopictakesthemosthoursintheschoolweek?

e Whatisthetotaltimespentatlunchandrecess,inhoursandminutes?

f Whatisthetotaltimespentinclass,assemblyandthelibraryatschool, inminutes?...inseconds?

8:45 Assembly Roll/Notes/ Messages Roll/Notes/ Messages Roll/Notes/ Messages Roll/Notes/ Messages

9:00 Mathematics English–Reading English–Reading Mathematics Integrated Sci/Geog/ Hist

10:00 English–Reading Mathematics English–Writing English–Speaking& Listening Mathematics

11:00 Recess

11:30 English–Writing Library Mathematics English–Writing English–Reading

12:30 English–Writing TheArts Mathematics Integrated Sci/Geog/ Hist English–Writing

1:30 Lunch

2:15 Language Physical Education Integrated Sci/Geog/ Hist TheArts Integrated Sci/Geog/ Hist

3:30 School finishes

2 Brooketook3hoursand25minutestocompletethemarathon.Ifshefinishedat 1145,whattimedidshestart?

3 ThisisaplannedtraintimetablefortheCentraltoCaulfieldline.Onetrainleaves Centralevery10minutesanditis4minutesbetweenstops.Copythetimetable andcompleteit,showingthearrivaltimesoftrainsateachdestination.

4 Ethanstartshikingat7:00a.m.Thehiketakeshim3hoursand30minutes,and hestopsfora15-minutebreakhalfway.WhattimewillEthanfinishthehike?

5 MaiLinhastocatchthe1645flighttoPerth.Sheneedstocheckinattheairport 60minutesbeforetheflight.MaiLinlives1hourand15minutesfromtheairport. WhattimewillthetaxineedtopickMaiLinup?

6 Frankhastoexercisefor35minutesbeforeschool.Ittakeshim45minutestoget readyforschooland25minutestogettothere.Whattimeshouldhegetupifhe wantstobeatschoolat0835?

12C Reviewquestions–Demonstrateyourmastery

1 Murrayisstudyingforhisfirst-aidcertificate.

OnMondayhewenttohisfirst-aidcoursefor3hoursand40minutes.

OnTuesdayhestudiedthefirst-aidmanualfor1hourand15minutes.

OnWednesdayhestudiedthemanualfor1hourand48minutes.

OnThursdayhewenttothecoursefor3hoursand5minutes.

OnFridayhetookhisfirst-aidexam.Itwentfor2hoursand57minutes. WhatwasthetotalamountoftimeMurrayspentgettinghisfirst-aidcertificate?

2 Howmuchtimehaselapsedbetween:

a 4:05a.m.and7:35a.m.onthesameday?

b 7:55p.m.and8:03p.m.onthesameday?

c 10:08p.m.and3:45a.m.thenextday?

d 6:28a.m.and3:45p.m.thatafternoon?

e 0545and1355thesameday?

f 1256and1326thenextday?

a HowlongdoesittaketotravelfromRockfordtoBlainey?

b SamlivesatRockford.SheneedstobeatWinstonat2:45p.mWhichtrainwill sheneedtocatch?

c HowlongwillSam’straintriptake?

4 Theschoolbusleavesat3:40p.m.Itstopsatthefirstbusstop15minuteslaterfor 2minutes.Thenittravelsforafurther8minutestothesecondstop.Whattimeis itwhenitgetstothesecondstop?

5 Rachelneedstocatchatrainthatdepartsat10:05a.m.Shemustarriveatthe station10minutesbeforedeparturetobuyaticketandfindherplatform.Thetrain stationis12minutesawayfromherhouse.WhattimeshouldRachelleaveher housetocatchthetrain?

12D Challenge–

Planyourdreamholiday

Resourcesrequired:accesstoacomputer

Youareplanningadreamtripforaweekaround Australiawithyourfamily.Choosethreedestinations youwouldliketovisit.Yourchallengeistocreatean itinerarythatincludesthefollowingelements.

Itinerary: Plananoverviewofthetripfromstarttofinish.Whatdatewillyoudepart andthenarrivehome?Wherewillyouplantospendeachday?

FlightSchedule: Decideonflightsandrecordwhenyourplanewilldepartandarrival timesateachdestination.Recordtheflightdurationsforeachflight.Makeaplanfor gettingfromeachairporttothecitycentre.

Activities: Foreachdestination,planatleastonedayofdailyactivities.Estimatehow longeachactivitywilltakeandusea24-hourtimeformatforeachactivity’sstartand endtimes.

Belowisanexampleofflightsandactivities.

Destination1:Sydney,Australia

Flight:DepartfromMelbourneat1000,arrivein Sydneyat1130(1.5-hourflight).35minutetrain departing1220fromtheairportandarriving 1255citycentre.

Activities:

• VisitSydneyOperaHousefrom1300to1500

• ClimbtheSydneyHarbourBridgefrom1530to1730

• EveningwalkatBondiBeachfrom1830to 2030

Destination2:Brisbane

Flight:DepartfromSydneyat0900,arrivein Brisbaneat1030(1.5hourflight).20minutetrain departing1108fromtheairportandarriving1128 citycentre.

Activities:

• VisittheLonePineKoalaSanctuaryfrom1230to1330

• ExploretheQueenslandMuseumfrom1415to1600

• StrollalongtheSouthBankParklandsfrom1745to1900

Destination3:Perth

Flight:DepartfromBrisbaneat1300,arriveinPerth at1600(3-hourflight,2-hourtimedifference). 35minutetraindeparting1630fromtheairportand arriving1705citycentre.

Activities:

• VisitKingsParkandBotanicGardenfrom1720to1900

• VisittheWesternAustralianMuseumfrom0800to1000

• ExploretheFremantleMarketsfrom1145to1500

3rd

Reflection:

Afterplanningyourtrip,recordyourthinking.

• Howdidyoucalculatethetotaltraveltimebetweendestinations?

• Howdidyoumanagethedifferenttimezoneswhenplanningyourflights?

• Howdidyoudecidehowlongeachactivityneeded?

• Howcantheskillsyouusedinthistaskbeusefulinreallife?

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12E Mathematicsin FirstNationscontexts

SonglinesDreaming

Whenyoulookatthosesortsofstories,youseetheconnectivitybetweenallofthe elements,betweenthesky,betweentheEarth,betweenthewater,between magnificentsacredsitesthatareinthelandscapethatconnectourpeoplethroughthis ancientwisdomandtheseancientstoriesinsong.

–DrAnnePoelina,NyikinaWarrwa

Kungkarangkalpa:The‘SevenSisters’Songline

The‘SevenSisters’Songline’tellsthesagaofanendlessjourneymadebyagroupof femaleAncestralbeings.Theyarepursuedbyapowerfulmythologicalfigurewhois, byturns,unpredictable,dangerous,driven,thwarted,desperate,andtricky.The Hunter,WatiNyiru,anAncestralshape-shifterorsorcerer,takesonmanyguisestotry totricktheSevenSistersastheytravelacrosstheland.

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ThisSonglinealsoservesasanavigationaltool,mappingthestarsofthePleiadesand connectinglandmarksacrossthelandscape.Thejourneyismarkedbyspecific landmarksthatguidethepeopleinnavigation.Thesistersarepursuedbyahunter acrossavastlandscape,andtheymustmanagetheirresources,plantheirroute,and usesacredlandmarkstoescape.

Activity1:DistanceGainedbytheHunter

TheSevenSistersandtheHunterbothtravelatdifferentspeeds.Thesisterstravelat 5kilometresperhour,whilethehuntermovesfasterat6kilometresperhour.They travelfor4hourswithoutstopping.After4hoursoftravel,howmanykilometreswill thehunterhavegainedonthesisters?

Activity2:JourneyofCreation

Kungkarrangkalpa(SevenSistersDreaming),2011,JudithYinyikaChambers,acryliconcanvas,763x1525x33mm. DonatedthroughtheAustralianGovernment’sCulturalGiftsProgrambyWayneandVickiMcGeoch.NationalMuseumofAustralia

Aspartofthedreamingstory,theSevenSistersembarkonasacredjourneyto escapetherelentlessadvancesoftheHunter.Theyaretravellingtoasacredcave 120kilometresaway,creatingrivers,mountains,andvalleysalongtheirpathtoelude him.Theirjourneyisguidedbythesonglines,andtheirspeedchangesbasedonthe typeofterraintheytraverseandcreate:

• Riverbeds:6km/hforupto5hoursaday.

• Mountains:2km/hforupto3hoursaday.

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• Valleys:4km/hforupto2hoursaday.

Eachday,theymustrestfor2hourstosingthelandintobeingandensuretheirpath confoundstheHunter.Theirtotaltraveltimeislimitedto10hoursperday,including rest.HowmanydayswillittaketheSevenSisterstoreachthesacredcave?They won’tneedtheir2hourrestattheendoftheirjourney.

Activity3:EscapingtheHunteratSacredSites

TheSevenSisterstravelthroughasacredandever-changinglandtoescapethe Hunter.Passingthroughrockholes,sandhills,creeks,caves,andspinifexlandscapes, theydrawstrengthfromthedesertwindsandthelivingland,allowingthemtorestfor only30minutesevery10kilometres,whilststilltravellingat5km/h.TheHunter, unabletoconnectwiththesacredforcesoftheland,walksfasterat6km/hbut requires1hourofrestevery10kilometrestorecoverhisstrength.IfboththeSisters andtheHuntertravel50kilometres,howmuchtimedotheSisterssavecomparedto theHunter?Expressyouranswerinminutes.

Activity4:UsingtheStarstoNavigate

Atnight,theSevenSistersusethestarsforguidance,followingthePleiades(their celestialrepresentation)inthesky.ThePleiadesisvisiblefor6hourseachnight,during whichtheytravelatasteadypaceof4km/h.However,astheterrainchanges,they mustslowdownto1km/hfor15minuteseachhourtonavigateobstacleslikegullies andcreeks.Additionally,everythirdnight,athickmistpartiallyobscuresthestarsfor thelast2hoursofvisibility,reducingtheirspeedto2km/hduringthisperiod.Howfar cantheSevenSisterstravelin6nightsundertheseconditions?Expressyouranswerin kilometres.

Activity5:FindingWaterintheLandscape

TheSevenSistersareembarkingonajourneyacrossadeserttoreachahiddenwater source30kilometresaway.Theybeginatawatersourceat0km,andthereare additionalwaterholesevery15kilometres.Theycancarry10kilometre’sworthof wateratatime.Tocompletetheirjourney,theymustuseastep-by-stepcaching strategy (seenotebelow) tostorewateratintermediatepointsthatensurestheynever runoutofwaterandalwayshaveenoughtoreturntoacacheorwatersourcesafely. Howcantheyaccomplishthis,andwhatisthetotaldistancetheymustwalkto succeed?

Note: Acachingstrategyisleavingsupplesofwateratdifferentpointsalong thejourney

CHAPTER

Usefulskillsforthistopic

• theabilitytonameandmeasurelinesandangles

• theabilitytodrawlinesandanglesusingaruler

Vocabulary

Oblique

• Horizontal

Revolution • Intersection

Perpendicular

Complementary

Ray

Supplementary

Vertical

Parallel

Vertex

Decideifyouthinkeachofthestatementsbelowaretrueorfalse. Bepreparedtojustifyyouranswer.

a Theanglebetweenthehandsofaclockat9:00isarightangle.

b Thetwolinesthatmaketheletter‘X’areparallel.

c Atrianglecanhavetworightangles.

d Thecornerofabookisusuallyarightangle.

e Abasketballcourthasatleasttwoexamplesofparallellines.

f Theanglebetweenthehandsofaclockat6:00isastraightangle.

g Thesidesofatrianglearealwaysstraightlines.

h Thetworailsofatraintrackareperpendicularlines.

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 Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles Linesandangles

Linesandanglesareallaroundus.Weusethemwhenwedraw,weusethem whenwebuildandweevenusethemingames.Architectsandbuildersuselines andanglesintheirwork.

Inmathematics,alineisalwaysastraightline.Itdoesnotincludecurvessuchas circlesorsquiggles.

Linesgoonforeverinbothdirections.Itisimpossibletodrawalinethatgoeson foreverbecauseweeventuallyrunoutofpaper.Soweusuallydrawpartofaline andimaginethatitgoesonforever.Sometimesweaddarrowstoshowthis.

Anangleisthemeasurementofaturn.Ifyouturn throughonerevolution,youhaveturned360° . Thereare360° ofturninacircle.

Whentwolinesmeet(orintersect)atapoint,we measurethenumberofdegreesyouwouldneedto turnfromonelinetotheother.Forexample,the anglebetweenthesetwolinesis35°

13A Linesandangles

Linescanbehorizontal,verticaloroblique.Obliqueisanotherwordfordiagonal orslanting.

Doyouknowhowtocheckwhetheranedgeishorizontal?Peopleoftenuseatool calledaspiritlevel.Haveyoueverseenorusedaspiritlevel?Howdoesitwork?

Doyouknowhowtomakeaverticalline?Builderssometimesuseatool calleda‘plumbbob’.Aplumbbobisastringwithapointedweighton oneend.Theweightusedtobemadeofaheavymetalcalledlead.The Latinnameforleadis plumbum andthisiswhereplumbbobgetsits name.

Touseaplumbbob,startbyfindingamarkonthewall,suchasanail hole.Thenholdoneendofthestringlevelwiththatmark.Whenthe plumbbobstopsmoving,haveafriendmarkthefloorbesidethepointed endoftheplumbbob.Thenrulealinefromthemarkonthewalltothe markonthefloorandyouhaveaverticalline.

Parallelandperpendicularlines

Pairsoflinescanberelated.Twoimportantrelationshipsareparallellinesand perpendicularlines.

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Twolinesareparalleliftheywillnotcrossnomatterhowfartheyareextended.

Whenwedrawparallellines,wedrawtwosmallarrowstoshowthatthelinesare parallel.Forexample,thesepairsoflinesareparallel.

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Thesepairsoflinesarenotparallel.

Youmightbeabletoseepairsofparallellinesinyour classroom.Chooseawallandlookatitssideedges. Nowimagineextendingthosetwoedgesintotheair. Theedgesareparallelandwillnotmeet.

Twolinesareperpendiculariftheyareatrightangles(90°)toeachother.

Whenonelineisperpendiculartotheother,wedrawasmallrightanglewherethe linesintersecttoshowthatthelinesareata90° angletoeachother.

Abuildermakessurethatthewallsareperpendiculartothefloorandadjacentwalls areperpendiculartoeachother.

Linesegments

Wehaveseenthatwecandrawpartofalinetomeanalinethatgoesonforever. However,sometimeswedrawpartofalineandreallymeanonlythepieceinstead ofthewholeline.Apieceofalineiscalleda linesegment.Theword‘segment’ meanspart.

Sometimeswedrawpartofalineandreallymeanhalfofaline.Ahalflineiscalleda ray.Araystartsatapointandthengoesonforeverinonedirection.Thinkofraysof sunshine.

Thearrowsontheendsofraysandlinesarenotalwaysused,butsometimesitis helpfultousethem.

Angles

Tomakeanangleweneedtworaysmeetingatapoint.Wedrawthisbyshowingtwo linesegmentsmeetingatapoint.Theraysorsegmentsarecalledthearmsofthe angle.Thepointwherethearmsoftheangleintersectiscalledthe vertex.Itis sometimeslabelled O

Turningthewholepicturearounddoesn’tchangetheangle. Changingthelengthofthearmsdoesn’tchangetheangle. Whenwecutthefirstslice fromacake,wemaketwo cuts.Thosetwocutsformthe arms ofanangle.Infact,they formthearmsoftwoangles; asmalleronebetweenthe armsandalargeroneoutside thearms.

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Anytwoarmswillproducetwoangles.Wedraw asmallcurvedarrowtomarktheangleweare talkingabout.

Therearemanytypesofangles.

LookatangleA.Theintersectinglinesareperpendicular,so angleAisarightangle.Nowlookattheotherangles.AngleB islessthanarightangle.

AngleCisgreaterthanarightangle.

13A Wholeclass LEARNINGTOGETHER

1 Grouptheseanglesaccordingtotheirtype.

2 Youcanusethehandsofananalogueclocktoshowangles.Beginatthe minutehand,thenimaginetheanglemadeasyoumoveclockwiseuntilyou reachthehourhand.Whattypeofangledothehandsofaclockmakewhen theclockshows:

3 Writethefollowingcompassdirectionsoncardsandplacethemonthe appropriatewallsoftheclassroom.

Facenorth,thenturntofaceeast.Whatturnhaveyoumade?(Aright-angle turn.)Repeatwithothercompassdirectionstoshowturnsthatareastraight angle,areflexangle,anacuteangleandanobtuseangle.

13A Individual APPLYYOURLEARNING

1 Nameeachtypeofangle.

2 Namethetypeofanglemadebythehandsofeachclock.Whattimeiseach oneshowing?

b c d e

3 Nametheanglethatmatcheseachclue.WhattypeofangleamI? Iamhalfarightangle. a Iamtwiceasbigasarightangle. b

Iamthreetimesthesizeof arightangle. c Iamhalfastraightangle. d

Iamhalfarevolution. e Iammorethanarightangle addedtoastraightangle.

4 Drawanexampleof: astraightangle a anobtuseangle b areflexangle c arightangle d anacuteangle e

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13B Measuringangles usingaprotractor

Howdowemeasuretheanglemadewhentwolinesintersect?

Lookatthesetwoangles.Wecannotusearulertomeasurethedistancebetweenthe arms;themeasurementcouldbethesame,butweknowthatoneangleis90° andthe otherisanacuteangle,whichislessthan90° Also,ifyoumovetherulerupordowntheangle,thelengthchanges. Thebestwaytomeasureanangleistouseaprotractor.

Measuringacuteangles

Aprotractorhastwosetsofnumbers:onesetontheinsideedgeandonesetonthe outside.Theinsidenumbersmeasureanglesfromtheright.

Tomeasureanangle,weputthe0° line alongonearmandthecentrepointofthe 0° lineonthevertexoftheangle.

Forexample,thisanglemeasures38

Theoutsidenumbersareformeasuring anglesfromtheleft.

Onearmisonthe0° line.Wemeasure theanglefromtheleftusingtheoutside numbers.

Forexample,thisanglemeasures30° . The angle is 30°.

Measuringobtuseangles

The angle is 110°.

Bothoftheseanglesaregreaterthana rightangle.

Measuringreflexangles

Tomeasureareflexangle,youmayneed torotatetheprotractor.

Thisgivesyoupartoftheangle(55°).To findthefullsizeoftheangle,younow needtoadd180° tothenumberof degreesshownontheprotractor.

The angle is 110°.

Solution

Useaprotractortomeasuretheangle.Placethe0° lineoftheprotractoronthe horizontalarmoftheangle.Makesurethatthecentrepointofthe0° lineisatthe vertexoftheangle.Readthenumberwheretheotherarmoftheangleispointing. Bothoftheseshow45° angles.

Solution

Rotatetheprotractorsothe0° lineonthe protractorisontheobliquearmofthe angle.Makesurethecentreofthe0° lineis atthevertex.Readthenumberwherethe otherarmoftheangleispointing.The angleturnsfromtheright,soweusethe insidescale.Thisisa60° angle.

Example4

Whatisthesizeofthisangle?

Solution

Turntheprotractorupside downsothatthe0° lineon theprotractorisonthe horizontalarmoftheangle, withthearcbelowtheline. Makesurethecentreofthe 0° lineisatthevertex. Measurehowmuchbigger than180° theangleisby readingtheinsidescale.The armispointingtoa60° angle. Nowadd180° to60° . 180° + 60° = 240° Thisisa240° angle.

13B Wholeclass LEARNINGTOGETHER

1 Workinpairs.Onestudentdrawsanacuteangleandtheirpartnerdrawsan obtuseangle.Swapangleswithyourpartneranduseaprotractortomeasure eachother’sangles.

2 Workinpairs.Drawareflexangleeach.Swapangleswithyourpartnerand useaprotractortomeasureeachother’sangles.

3 Estimatethesizeofthesemarkedangles,thenuseaprotractortomeasure eachone.

13B Individual APPLYYOURLEARNING

1 Writethesizeofeachangle,thennamethetypeofangle.

2 Useaprotractortodrawtwointersectinglinesthatmaketheseangles.Mark theanglewithacurvedarrowtoshowclearlywhichangleistheanswer.

3 Drawasketchoftheseanglesthenuseaprotractortomeasurethem.How closewasyoursketchtothecorrectangle?

Findingunknownangles

Youdonotneedtouseaprotractortofindthesizeofeveryangle. Sometimes,theangleyouneedtomeasureisrelatedtooneyoualreadyknowabout. Whentwolinescross,wecanseefouranglesatapoint.

Wewillinvestigatehowanglesatapointarerelated. Inmathematicsweuselettersofthealphabettolabel anglesbecauseithelpsusknowwhichangleweare talkingabout.

Complementaryangles

Thethreelinesontherightintersectandwecanseethree angles,markedA, B,andC.

WecanseethatangleAisarightangle(90°).

AnglesBandCmakeupanotherrightangle(90°).They arecalled complementaryangles becausethetwoangles togethermakearightangle.

IfweknowthesizeofangleC,thenwecanworkoutthesizeofangleB.

IfangleCis35°,angleBis90° 35° = 55°

Supplementaryangles

Whentwolinescross,wecanseefourangles.

Lookatthehorizontalline.

WecanseethatanglesAandBmakeastraightangle(180°).

AnglesAandBarecalled supplementaryangles becausethe twoanglesaddupto180°.IfangleAis135°,thenangleBis 180° 135° = 45° AnglesCandDarealsosupplementary angles.

Nowlookattheother(oblique)line.TheanglesAandDaresupplementarybecause theymakeastraightangle.SoaretheanglesBandC.

Oppositeangles

Whentwolinesintersect,thetwooppositeanglesare equal.

AnglesAandCarethesamesize.

Thisisbecause:

angleA + angleD = 180° and angleC + angleD = 180°

AnglesBandDarealsothesamesize.

IfangleAis145°,thenangleCis145° .

IfangleDis35°,thenangleBis35°

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Anglesaboutapoint

Whenthreeraysmeetatonepoint,wegetthreeangles.The threeanglesaddupto360°

angleA + angleB + angleC = 360° IfangleAis60° andangleBis160°,thenangleCis:

Example5

a AngleAisoppositeananglethatmeasures60°,soangleA = 60° AngleBisoppositeananglethatmeasures120°,soangleB = 120°

b AngleA + 145° + 160° = 360° SoangleA = 360° 160° 145° = 55° 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 RememberRemember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember RememberRemember Remember Remember Remember Remember Remember Remember Remember Remember Remember Remember

Twoanglesthataddupto90° arecalledcomplementaryangles. Twoanglesthataddupto180° arecalledsupplementaryangles. Whentwolinescrosseachother,theoppositeanglesareequal. Whenthreelinesmeetatonepoint,theanglestheymakeadd upto360°

1 Workinpairs.

• Thefirststudentdrawsarightangle.

• Theirpartnerdrawsalinefromthevertexcuttingtherightangleintotwoangles andmeasuresoneofthenewangles.

• Thefirststudentthenworksoutthesizeofthecomplementaryangleby subtractingfrom90°

2 Workinpairs.

• Thefirststudentdrawsastraightangleandmarksavertexonit.

• Theirpartnerdrawsalineoutfromthevertexandmeasuresoneofthe twoangles.

• Thefirststudentthenworksoutthesupplementaryanglebytakingaway from180°

3 Workinpairs.

• Thefirststudentdrawstwolinescrossingeachother.

• Theirpartnermeasuresoneoftheangles.

• Thefirststudentthenworksoutwhichangleisthesamesize,usingtheprinciple thatoppositeanglesareequal.

• Howmanypairsofoppositeanglescanyoufind?

• Measuretheanglestocheckthattheyareequal.

4 Asaclass,discusshowyoucanfindthesizeofalltheanglesifyouknowthesize ofoneoftheangles.Drawthisdiagramandmarktheangles.

a Howmanydegreesaretherewhenallofthe anglesareaddedtogether?

b FindangleA.

c FindangleC.

d FindangleB.Canyoudothisinmorethan oneway?

1 Namethecomplementaryangles.

2 Namethesupplementaryangles.

3 Theseanglesarenotdrawntoscale.Findtheunknownanglewithoutusing aprotractor.

4 Withoutusingaprotractor,writethesizeoftheangleB,thenworkoutthesizeof anglesCandD.

1 Drawasimplelinedrawingofahouse.Useoneormoreofthesewordstodescribe thelinesinyourdrawing.

horizontalverticalparallelperpendicular

2 Choosefromthefollowingwordstodescribetheangles a to f below.

rightangleobtuseangleacuteangle straightanglereflexangle

3 Drawtworaystomakethesekindsofangles.Marktheanglewithacurvedarrow.

Acuteangle a Straightangle b Rightangle c Reflexangle d Obtuseangle e

4 Useaprotractortomeasuretheseangles.

5 Useaprotractortodrawtheseangles.

6a Namethecomplementaryangles.

b Namethesupplementaryangles.

c IfangleCisequalto45°,whatsizeisangleD?

d IfangleBisequalto38°,whatsizeisangleA?

7 For a to f,describethekindofangleshownandwritedownitssize.

g Whichisthesmallestangle?

h Whichisthelargestangle?

i Whatisthedifferencebetweenthelargestandsmallestangles?

8 Findtheunknownangle without usinga protractor. 120° ?

9 WritethesizeofangleB,thenworkoutthesize ofangleCandangleD.

MatchstickFish

1 Use8craftstickstomakethisfish.Showthefishswimmingtotheleftbymoving only3craftsticks.

2 Afteryouchangethefish’sdirectionbymoving3matchsticks,identifyallanglesin yournewfishshapewherematchsticksmeet.Recordtheirmeasurementsand classifythemasacute,right,obtuse,orstraight.

3 Chooseapointwherethreematchsticksmeet.Measuretheanglesaroundthat point.Dotheyaddupto360°?Explainwhy.

4 Identifyandprovewhichanglesareverticallyoppositeandequal.

5 Createanewfishshapeusing8matchsticksthatincludesatleastonerightangle andonepairofverticallyoppositeangles.Labelandmeasurethem.

Usefulskillsforthistopic

• identifyandnamecommon2-Dshapesand3-Dobjects

• identifysymmetryinshape

• describeandperformtransformations

• connectnetsto3-Dobjects

Vocabulary

Isosceles

Obtuse

Parallelogram

Pentahedron

Scalene

Quadrilaterals

Trapezium

Polyhedron

1 Thefourprefixesbelowareoftenusedtodescribeshapes.Findoutwhateach prefixmeans,thenlistasmanywordsasyoucanbasedoneachprefix. Howdoesthemeaningoftheprefixconnecttothemeaningoftheword?

Whatotherprefixescanyounameassociatedwithshapesandwhatis theirmeaning?

2 Nameeachshape:

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Inthischapterweexploretheworldofshapewhichhelpsusunderstandthe objectsandstructuresaroundus.

Webeginbylookingat 2-Dshapes,learningabouttheirpropertiessuchassides, angles,andsymmetry.Thishelpsustorecogniseanddescribetheflatshapeswe seeeveryday.Understanding 3-Dobjects helpsusvisualiseandcomprehend simpleobjectslikeboxestocomplexstructureslikebuildings. Netsandcross sections arewaystorepresent3-Dobjectsin2-D.Thesehelpusunderstandhow 3-Dobjectsareconstructedandhowtheycanbedeconstructed.

Thischapterincludes Symmetry and Transformations.Whenwerecognise symmetrywecanappreciatethebalanceinbothnaturalandhumanmade designs.Transformationsshowushowshapescanchangepositionorsizewhile maintainingtheirproperties.

Shapesareeverywhere,andunderstandingthemhelpsusappreciatethebeauty andusefulnessoftheworldaroundus.

14A Two-dimensionalshapes

Webeginbylookingattwo-dimensionalshapescalled polygons.Apolygonisa two-dimensionalshapeenclosedbythreeormorelinesegmentscalledsides.Exactly twosidesmeetateachvertexandthesidesdonotcross.

Onecommonplacewhereweseepolygonsisonstreetandtrafficsigns.

Thenamesforpolygonsvary,dependingonhowmanysidestheyhave,orhowmany anglestheyhave,orboth.

Thinkofsomewordsthatstartwiththeprefix‘tri’.Forexample,atricyclehasthree wheels,andatrilogyisthenameforaseriesofthreebooks.

Theprefix‘tri’means‘three’.Soatrianglehasthreeangles.Italsohasthree straightsides.

Equilateraltriangles

Anequilateraltrianglehasequalangles.Equilateraltriangles arealsocalled‘regulartriangles’.

Isoscelestriangles

Anisoscelestrianglehastwoequal‘legs’orsides.Every equilateraltriangleisalsoisosceles.Thetwosmallmarkson thesidesindicatethattheyarethesamelength.

Scalenetriangles

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Ascalenetriangleisoneinwhichthesides havedifferentlengths.

Right-angledtriangles

Aright-angledtrianglehasarightangleasoneofitsangles.

ThelittlesquareatvertexBmeansthattheangleis90◦ .

Aright-angledtrianglecanbeisosceles.

Atrianglecannotbebothequilateralandright-angled.

Quadrilaterals

Ashapecalleda quadrilateral hasfoursides.Thesidesmustnotcrossover.Italsohas fourcornersor vertices.

Bothasquareandarectanglehavefourverticesandfoursides.Likeallpolygons, quadrilateralshaveseveralimportantfeatures.Nosidemaycontainmorethantwo vertices,orcrossanotherside.Theremustnotbeanygapsinthesidesofashape.

Anglesinaquadrilateral

Aquadrilateralhasfourangles.Whatdotheyaddupto?

Youcanfindtheanswertothisbydrawingaquadrilateralandseparatingitintotwo triangles.Remember:theanglesofatrianglealwayssumto180◦

Parallelogram

Aparallelogramisaquadrilateralwithoppositesides parallel.

Itlookslikea‘pushedover’rectangle.

Rectanglesandsquaresarespecialkindsof parallelograms.Theyhavefourrightanglesaswellas oppositesidesparallel.

Trapezium

Atrapeziumhastwosidesthatareparallel.Youmighthaveseena tableatschoolwiththisshape.Itisatrapezium.

Thesidesinatrapeziumcanbesameordifferentlengths.

Rectangle

Arectangleisaquadrilateralinwhichalltheanglesarerightangles. Theoppositesidesofarectanglehavethesamelength.Thesesidesarealsoparallelto eachother.

Propertiesofarectangle

1 Allanglesarerightangles.

2 Oppositesidesareparallel:

3 Oppositesideshavethesamelength:

Square

Asquareisaquadrilateralwithallitsanglesequalandallitssidesthesamelength.So asquareisaspecialtypeofrectangle,withallsidesthesamelength.

Pentagons

ThewordpentagoncomesfromtheGreekwords penta meaning‘five’and gon meaning‘angle’.Soapentagonhasfiveangles,fiveverticesandfivesides.Itssides mustnotintersectexceptwheretheymeetatthevertices.

14A Wholeclass LEARNINGTOGETHER

1 WhatamI?

Iamaregularquadrilateral.Thatmeansmysidesarethesamelengthandmy anglesareequal.

Iamashapeyoualreadyknow. Ihaveequalanglesandequallengthsides. Allmyanglesare90◦ . WhatamI?

2 Drawingarectangle

a Useyourrulertodrawaline.Marktwopoints onthelineanduseyourprotractorto constructtworightangles.

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b Markofftwolengthsof5cm,asshown.Usea rulertojointhetwonewpoints.Whatshape haveyoumade?

4 Whichoftheseshapesareirregularpentagons?Why?

5 a Drawfourdifferenttriangles.Makesureyouincludeatleastoneright-angled triangleandatleastonetrianglewithanobtuseangle.Labeltheverticesof eachtriangleA,BandC.

b Copythistable.

CarefullymeasuretheinsideangleatA,BandCofeachofyourtrianglesfrom part a onthepreviouspage.Writetheanglesinthetable,thenfindthesumof thethreeanglesineachtriangle.

c Nowfillintheblankinthisstatement. Ifweadduptheanglesattheverticesofatriangle,thesumis____.

6 Canatrianglehavetworightangles?Explainyouranswer.

7 Canatrianglehaveareflexangle?Explainyouranswer.

14A Individual APPLYYOURLEARNING

Acute-angledtriangles

1 Anacuteangleisananglethatislessthan90◦ .Thisisan acuteangle.

Anacute-angledtrianglehasallangleslessthan90◦ .Whichoftheseare acute-angledtriangles?

Obtuse-angledtriangles

2a Anobtuseangleisananglethatislarger than90◦ .AngleDontherightsideisan obtuseangle.

Usearulerandprotractortodrawanobtuseangle.Labeltheobtuseangle.

b Anobtuse-angledtrianglehasone anglelargerthan90◦ .Inthis obtuse-angledtriangle,angleBis greaterthan90◦

Whichofthesetrianglesareobtuse-angledtriangles?

3 Writethelabelsthatmatcheachshape.Someshapeshavemorethanonelabel, andlabelscanbeusedmorethanonce.

rectangle,square,hasfoursidesofequallength,quadrilateral,notaquadrilateral, hasfourrightangles

4a Canatrianglebeacute-angledandisosceles?Ifyouthinktheansweris‘yes’, drawone.

b Canatrianglebeobtuse-angledandisosceles?Ifyouthinktheansweris‘yes’, drawone.

c Canatrianglebeobtuse-angled and equilateral?Explainyouranswer.

5 Usearuler,apencilandaprotractortoconstructarectanglewithsidelengths 4cmand7cm.Checkthatallanglesarerightangles,andthatoppositesidesare thesamelength.

6 Drawapentagonwithequalanglesbutunequalsides

Seeifyoucandrawapentagonthathasallofitsangles108◦ ,buthassidesof differentlengths.(Hint:Startwithalinesegment,makeaturnof108◦ andthen drawanotherlinesegment.)

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14B

Three-dimensionalobjects

Inthissectionwelookatthree-dimensionalobjects.Everythingaroundusexists inthreedimensions.Soyouareathree-dimensionalobject,andachairisa three-dimensionalobjecttoo.Inmathematics,werefertothree-dimensionalobjects as solids

Weseealotofsolidswhenwegoshoppingatthesupermarket.Peopleusuallycall thembyothernames,suchas cylinder,triangularbox,rectangularbox,andsoon.

Lookattheshapesabove.Inwhatwayaretheysimilartotheshapesbelow?Inwhat wayaretheydifferentfromtheshapesbelow?

Thinkofsomethingsyoubuyatthesupermarketthatlookliketheseobjects,orthat aresoldinboxesorpacketsshapedlikethese.

Polyhedra,prisms,pyramidsandcylinders

Theword polyhedron ismadeupfromtheGreekwords poly,whichmeans‘many’, and hedron,whichmeans‘face’.Sopolyhedronmeans‘manyfaces’.Apolyhedronisa three-dimensionalobjectwithflatfacesandstraightedges.Thefacesarepolygons. Theyarejoinedattheiredges.Thepluralofpolyhedronis polyhedra,sowecanhave onepolyhedron,andtwoormorepolyhedra.

Whenwedescribeapolyhedron,weareinterestedinits properties,orfeatures.

Eachflatsideofapolyhedroniscalleda face.Wheretwo facesmeet,yougetanedge.Anedgeisalinesegment. Thesharpcorneronapolyhedronwheretwoormore edgesmeetiscalleda vertex.Thepluralofvertexis vertices.

Wegivepolyhedraspecialnamesaccordingtohowmany facestheyhave.Thesmallestnumberoffacesthata polyhedroncanhaveisfour.

Tetrahedra

face edge vertex

TetrahedronisfromtheGreekwords tetra,whichmeans‘four’,and hedron,whichmeans‘face’.Tetrahedronshavefourfaces.

Pentahedra

Herearetwodifferentpentahedra. Penta means‘five’,sotheseobjectshavefivefaces.

Regularpolyhedra

Forapolyhedron,theword‘regular’meansthatallofitsfacesareidenticalregular polygonsandthesamenumberoffacesmeetateachvertex.Therearefiveregular polyhedra,alsocalledplatonicsolids.Acubeisaregularpolyhedron.

Cube

Acubehas6faces,allidenticalsquares.Threefacesmeetat eachvertex.

Regulartetrahedron

Aregulartetrahedronhas4faces. Tetra meansfour.Theyare identicalregulartriangles(equilateraltriangles)and3facesmeetat eachvertex.

Thethreeremainingplatonicsolidsare:

octahedrondodecahedronicosahedron

Prisms

A prism isapolyhedronwithabaseandatopthatarethesameandwhoseother facesareallparallelograms.

Base and top are the same.

Other sides are parallelograms.

Whentheparallelogramsareallrectangles,thisisknownasarightprism. Aprismgetsitsnamefromtheshapeofitsbase.Theprismbelowhasarectangular base,soitiscalleda rectangularprism

Becausearectangularprismhas6faces,itisahexahedron.

Cylinders

Thissolidiscalleda cylinder.Ithasacircularbaseandtop. Cylindersarenotprisms,astheydonothaverectangularfaces,and theyarenotpolyhedra,asnotalltheirfacesareflatandnotalltheir edgesarestraight.

Pyramids

Apyramidisapolyhedronthathasapolygonforitsbaseandallofitsotherfaces aretrianglesthatmeetatonevertex.Thepointatwhichthesefacesmeetiscalled the apex

Conesarenotpyramids.

Thesolidontherightisatriangular-basedpyramidbecauseithasa triangleforthebase.Itisalsoatetrahedronbecauseithas4faces.

Thisisahexagonal-basedpyramid.Itisanothersolidthathastwo names:onenamefortheshapeofitsbaseandonenameforthe numberoffaces.Ithas7faces,soitcanalsobecalledaheptahedron.

1 Lookatthesolids a to c below.Theyallsitnicelyonaflatsurface.Collectclassroom solidslikethoseshownandplacethemonaflatsurface.Sketcheachsolid,then drawitsbase.Nametheshapeofthebase.

2 Workinpairsusingaclassroomsetofsolids.Copyandcompletethistableforat leastthreeofthepolyhedra,makingsureyouhaveatleastonepyramidandone prism.(Remember:spheres,conesandcylindersarenotpolyhedra.)Shareyour resultswiththeclass.

3 Investigateinterestingthree-dimensionalobjectssuchasthepyramidsofEgypt, China,KoreaandIndonesia.Prepareashortreportfortheclassaboutthe mathematicsinvolved.

14B Individual APPLYYOURLEARNING

1 This3Dshapeisa cone.Inwhatwayisitsimilartoacylinder?

2 Copyandcompletethistableforeachobject.

Anetisaflatshapethatcanbefoldeduptomakeathree-dimensionalobject.Every polyhedroncanbecutintoanet.

Ifwetakeanemptybreakfastcerealboxandcutaroundthreeofthetopedgesand downthefourverticaledgeswecanfolditdownflat.Thisgivesusanetofthebox.

Ifyoutookacubeand‘unfolded’it,youwouldhave6squaresjoinedinanet.

Becausethecubehas6squarefaces,thenetmusthave6squares.Therearemany possiblenetsforacube.

1 Whichnet(A to D)matchesthispolyhedron?

2 Whichofthesepentominoes(A to I)isalsoanetforanopenbox?(Anopenbox islikearectangularprismwithonefacemissing.)Thereismorethanonecorrect answer.

1 Labeleachnetonthefollowingpageusingoneofthesenames: square-basedpyramidpentagonalprismcube dodecahedronpentagonal-basedpyramidtetrahedron octahedron

2 Drawanetforeachsolid.

14D Symmetryand transformation

Ifyoudrewalinedownthemiddleofyourface,youwouldseethatthetwohalves matchupexactlyacrosstheline.Thisiscalled symmetry.Ifyouplaceasmallmirror alongthedottedlinesbelow,youwillfindthattheimageinthemirrorcompletes thepicture.

Weseepatternsallaroundus.Manypatternsaremadebyshapesfittingtogether.

Rotation, reflection and translation aresomeofthedifferentwayswecantransforma two-dimensionalshape.

Inthissectionwecontinuetodiscovermoreaboutthepropertiesoftwo-and three-dimensionalshapes.Welookatsymmetryingeometryandthinkofhowthis mightapplyinnature.Weinvestigatetheeffectofmovingtwo-dimensionalshapes andvisualisethesetransformations.

Symmetryoftwo-dimensionalshapes

Ifwestandupstraight,theverticallinedownthecentreofourfaceorbodydividesus intotwoalmostidenticalpieces.Inmathematics,whenthepiecesofa two-dimensionalshapematchupexactlyacrossastraightline,wesaytheshapeis ‘symmetricalabouttheline’.

Drawanisoscelestriangleonapieceofpaper,thencutitout.Foldtheright-handside ofthetriangleoversoitliesexactlyontheleft-handsideofthetriangleandmakea creasedownthecentreofthetriangle.Thehalvesofthetriangleoneithersideofthe foldlineshouldmatchexactly.Thefoldlineiscalledthe axisofsymmetry orthe line ofsymmetry.Useaprotractortomeasuretheangleswherethefoldlinemeetsthe base.Theyshouldbe90◦

Theisoscelestriangleissymmetricalaboutthefoldline.

Someshapeshavemorethanonelineofsymmetry.Theseshapeshavetwolinesof symmetry.

Thisshapehasthreelinesofsymmetry.

Transformationandtessellation

Rotation,reflectionandtranslationaredifferentwayswecantransforma two-dimensionalshape.

Rotation

Werotateashapewhenweturnitthrough anangle.

Thediagramattherightshowsanarrow shapepointingupwards.Theshapehasbeen rotatedclockwisearoundthereddotthree times,eachtimeby90◦ .Theword‘image’has beenusedtolabeltheshapeineachnew position.

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Wecanrotateclockwiseoranticlockwiseabout apoint.

Thistrianglehasbeenturned90◦ inaclockwise directionabout O

Thistrianglehasbeenrotated90◦ anticlockwise.

Therectanglehasbeenrotated90◦ inaclockwisedirection.

Reflection

Areflectionisatransformationthatflipsafigureaboutaline.Thislineiscalledtheaxis ofreflection.Agoodwaytounderstandthisistosupposethatyouhaveabookwith clearplasticpagesandatriangledrawnononepage,asinthefirstdiagrambelow.If thepageisturned,thetriangleisflippedover.Wesayithasbeenreflected;inthiscase theaxisofreflectionisthebindingofthebook.

Thisshapehasbeenreflectedintheverticalline.

Example2

Theimageisthemirrorimageoftheshapeontheleft.Ithasbeenreflectedinthe verticalline.

Translation

Whenwetranslateashape,weslideit.Wecanslideitleftorright,upordown.

Transformationsmovetheshapewithoutrotatingit.

Thisshapehasbeen translatedhorizontally. image

Thisshapehasbeen translatedvertically.

Example3

Howhasthisshapebeenmoved?

Solution

Theshapehasbeentranslatedhorizontally.

Tessellation

A tessellation isatilingpatternmadebyfittingtogethertransformationsofa two-dimensionalshapewithnogapsoroverlaps.Thetessellationcancontinueinall directions.

Startwithanequilateraltriangle.

Wecanrotateit180◦ andtranslateitsothetrianglesfittogether perfectly.Thetilinggoesonforever.Wesaythattheequilateraltriangle tessellates

Theshapeusedinthepatternbelowisnotatessellatingshapebecausewecannot rotateandtranslateittofillupthewholespacewithoutgapsoroverlaps.

Example4

Willthisshapetessellate?

Solution

Yes,thisshapewilltessellate.Itcanberotated180◦ andtranslatedsothepieces fittogetherwithoutanygapsoroverlaps.

Thepatterncanbecontinuedhorizontallyandverticallyasfarasyouwish.

1 Howmanylinesofsymmetrydoeseachquadrilateralhave?Explainwhyeach shapehasthatnumberoflinesofsymmetry.

2 Describethetransformations.

3 Writethealphabetincapitalletters.Listthelettersthathaveatleastonelineof symmetry.

4 Useattributeblocksorpatternblocks.

a Chooseashape(notahexagon)thatyouthinkwilltessellate.Showthatthe shapetessellatesbyputtingtogetheratleast10tracingsoftheshape.

b Measureorcalculatetheanglesaboutapointwithinyourtessellatingpattern.

5 Useblockstomakeatilingpatternwithtwoormoredifferent-shapedtiles.Which shapesdidyouuse?Measuretheanglesaboutapointwithinyourtessellating pattern.

6 Usingadigitaltoolcreateatessellationpatternwithquadrilaterals.Arrangethe quadrilateralssothattheyfittogetherwithoutanygapsoroverlaps.

• Describethestepsyoutooktocreatethetessellation.

• Explainwhyregularquadrilateralsworkwellforthispattern. 14D Individual APPLYYOURLEARNING

1 a Draw5regularpolygonsofdifferentsizes.

b Markdottedlinesonthepolygonstoshowallthelinesofsymmetry.

2 Describethetransformationoftheseshapes.

3 Drawtheseshapes,thendrawwhattheylooklikeaftertheyhavebeenreflected intheline.

4 Thisshapeismadefrom4identicalsmallsquares. Ithas4linesofsymmetry.

Use4identicalsquarestomakeashapethathas:

a 1lineofsymmetry

b 2linesofsymmetry

Use5identicalsquarestomakeashapethathas:

c 1lineofsymmetry

d 4linesofsymmetry

e 0linesofsymmetry

5 Drawtheseshapes,thendrawwhattheylooklikeaftertheyhavebeentranslated.

a Translatehorizontally

b Translatevertically

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6 Foreachoftheseshapes:

• rotatetheshape90◦ clockwise

• drawtheimage

• repeattheabovestepstwice.

7 Selectoneshapefromyourclasssetofshapesthatwilltessellate.Drawa tessellationusingtheshape.Colourtheshapestoshowthepatternyouhave made.

1 Useyourrulertodrawanisoscelestrianglethathaslongestsidesoflength6cm.

2 Usecompassesoraloopofstringtodrawtwosmallequilateraltriangles.Make eachtriangleadifferentsize.

3 Useaprotractortodrawanequilateraltrianglewithanglesof60◦ .Youcanhave sidelengthsofupto8cm.

4 Peterhas36metresofstring.Heneedstouseallofthestringtomarkoutsix equilateraltrianglesontheground.Howlongwillthesidesofeachtrianglebe? Drawadiagramtoshowyouranswer.

5 Drawthesescalenetriangles.

a Ascalenetrianglethathasarightangle

b Ascalenetrianglethathasa40◦ angle

c Ascalenetrianglethathasonesidehalfaslongasoneoftheothersides

6 Aretheanglesmarkedwithasmileyface acuteorobtuse?

7 Unscrambleeachword,thenmatcheachnametothecorrectshape(A to C). burhsom

8 Matcheachnetwiththecorrectlabel.Somenetshavemorethanonelabel. rectangularprismhexagonal-basedpyramidoctagonalprism cubepentagonal-basedpyramidhexahedron triangular-basedpyramidheptahedrontetrahedron octahedron

9 Usingadigitaltooldesignasymmetricalpatternthatcouldbeusedforatilefloor orwallpaper.

• Describetheshapesandcolorsyouusedandexplainhowyouensuredthe patternissymmetrical.

• Howdoessymmetrymakethepatternmoreappealing?Additionally,incorporate tessellationandtransformationintoyourdesign.

• Explainhowyouusedtessellationtocreatearepeatingpatternandhow transformationsliketranslations,rotations,andreflectionswereappliedto maintainsymmetryandenhancetheoveralldesign.

14F Challenge–Ready,set,explore!

Designthinking

1 Apentominoisanarrangementoffivesquares.Thesquaresmustbearrangedso thattheyhaveacompletesideincommon.

Thismeansthattheycantouchlikethis orthis ,butnotlikethis or this

Thepentominobelowhasbeenflippedorrotated,butitcanonlybecountedas onepentomino.

Howmanydifferentpentominoescanyoumake?Usegridpapertodrawyour pentominoes.

2 Becomeanarchitectbyusing3Dobjectsinahousedesign.

a Sketchahousethatusesthree-dimensionalobjectsinitsdesign.Includeatleast oneofeachofthese:triangularprism,rectangularprism,cube, hexagonal-basedpyramid,cylinder.Youcanusemorethanoneofeachobjectif youwish.Youcanalsouseother3Dobjects.

b Usepaperorcardboardtoconstructthe3Dobjectsyouneedforyourhouse design,thenusetheobjectstoconstructyourhouse.Labelallthe3Dobjects youusedtoconstructyourhouse.

c Sketchyourhousedesign.Thinkabouthowtodrawathree-dimensionalshape onpaper.Whichanglewillyoudrawitfrom?Howwillyoushowperspective?

3 Explorehowtessellationisusedindifferentculturesaroundtheworld.Chooseone exampleoftessellationinartorarchitecturefromaspecificculture,suchasIslamic geometricpatterns,ancientRomanmosaics,orJapanesetraditionaldesigns.

a Describethetessellationpatternyouchoseandexplainhowitreflectsthe culture’sartisticandarchitecturalstyle.

b Createyourowntessellationpatterninspiredbytheexampleyouresearched andexplainthestepsyoutooktodesignit.

Usefulskillsforthistopic

• somepriorexperienceusingmaps

• theabilitytousecoordinatesanddirectionallanguagetodescribepositionand movement

Vocabulary

Gridreference • Cartesianplane

Orderedpair

-axis(horizontal)

y-axis(vertical) • Coordinates

Carmelisfacingnorth.Inwhichdirectionwillshebelookingifsheturns:

1 45degreestoherright?

2 90degreestoherleft?

3 135degreestoherleft?

4 135degreestoherright?

5 45degreestoherleft?

6 180degrees?

7 90degreestoherright?

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Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates Mapsandcoordinates

Forthousandsofyears,peoplehave mademapsoftheirsurroundings.

Earlymapsreliedonwhatpeoplecould seeandmeasure,andsometimesthey weren’tveryaccurate.

Today’smapsarebasedonsatelliteimages.Theyareveryaccurateandarebased onphotostakenfromspace.

15A Readingmaps

ThemapofEuropeandpartofAfrica belowisdividedintocolumnsandrows anduseslettersandnumberstolocate placesonthemap.

Thecombinationofaletterandanumberto describeapositiononamapiscalleda grid reference.Eachsquareonthemaphasits owngridreference.Forexample,B3isthe squareincolumnB,row3.Thegrid referenceforIcelandisA2.Whichcountries arelocatedatC4?

Example1

ThereareseveralgridreferencesthatcouldbeusedtolocatepartsofFranceon themapabove.Whatarethey?

Solution

FrancecrossescolumnsBandC,andisinrows5and6.Thegridreferencesfor FrancecouldbeB5,C5,B6andC6.

Example2

WhichcountrieswouldyoufindatD2?

Solution

TraceyourfingerupcolumnDuntilitmeetsrow2.Norway,SwedenandFinland areatD2.

Usethemaponthepreviouspageforthesequestions.

1 Whichcountryhasthegridreference:

2 Writethegridreferencethatcouldbeusedtofind:

1 OnthemapofManlyabove,whatwouldyoufindat:

2 Writethegridreference(s)thatwouldlocate:

15B TheCartesianplane

Anumberlinecanbeusedtoshowbothpositiveandnegativeintegers.

Ifwetakeasheetofgridpaperanddrawtwonumberlinesonit,atrightanglesto eachother,wehavea Cartesianplane

Theaxesarecalledthe coordinateaxes.Theyare namedaftertheFrenchmathematicianand philosopherRenéDescartes(1596–1650). Descartesintroducedcoordinateaxestoshowhow algebracouldbeusedtosolvegeometricproblems.

ThehorizontalaxisoftheCartesianplaneiscalled the x-axis.Theverticalaxisiscalledthe y-axis.The axes intersectatzero.

Axes,thepluralofaxis,ispronounced‘axees’,not likethepluralofaxe.

AnypointontheCartesianplanecanbedescribedusingtwonumbers.Thefirst numbertellsushowfarwemoveacrossalongthehorizontalaxis.Thesecondone tellsushowfarwemoveupordowntheverticalaxis.Thesearecalledthecoordinates ofthepoint.OntheCartesianplaneonthepreviouspage,thepointAislocated at ( 4, 7)

Thecoordinates arean orderedpair.Wecallitanorderedpairbecauseitisapairof numbersandtheorderinwhichtheyaregivenmakesadifference.Anorderedpairis writteninbrackets.The x-coordinateisalwayswrittenfirst,thenacommaandthen the y-coordinate.Forexample, ( 4, 7)

Example3

RuleupandlabelaCartesianplane,thenplotandnamethesepoints.

W = (2, 1)

X = ( 2, 2)

Y = (1, 2)

Z = ( 3, 2)

15B Individual APPLYYOURLEARNING

1 WritethecoordinatesofthepointsE,F,G,H,I,J,KandL.

2 DrawyourownCartesianplanewithaxesmarkedfrom6to 6,asshownabove. MarkthesepointsontheCartesianplanewiththeletterandadot.

L =(3, 2) M =(−2, 0) N =(4, 5) P = (−5, 4)

Q = (5, 0) R = (0, 3) S =(1, 3) T = (0, 4)

U = (0, 0) V = (−4, 2)

3 DrawanotherCartesianplaneandnumbertheaxesfrom6to 6.Ploteachsetof orderedpairs,anddrawadotforeachpoint.Useyourrulertojointhedots,then namethegeometricalshapethatyouhavecreated.

a (3, 2), (5, 3), (3, 4), (1, 3)

b (−3, 1), (−3, 5), (−5, 5), (−5, 1)

c (−5, 2), (−5, 2), (−1, 6), (4, 2), (5, 3), (0, 6)

d (5, 1), (5, 1), (3, 1), (3, 1)

e (1, 3), (1, 5), (−2, 4)

f (2, 1), (1, 2), (−2, 2), (−3, 1)

g (4, 5), (4, 2), (−1, 2)

Uncorrected 3rd sample pages • Cambridge University Press & Assessment © • Evans, et al 2026 • 978-1-009-76084-3 • (03) 8671 1400 UNCORRECTED

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h Atwhatpoint dothediagonalsintersectinparts a, b and d?

2 Writethegridreferencesfortheselocations.

a SemaphorePoint

b DogRock

c TownJetty

d PrincessRoyalFortress

e BrigAmity

f Reservoir

g TheroundaboutwhereMiddletonRoadandCampbellRoadintersect

h AlbanyPrimarySchool

3 Followthesedirections.StartatA1ontheAlbanyHighway.Travelsouth-eastto YorkStreet.TurnrightintoYorkStreet,gothroughtheroundabout,thenturnleft onPrincessRoyalDrive.AsyougoalongPrincessRoyalDrive,whatwillyousee onyourright?

4 Selectalocationofyourownandwriteasetofdirectionstogetthere.Swapyour directionswithafriendandseeifyoucanfindeachother’slocations.

5 DrawaCartesianplanewitheachaxisnumberedfrom 10to10.Plotthe followingorderedpairs.Nowuseyourrulertojointhepairsintheordertheyare written.Namethegeometricalshapethatyouhavemade.

AdventureinNumberland

Imagineyouareanexplorerinamagicallandwhere thenumbersonthemapcanchangebetween positiveandnegative.Youstartyourjourneyatthebase camplocatedatpoint (0, 0) onacartesianplane.Your missionistoreachthetreasurehiddenatpoint (10, 5)

Ateachstepofyourjourney,youcanmove:

• 3stepsforwardorbackwardparalleltothe x-axis. or

• 2stepsupordownparalleltothe y-axis.

However,therearealsoenchantedbarriersalongyourpath:

• Ifyoumovethroughpoint (4, 2),youmustmove4stepsbackwardonthe x-axis.

• Ifyoumovethroughpoint (7, 3),youmustmove3stepsdownthe y-axis.

• Ifyoupassthroughpoint (1, 2),youcanmove5stepsforwardonthe x-axisforfree.

Canyoufindaroutetoreachthetreasureat (10, 5) while navigatingthroughthese enchantedbarriers?

Explain yourstrategyandthestepsyoutaketoensureyoureachyourdestination.

Usefulskillsforthistopic

• collectingdataandrepresentingitinadisplay

• interpretingdatatables,pictographsandbargraphs

• identifyingthemodefromdatacollected

Vocabulary

Thegraphsbelowwereaccompaniedbythefollowingtextinanewspaperarticle: ‘Companyprofitslookmuchbetterin2025whencomparedwith2018’.

• Whatdoyounoticeaboutthesegraphs?

• Whyisthetextthataccompaniesthegraphmisleading?Discuss.

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Howcanwefindoutabouttheworldaroundus?Onewayistocollect information,organiseit,thenstudytheresults.Collectingandstudying informationinthiswayiscalled statistics.Peoplewhogatherandanalyse statisticsarecalled statisticians.

Mostpeopleusetheword population todescribethenumberofpeoplelivingin oneplace.Statisticiansusethewordpopulationtodescribeagroupthattheyare interestedinstudying.Forexample:

• peoplewholiveinAliceSprings

• iPhonespurchasedinTasmania in2024

• canetoadsintheNorthern Territory.

Howwouldyoufindoutaboutcanetoadsin theNorthernTerritory?Itwouldnotbepossible tocounteverycanetoad.Soastatisticianwouldcollect informationaboutasmallergroupwithinthepopulation,suchas thenumberofcanetoadsinonesquarekilometreoflandnear Maningrida,inArnhemLand.Thissmallergroupisknownasa sample

Whenthestatisticianhassomeinformationfromthesample,theythentryto makepredictionsabouttheentirepopulationofcanetoads.

Theinformationthatwegatheraboutapopulationiscalled data.Ifwehave manyquestionstoanswer,weneedtogatherdataaboutdifferentaspectsofour population(orsample).Thedatacanbeorganisedandpresentedintables,charts andgraphs,andweinterpretthatinformationinordertomakesomeconclusions andrecommendations.

Darwin Maningrida

Alice Springs

Thestatisticalprocess

Whenweplanastatisticaldatainvestigation,weneed todecideontheproblemwearegoingtoinvestigate andthenposesomequestionsthatwemightlike answersfor.Forexample,iftheschoolcanteenwants tostartsellingfrozenyoghurtwemightask‘What flavoursoffrozenyoghurtdoYear6studentslike?’and ‘Whoarethepeoplewhowillbuyfrozenyoghurt?’.

Next,wethinkaboutwhatdataweneedtocollecttoanswerthosequestions. Therearemanywaystocollect,organise,andpresenttheinformation,sothere aremanychoicestobemade.

Finally,welookatthedataorganisedandpresentedintables,charts,andgraphs andinterpretthatinformationtomakeconclusionsandrecommendations.Inour frozenyoghurtexampleabove,wemightusetheinformationwehavecollected tosuggesttothecanteenstafftheflavoursthatYear6studentslike.

Section16Fexplainsthestatisticaldatainvestigationprocessinmoredetail,and youwilluseittocarryoutyourowndatainvestigations.

Typesofdata

Thereareseveraltypesofdata.Foreachtype,therearedifferentthingsto considerwhencollectingandrecordingthedata,andwaystopresentit.

Onetypeofdataisdatathatwecan count.Weget countdatawhenweinvestigatesituationssuchas:

• thenumberoftreesindifferentbackyards

• thenumberofgoalsscoredinanetballmatch

• thenumberofjellybeansinapacket.

Anothertypeofdataisdatathatwecan measure,

• theheightofstudents inyourclass

• theageofstudentswhentheyfirstrodeabikewithouttrainingwheels

• theamountofwaterleftinstudents’drinkbottlesafterlunch.

Thenthereisdatathatbelongsin categories.Sometimesthereisachoicetobe madeaboutwhichcategorythedatabelongsto.Categoricaldataincludes:

• typesofhouses

• coloursofcats

• hairstyles.

16A Usingquestions andcollectingdata

Tablesareusedtorecordandpresentdata.Theinformationinatableisorganisedso thattheimportantideascanbeunderstoodeasilyandquickly.

Jamesinterviewedhisclassmatesandwrote downthedifferentwayseachpersoncame toschool.

Jameswantedtodrawsomeconclusionsabout theinformationinhislist,butthepatternswere noteasytosee.Hedidnotfindhislist veryuseful.

SoJamesorganisedhisinformationintoatable, usingtallymarkstorecordhisdata.

Eachstrokeinatallystandsforoneitem.Thefifthstrokeismadeacrossagroupof four.Thismakesiteasytocountbyfivestoworkouthowmanyareinatally.

Jamesthencountedthenumberoftimesthathehadreceivedthesameanswer.Thisis calledthe frequency

NowJamescanusethedatatoanswermanydifferentquestions. Forexample:

• Howmanystudentswereinterviewed?

30studentswereinterviewed.

(Addthenumberofstudentsineachcategorytofindthis.)

• Howmanystudentscaughtthebustoschool?

8childrencaughtthebustoschool.

• Whatfractionoftheclasstravelledbycar?

10 30 or 1 3 oftheclasstravelledtoschoolbycar.

• Whatpercentageoftheclasswalkedtoschool?

Percentage = 3 30 × 100 1 %= 10% 10% oftheclasswalkedtoschool.

Sometimesweneedtocomparethedatafromtwodifferentgroupsofpeople.Wecan showthedataforeachgroupinthesametablebyusingatwo-waytable.

Example1

Elizaaskedthestudentsinherclasshowtheytravelledtoschool.ThenElizaand Jamesbothshowedtheirdatainthesametwo-waytable:

Travellingtoschool

a HowmanystudentsdidElizainterview?

b Howmanystudentstravelledtoschoolbybikeineachclass?

c Whatfractionofeachclasscametoschoolbybus?

d Whichclasshadthelargerpercentageofchildrencomingtoschoolbybus?

Solution

a Byaddingthenumbersineachcategory,wegetatotalof25studentsfor Eliza’sclass.

b InJames’sclass,9studentstravelledtoschoolbybike.InEliza’sclass, 3studentstravelledtoschoolbybike.

c 8 30 or 4 15 ofJames’classtravelledbybus. 7 25 ofEliza’sclasstravelledtoschool bybus.

d Weknowthat8studentsinJames’sclasscamebybus.First,weconvertthis toapercentage.

Percentage = 8 30 × 100 1 % = 26.7%

26.7% ofthestudentsinJames’sclasscametoschoolbybus.

InEliza’sclass,7studentscamebybus.

Percentage = 7 25 × 100 1 % = 28%

28% ofthestudentsinEliza’sclasscametoschoolbybus.

SoeventhoughagreaternumberofpeopleinJames’sclasscametoschoolby bus,agreaterpercentageofEliza’swholeclasscamebybus.Percentageisa usefultoolforcomparingtherelativesizeofgroupswithingroups.

16A Wholeclass

LEARNINGTOGETHER

1 Workinpairstocollectthedataforthisscenario.Theschoolcanteenisgoing tointroducefrozenyoghurttreats.Youhavetofindoutwhichfrozenyoghurt flavoursshouldbestocked.

a Writethequestionthatyouwillaskyourclassmates.

b Drawupatablethatwillletyouusetallymarkstocollectyourdata.Make surethatyouhavespaceonyourtableforatotal.

c Surveyeachmemberofyourclasstogatherthedata.

d Makefivestatementsaboutyourdata.Whatrecommendationswillyou maketothecanteen?

2 Measureandrecordthelengthoftheleftfootofeachmemberofyourclass. Yourmeasurementshouldbecorrecttothenearestcentimetre.

a Drawupatablefortheresults,includingacolumnfortalliesandacolumn forthefrequency.

b Usethedatainyourtabletoanswerthesequestions.

• Whatistheshortestfootlength?

• Whatisthelongestfootlength?

• Whichfootlengthoccurredthemostoften?

16A Individual APPLYYOURLEARNING

1 Imagineyouwanttofindaccuratedataforquestions a to e below.Selectthe mostappropriategroupofpeopletosurveyforeachquestionfromthislist.

Pre-schoolchildrenPrimary-schoolstudentsMums Plumbers Librarians Adultsover18

a Whattimeofdaydopeopleusetheircars?

b Whatisthebesttypeofpipetousefordrains?

c WhatisyourfavouritesongbyTaylorSwift?

d Whichschoolsubjectisthemostfun?

e Whatisthemostpopularbookforteenagers?

2 Writeaquestionthatyoucouldasktogetdataaboutthesetopics.Whocould youask?

a Thecostofweeklygroceries

b Thedifferenttypesoflibrarybooksborrowedinoneweek

c Thebestfoodfordogs

d Themost-watchedTVnewsservice

e Thefavouritebreakfastcereal

3 TheYear6classatMtBotanicSchoolcountedthenumberofAustraliannative treesintheirtwolocalparks.Thistableshowstheirresults.

a Copyandcompletethetablebyfillinginthefrequencycolumnforeach typeoftree.

b Howmanytreesarethereineachpark?

c Foreachpark,writeeachtypeoftreeasafractionofthetotalnumber oftrees.

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d Whichparkhasthegreaterpercentageofbottlebrushtrees?

16B Mode,medianandmean

Whenwemakestatementsaboutdatathatwehavecollected,weoftenwanttosay whichitemisthemostpopular,whichitemisinthemiddleandwhichitemisthe average.Therearemathematicalwordstodescribethesethreeideas.Theyaremode, medianandmean.

Mode

Howdowefindoutwhichitemisthemostpopular?Orthemostcommon?Orthe favourite?Allofthesequestionsareaskingthesamething.Theywanttoknowwhich valueoccursthemostoftenorhasthehighestfrequency.Thisvalueiscalledthe mode Thereisaneasywaytorememberthis. Mode istheFrenchwordfor‘fashion’,anditis alsothemostfashionable(ormostpopular)valueinasetofdata. Sometimestwovaluesareequallypopularandalltheothersarelesspopular.Inthis case,wetakebothvaluestobethemode.

Example2

AgroupofYear6studentsatYorkSchoolrecordedtheirshoesize.

Usethedatatocalculatethemode.

Solution

Putthedataintoafrequencytable.

Themodeissize4,becausesize4shoesoccurmostofteninthisclass.

Median

Whenasetofvaluesisarrangedinorderoftheirsize,the‘middlevalue’isthe median Herearetheagesofagroupofchildreninorder,fromyoungesttooldest.

10, 10, 11, 11, 11, 11, 11 , 11, 12, 12, 12, 12, 13

Thissetofdatahas13values.Theseventhvalueisthemiddlevalue,asithas6values oneitherside.Themedianageis11.

TenstudentsatSnapperPointSchooldecidedtoworkouttheirmedianage.Their ageswere:

9, 9, 10, 10, 10, 12 , 12, 13, 13, 14

Thissetofdatahasanevennumberofvalues,withtwomiddlevalues:10and12.

Tocalculatethemedian,weneedtofindtheaverageof10and12.

Median = 10 + 12 2 = 11

Sothemedianageis11,eventhough11doesnotoccurinthissetofdata. Thereisaneasywaytorememberwhat‘median’means.Thinkofthemedianstrip thatrunsalongthemiddleofaroad.Themedianisalwaysinthemiddle,withanequal numberofvaluesoneitherside.

Mean

Youmighthavealreadyheardofthe mean,andknowthatitisalsocalledthe average Weusethemeanwhenwewanttomakeasimplestatementabouttheaveragevalue inapopulationorsample.Tocalculatethemean,weaddup(orsum)thevalues,then dividebythenumberofvalues.

Mean = sumofvalues numberofvalues

Sometimesyouwillseereportslikethis.

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ThisdoesnotmeanthatAustralianfamilieshave2wholechildrenandanother0.4 ofachild.

Itmeansthatmanyfamiliesweresurveyed.Someofthosefamilieshadnochildren, somehad1, 2or3childrenandothershadperhaps5or9children.Theremighthave beenfamilieswithothernumbersofchildren,too.

Forexample,thefamiliesofKidStreethave1, 7, 1, 0and3children.Themean (oraverage)numberofchildrenineachhouseinKidStreetis:

Mean = sumofvalues numberofvalues = 1 + 7 + 1 + 0 + 3 5 = 12 5 = 2 4

SothemeannumberofchildrenineachfamilyinKidStreetis2 4children.

Example3

Twenty-sixYear4studentsrecordedthenumberofchildrenintheirfamilies.

Usethisdatatocalculate:

a themode

b themediannumberofchildreninastudent’sfamily

c themeannumberofchildreninastudent’sfamily(roundedtotwodecimal places)

a Sortthedataintoafrequencytable.

Thevaluethatoccursthemostoftenis2.Sothemodeis2.

b Arrangethedatainorderfromsmallesttolargest,thenfindthemiddlevalue.

Thereisanevennumberofvalues,sowetaketheaverageofthemiddle twovalues. Median = 2 + 3 2 = 2.5

Themediannumberofchildreninastudent’sfamilyis2.5. c Sumofvalues

Mean = sum ofvalues numberofvalues = 77 26 = 2.96

Themeannumberofchildreninastudent’sfamilyis2 96.

16B Wholeclass LEARNINGTOGETHER

1 ThecanteenatRiverinaSchoolsellsfrozenoranges.Overfiveweeks (25schooldays),thenumberoffrozenorangessoldperdaywasrecorded. Thistableshowsthetallieddata.

Frozenorangesales

a Copythetableandcompleteitasaclass.

b Calculatethemode.

c Calculatethemean.

d Whatisthemedian?

e Discussthesequestionsasawholeclass.

• Howmanyorangesdoyouthinkthecanteenstaffshouldhaveinthe freezeratthebeginningoftheday?

• Thinkaboutthecalculationsyouhavemade.Wouldthemodebeagood valuetorelyupon?

• Doyouthinkthecanteenshouldhavea‘safetymargin’toavoid runningout?

16B Individual APPLYYOURLEARNING

1 Calculatethemode,meanandmedianforeachsetofdata.

a 1, 2, 2, 3, 4, 5, 5, 5, 6, 7, 9

b 2, 7, 4, 3, 5, 6, 2, 3, 4, 3

c 10, 100, 100, 10, 1000, 10, 1, 10, 1000

d 23, 12, 45, 12, 34, 33, 67, 59, 72, 43

2 ShelleyandDarrenkeptarecordofhowmuchtheyspenteachdayonpetrol, foodandotherexpensesduringtheirholiday.

Monday: $43.00Tuesday: $138.00Wednesday: $560.00

Thursday: $120 00Friday: $43 00

a Whatisthemode?

b Calculatethemedian.

c Calculatethemean.

d HowmuchperdayshouldShelleyandDarrenbudgetfortheirnext holiday?Shouldtheyrelyonthemode,meanormedian?Explainyour answer.

3 ThestudentsinMsMadison’sclasseachmeasuredtheirhandspanin centimetres.Herearetheresults.

14, 15, 13, 18, 20, 16, 17, 14

Write‘True’or‘False’toanswerthesestatementsaboutthehandspans.

a Themodeis20cm.

b Thereisnomode.

c Themodeis14cm.

d Themedianis15 5cm.

e Themeanis 15 875cm.

f Theaverageis 127cm.

4 a Themeanofasetofnumbersis12.Thesumofthenumbersis72.How manynumbersareintheset?

b Themeanofasetoffournumbersis10.Thenumbersarealldifferent. Whatmightthenumbersbe?Giveatleasttwopossibilities.

c Tahliahadfivetests.Eachtestwasmarkedoutof50.Inthefirstfourtests hermarkswere45, 42, 43and48.IfTahlia’saveragetestmarkwas44, whatwasthemarkonherfifthtest?

16C Presentingand interpretingdata

Dotplots

Adotplotisusedforcountdata,whereonedotdrawnaboveabaselinerepresents eachtimeaparticularvalueoccursinthedata.Dotplotsareusefulwhenwewantto seeataglancewhatthedatashows.

Example4

MrMudge’sYear6studentsrecordedhowmanyofthesevenbooksinthe Wizard seriestheyhadread.

7,6,1,6,0,7,3,4,5,2,6,7,7,1,4,0,2,5,7,7,7,7,7,2,0,3

a Organisethedataintoafrequencytable.

b Createadotplotforthedata.

Solution

(continuedonnextpage)

Columngraphs

Acolumngraphusescolumnsofdifferentlengthstocomparedifferentquantities.The columnscanbeeitherverticalorhorizontal,andtheymakeiteasytocompare differentvalues.Columngraphsarealsoknownasbargraphsorbarcharts. Thistableshowsdatacollectedaboutthenumberofdifferenttypesofbirdsfoundina ‘sample’siteatBoolLagoon,SouthAustralia,onChristmasDay,2006.

Thiscolumngraphshowsthesameinformationasthetable.

Bool Lagoon waterbird numbers

Type of bird

Number of birds

Thereisatitleatthetopofthecolumngraphsothatpeoplewillknowwhatitshows. Thecolumngraphhasahorizontalaxis(the x-axis)andaverticalaxis(the y-axis). Eachaxishasatitlesothatyouknowwhichdataarebeingcompared.The y-axison thischarthasascaleinmultiplesof10.

Side-by-sidecolumngraphs

Often,morethanonepieceofinformationiscollectedforeachcategory.Column graphscanbeusedtoshowdatafromtwocategoriesonthesamegraph.Thisisuseful whenwewanttocomparedatafromdifferentcategories.

Example5

Thistablegivestheamountofgovernmentspendingperpersononeducationand healthinfourtownsforoneyear.AllamountsaregiveninthousandsofAustralian dollars($000s).

Presentthisinformationinaside-by-sidecolumngraph.

Linegraphs

Linegraphsarecreatedbyplottingpoints,thendrawinglinesegmentstojointhe pointstogether.Linegraphsareoftenusedtodisplaydatasuchastemperature,where thedatagoesupanddown(orfluctuates)overaperiodoftime.Thelinejoiningthe pointsgivesusanideaofupanddownchanges.

Example6

ThesetemperatureswererecordedinMilduraononeJulyday.

a Displaythedatainalinegraph.

b Whatisthedifferencebetweenthetemperaturesat0600and1500?

c Describewhathappenedtothetemperatureovertheday.

b At0600thetemperaturewas3◦ C.At1500thetemperaturewas11◦ C.The differenceintemperatureis8◦ C.

c Thetemperaturerosegraduallyuntil1200,thenitdroppedby4◦ Cinone hour.Itthencontinuedtodropgradually,butnotasfast.

Piecharts

Piechartsareanotherwaytorepresentdata.Justasthenamesuggests,apiechartis likeawholepiethathasbeencutintodifferentportions,orpieces.

Piechartsareoftenconstructedusingpercentages.Thesizeofeachsliceofpieisin proportiontoitspercentageofthewholepie.Forexample,25%willberepresented byaslicethatis25%ofthewholepie,and50%willberepresentedbyaslicethatis 50%ofthewholepie.

Thispiechartrepresentsthenutritionalvalueofapizza.Fromthepiechartwecansee that33%ofacheesypizzaiscarbohydrateand12%ofitisfat.

Example7

Patrickwantedtoknowhowmuchtimehespenton differentactivitiesoverthecourseofoneday.He timedeachactivity,thenmadeapiecharttoshow thepercentageoftimehespentoneachactivityover oneday.

HowmuchmoretimedidPatrickspendsleepingthan atschool?

Time spent over one day

Solution

Patrickspent42% ofhistimesleepingand27% ofhistimeatschool.

42% –27 %= 15%

Patrickspent15% moreofhistimesleepingthanatschool.

16C

Wholeclass LEARNINGTOGETHER

1 Collectinformationaboutthefavouriteschoolsubjectofeachmemberof yourclass.Organisethisinformationintoafrequencytableanddisplayyour datainacolumngraph.

Askquestionsaboutyourgraph,suchas:

• Whichsubjectismostpopular?

• Whatpercentageofstudentslikeonesubject?

• Howmanystudentsarethereintotal?

• Whatcanyouconcludefromyourdata?

2 ThispiechartshowssalesofpastaattheTopValueSupermarket. Usetheinformationinthepiecharttoanswerthe questions.

a Whichwasthebest-sellingpasta?

b Halfofthepastasoldwasspaghetti.Thisis50% ofthetotalsales.Whatpercentageofsaleswas spiralpasta?

c Whichtwotypesofpastasoldthesameamount?

d Whatpercentageofsaleswasshells?

e TopValueSupermarketsold24packetsofspaghetti. Howmanypacketsofspirals,shellsandpennedid theysell?

3 Thislinegraphisgraphingtemperaturesoveraweek.

Discusshowthedatawouldbedisplayed.

a Whatelementswouldweneedtoaddtounderstandthegraph?

b Suggestatitleforthegraph,andtitlesforthe x-and y-axes,thenlabelthem.

c Whereshouldthedaysbewritten?

d Suggestthetemperaturescale.

Pasta sales

4 Lookatthegraphbelowandanswertrueorfalsetothestatements.

a ThegraphshowsthetemperaturetrendsinJapanandVietnamoveraspecific period.

b ThetemperaturesinJapanaregenerallylowerthanthoseinVietnamduring thesameperiod.

c ThegraphindicatesthatVietnamhascolderwintersthanJapan.

d ThetemperaturesinJapanandVietnamremainconstantthroughouttheyear.

e ThegraphindicatesthatbothJapanandVietnamexperiencetheirhighest temperaturesduringthesummermonths.

Individual APPLYYOURLEARNING

1 a FourstudentsweresurveyedaboutthenumberofhoursofTVtheywatched eachday.Thistableshowsthedata.Copyandcompletethetable.

Student Numberofhours

Marco

Anelle

Felicity

Trent

b Drawacolumngraphtorepresentthedata.

2 a Harrietdrewupthefollowingdatatableforthemoneysheearnedfromher holidayjob.

Drawalinegraphtoshowthedata.Writeatitleforthegraph,andlabelthe x-and y-axes.

Moneyearned

b HowmuchmoneydidHarrietearnin4hours?

c HowmuchmoneydidHarrietearnin10hours?

d HowmuchmoneydidHarrietearnin3.5hours?

3 Nelliesoldguitars.Shemadeadotplottorecordthenumbersoldeachday.

a Whatwasthegreatestnumberofguitarssoldononeday?

b Whatwasthemostfrequentlyoccurringnumberforguitarssold inoneday?

c OnhowmanydaysdidNelliecollectdata?

d Howmanyguitarsweresoldintotal?

e Whatwasthemeannumberofguitarssold?

TheLancasterfamilyhavewatertanksastheironlysourceofwater.

At4a.m.,oneoftheirtankshad300litresinit.Afteritrainedfrom5a.m.until 6a.m.,thevolumeofwaterinthetankwas400litres.

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

At8a.m.theLancasterswenttoworkandschool. Itrainedfrom2p.m.to3p.m.andthetankreceived70litresofwater.

At5p.m.theLancastersreturnedhome.Theyused1litretomake coffeeandtea.

At6p.m.theywateredtheirgarden,using74litresofwater.

a Copyandcompletethefollowingtable.

b Presentthisdatainalinegraphbyplottingthepointsandjoiningthemwith linesegments.

c Whatwasthevolumeofwaterinthetankat8a.m.?

d Howmuchwaterwasinthetankat6p.m.?

5 Meaganislearningtodrive.Shekeepsalogofthenumberofhoursshedrivesthe familycar.

Monday:1.5hoursTuesday:1hourWednesday:0hours Thursday:0 5hoursFriday:1hour

a Drawabarcharttorepresentthedata.

b CalculatethemeannumberofhoursthatMeaganspentdriving overtheweek.

c Iflearnerdriversshoulddriveanaverageof1hourperday,howmuchextra timeshouldMeaganspenddrivingonSaturdayandSundaytomakeher averageequalto1?

6 Jakerecordedthenumberoflollysnakesineachpacketheopened.

a Createadotplotforthisdata.

b Whatisthemostfrequentlyoccurringnumberoflollysnakesinapacket?

c Whatisthesmallestnumberoflollysnakesfoundinapacket?

7 Hannahrecordedhowmuchtimeshespenton differentactivitiesover24hours.Shepresented herdataasapiechart,recordingwholehours insteadofpercentages.

a Howmanyhourswerespent not sleeping?

b Whichtwoactivitiesusedthesameamount oftime?

c HowmanyhoursdidHannahspendreading andrelaxing?

d WritetheamountoftimeHannahspent relaxingasapercentageofherwholeday.

8 ThispiechartshowsdataaboutanimalscaredforattheJabiruanimalshelter.

a Ifthereare100animals,howmanyfat-taileddunnartsarethere?

b Ifthereare100animals,howmanyArnhemleaf-nosedbatsarethere?

c Ifthereare100animals,howmanyblackwallaroosarethere?

d Ifthereare200animals,howmanyKakadudunnartsarethere?

e Ifthereare50animals,howmanywesternquollsarethere?

cared for

9 Createapiecharttorepresentthisdata.

Animals

16D Datainthemedia

Dataisreallyimportantbecauseithelpsusmakesmartdecisions.Forexample, businessesusedatatodeterminewhatcustomerslike,andgovernmentsusedatato decidehowtohelppeople.Whenwelookatdata,wecanunderstandthingsbetter andmakemorelikelytobesuccessfulchoices.

Butsometimes,datacanbetrickyandnottellthewholetruth.

Twotypesofdatacanbecollectedforarangeofpurposes.

• Primarydataisdatayoucollectyourselfsuchasaskingyourclassmatesdirectly abouttheirfavouritesubjects.

• Secondarydataisdatathathasalreadybeencollectedsuchasreadingareportor lookingupinformationonline.

Bothtypesofdataareuseful,butitisimportanttoknowwheretheinformationcomes fromandhowitwascollected;thishelpsyouunderstandhowreliableandaccurate thedatamightbe.

Datacanbecollectedintwodifferentways:

• Census survey,whichcollectsdatafromeverypersoninagrouporpopulation.A censussurveywillbemoreaccuratebutwilltakealotmoretimeandeffortandis notalwayspossible.

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• Asample surveycollectsdatafromasmallgroupofpeoplewithinalargergroup.It isquickerandeasiertousebutneedstobecarefullydesignedtoensureaccuracy. Choosingagoodsampleisimportantsothattheresultsarereliableandrepresentive ofthelargergrouporpopulation.

Sometimes,thewaydataispresentedinthemediacangiveusthewrongimpression abouttheinformationbeingpresented.Whenamediareportsays‘Statisticshave shown…’youshouldbeaskingquestionssuch:

• Whichstatisticsarebeingquoted?

• Howwasthedatacollected?

• Howlargewasthesample?

• Aretheresultsbeingreportedaccurately?

Example8

TheDungareeDailyNewspublishedanarticleonwhythelocalskateparkshould beclosed.

Whenthearticlewaswritten,thepopulationofDungareewas9500.

Asurveywasconductedwith250residents,andtheresultsweredisplayedina graphtosupportthepark’sclosure.Basedonthisreport,thelocalcouncilstated, ’ThemajorityofDungareeresidentswouldlikethelocalskateparktobeclosed.’

a Wasthedatasourcecollectedprimaryorsecondary?

b Wasthisacensussurveyorsamplesurvey?

c Isthelocalcouncil’sstatementatruereflectionofDungaree’spopulation?

d InwhatwaywerethesurveyresultsnotafairreflectionofDungaree’s population?

Dungaree Skate Park Closure Sample Survey

a Primary

b Sample

c No,onlyasmallsampleofthepopulationwassurveyed.

d Thesamplesurveywasonlycollectedfromadults.Noteenagersorchildren wereasked.

Wholeclass LEARNINGTOGETHER

TheheadteacheratSilverCreek PrimarySchoolwantedtoreduce lunchplaytimefrom1hourto 45minutes.Therewere350students whoattendedtheschool,andthe headmastersurveyedoneYear2 classof30studentsandtheir parents,andallthe20teachers.

Theheadteachershowedagraphof theresultstothewholeschooland recommendedthatlunchplaytime beshortened.Ataschoolassembly, theysaid,‘TheSilverCreekPrimary

Schoolcommunityhasstronglysupportedchanginglunchplaytime.Startingnext term,ourlunchplaytimewillbe15minutesshorter.’

Questionsfordiscussion

1 Wasthedatacollectedprimaryorsecondarydata?

2 Wasthisacensussurveyorasamplesurvey?

3 Istheheadteacher’sstatementatruereflectionoftheSilverCreekschool community?Giveareasonforyouranswer.

4 Whatwouldbeafairerwaytocarryoutthesurvey?

Shorter lunchtime play survey – Silver Creek Primary School

16D Individual APPLYYOURLEARNING

1 Arethesurveysbelowlikelytobecensusorsamplesurveys?Explain youranswer.

• Asoftdrinkcompanyasks2000peopleiftheypreferOrangeFizzorApple Sparkle.

• Everyhouseholdinatownisaskedabouttheirrecyclinghabits.

• Aheadteacherasksallstudentsabouttheirpreferencefortheschool uniform.

• Alocalsportclubasks50supportersiftheyshouldupdatetheirfootball jerseys.

2 ThePEteacheraskedalltheYear6studentstoraisetheirhandstochoose whichsportstheywantedtoplayduringPElessons.Hewantedtopicktwo sportsforthewholeyearlevel.Aftereveryonevoted,hemadeapiegraph toshowtheresults.

a WhattypeofsurveydidthePEteacherconduct?

b Whattwosportswerechosenfortheyearlevel?

c Wasthisafairwaytocollecttheinformation?

d Whydoyouthinktheteacherchoseapiegraphtoshowresults?

e Whatwouldbeabetterwaytodisplaytheresults?

f Couldhemakeaninformeddecisiontoselectthetwosports?

3 Imagineanewsarticleclaimsthat90percentofpeopleloveanewmovie basedonasurvey.However,thesurveyonlyasked10people,and9ofthem saidtheylovedthemovie.Howmightthisdatabemisrepresented,andwhat shouldyouconsidertounderstandthetruepopularityofthemovie?

16E Reviewquestions–

1 Mikecollectedgolfballsatthelocalgolfcourseandsoldthembacktotheplayers for $1each.Thistableshowsthenumberofgolfballshecollecteddailyfor twoweeks.

Foreachweek,calculate:

a themediannumberofballscollected

b themeannumberofballscollected.

2 Year6studentsatBarryStreetSchooltookpartinasurvey.Eachstudentwas askedtwoquestions:

• HowmanyhoursofTVdidyouwatchlastweek?

• Howmanyhoursdidyouuseacomputerlastweek? Herearetheresults.

a Howmanychildrenwereinterviewed?

b WhatisthemodeforTVwatching?

c Whatisthemodeforcomputeruse?

d Calculatethemeannumberofhoursforeachdataset.

e WhatisthemeannumberofhoursspentwatchingTVorusingacomputer?

f Calculatethemediannumberofhoursoftelevisionwatched.

g Calculatethemediannumberofhourschildrenusedacomputer.

Anewspaperpublishedareportonmobilephone useforstudentsunder13yearsofage.Thetitle was’Studentswithmobilephonesdistract students.’Thenewspaperhadcontacted 50randomstudentsfromaschoolwith 1200students,20parentsand30teachers.

a Didthenewspaperrunacensusorsamplesurvey?

b Whatotherinformationwouldyouliketoknowtojudgeifthisisafairtitle?

c Whatmighttheproblemsbewiththedatausedtosupportthisstory?

d Ifyouweretoconductyourowninvestigationonthistopic,whatquestions wouldyouask,andwhatmethodswouldyouusetogatherdata?

4 Mikawantedtoselladollontheinternet.She watchedasthedoll’spriceincreased,andrecorded thepriceeachhalfhouruntiltheauctionclosed at7p.m.

a Showthisdataonalinegraph.

b Whatwasthefinalpriceforthedoll?

c Whathappenstothegraph’slinewhenthe priceremainsconstant?

d Whatwasthepriceat4∶30p.m.?

e IfMikacheckedthepriceat6∶15p.m.,what pricemightshehaveseen?

5 Tomcountedthenumberofyoungborntosomeofthefemalenail-tailwallabies inTauntonNationalParkin2001andin2011.

a Createadotplotforeachsetofdata.

b Whatdoyounoticeaboutthenumberofyoungbornin2001compared to2011?

16F Challenge–Ready,set,explore!

Statisticaldatainvestigation

Understandingstatisticsanddataisessentialbecauseithelpsusmakesenseof informationinnewspapersandonTV.Manyjobs,likebeingafootballstatisticianora biologist,usestatisticstounderstandtheworld.

Therearethreetypesofdata:

• datathatwecan count,suchasthenumberofjellybeansinapacket

• datathatwecan measure,suchastheheightofstudentsinyourclass

• datathatbelongsin categories,suchashairstylesorthecolourofcars.

Tolearnhowdataiscollectedandshown,doingsomeactivitiesyourselfisbest.Here’s asimpleprocesstofollow:

• Plan:Weplantheinvestigation,designquestionstostarttheinvestigationand identifythetypesofdatathatcouldbeinvolved.

• Collect:Wecollectandcheckthedata.

• Process:Wepresentandinterpretthedata.

• Discuss:Wediscusstheresults.

Plan:Identifyingissuesandplanningtheinvestigation

Inthefirstpartofthedatainvestigationprocess,wedecideonthetopicwewantto investigateandtheissuesrelatedtothattopic.Thenwedesignquestionsthathelpus findoutabouttheissues.

Weaskthefollowingquestions.

• Whatdowewanttofindoutabout?

• Whatdatacanweget?

• Howdowegetthedata?

Let’slookatthisstepthroughanexample.

MsDraper’sclass6Dreadanewspaperreportaboutthelongeststripofpapertorn fromalollywrapper.Thestudentswonderiftheycouldcollecttheirowndata. Theirinvestigationwillinvolvemeasurementdata,asthelengthistheimportant ideahere.Tocollectthedata,theywillaskeachstudentintheclasstotearapiece ofpaperintothelongeststriptheycan,andmeasurethestrip.

Theydecidetogiveeachstudenta5cm × 5cmsheetofpaper. Beforetheycollectthedata,thestudentsidentifyseveralissues.

• Theremightbedifferentwaystotearthestrip.

• Theymayneedtoallowacertainnumberofattempts.

• Whowillcheckthemeasurements?

• Whattools,ifany,canbeused?

Thestudentsdecideonthefollowingrules.

• Thestripcanbetorninanywaythepersondecides.

• Notoolsotherthanthehandscanbeusedtotearthestrip.

• Eachpersoncanhavethreeattempts.Ifthestripbreaks,thenthatmeasurement isrecordedas0.Dataforuptothreeattemptsmaybesubmitted.

• Oneotherpersonmustalsomeasurethestriptocheckthemeasurement. Theclasscreatetheirquestions.

• Whatisthelongestcontinuousstripofpaperthancanbetornfrom a5cm × 5cmpieceofpaperbypeopleinourclass?

• Whatisthemodeandmeanforthisdata?

• Howcanweusethisinvestigationtopractisewhatwehavelearntaboutdata collectionandinvestigation?

Next,thestudentsdecidewhatdataistobecollected.Thenameofeachperson, thelengthoftheirmeasurementtothenearestcentimetreandthenameofthe ‘checker’willberecordedinatable.

Collect :Collecting,handlingandcheckingdata

Oncewehavedecidedwhattocollectandhowitwillbecollected,wecanproceed withthedatacollection.Tableswithtalliesareausefulwaytocollectdata.

Thedatacollectedby6Dlookedlikethis:

Process:Exploringandinterpretingdata

Weneedtoorganisethedatausingfrequencytablesandpresentitinawaythathelps usunderstandit.Wecanworkoutthemode,medianandmean,andmakestatements fromthedatathathelpustounderstandit.

Thestudentsin6Dcreatedafrequencytableanddotplotfortheirdata.

27282930313233343536373839404142434445

Thestudentssawfromthedotplotthatthemodewas35.Thismeansthatthe mostfrequentlyoccurringvalueforthelengthofthetornpaperstripwas35cm. Theyalsoobservedthatthelongestlengthofpaperstriptornfroma5cm × 5cm squarewas45cm,andcongratulatedArthuronhiseffort.

ThemeanforthedatafromMsDraper’sclasswascalculatedbyaddingeachvalue anddividingbythenumberofvalues.Thezerovalueswerenotincluded.

Sumofvalues = 37 + 35 + 29 +

Mean = sumofvalues numberofvalues = 1785 53 = 33.67924 … cmor34cm (roundedtothenearestcentimetre)

Discuss:Discusstheresultsandposenewquestionsthatarisefromthem

Oncewehavedrawnsomeconclusionsfromthedata,wemightrealisethatfurther questionsmustbeanswered.Wemightorganisetheexistingdatadifferentlyorcollect moredata.Atthispoint,itisimportanttodiscusstheuseofgraphs:Werethegraphs chosenthebestforthispurpose?

Thestudentsin6Drealisedthatincludingthezerovalues(obtainedwhenapaper stripbroke)wouldlowertheirmean.Theyalsoknewthatpiegraphsandlinegraphs wereinappropriatechoicesforpresentingthisdata.

Chooseaninvestigationandfollowtheprocessabove

1 Favouritetelevisionprograms

Doboysandgirlshavedifferentpreferences?

Doyoungchildrenwatchdifferentprogramstoolderchildren?

2 Hoursspentdoingafter-schoolactivities

Isthereaparticulardayoftheweekwhenmorepeopledomoreactivities?

CHAPTER

Usefulskillsforthistopic

• Identifyanduselanguagerelatedtochanceandprobability.

• Listpossibleoutcomesofchanceexperiments,especiallythosewithequallylikely outcomes.

• Predictthecomparativelikelihoodofdifferentoutcomesforfamiliarevents.

Vocabulary

Probability • Possibilities • Chance • Comparative • Outcomes • Prediction • Let’sengage Let’sengage Let’sengage Let’sengage Let’s engage 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 Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage Let’sengage

Imagineyouarerollingasix-sideddie. Whatistheprobabilityofrollinganumbergreaterthan4?

• Howwouldyou describethisprobabilityusingfractions,decimals,andpercentages?

• Canyouthinkofothereverydayeventswhereyoucanapplytheseconceptsto estimatethelikelihoodofdifferentoutcomes?

Probability Probability Probability Probability ProbabilityProbability Probability Probability Probability ProbabilityProbability Probability Probability Probability ProbabilityProbability Probability Probability Probability Probability Probability ProbabilityProbability Probability Probability Probability Probability Probability Probability ProbabilityProbability Probability Probability Probability Probability Probability ProbabilityProbability Probability Probability Probability Probability Probability Probability Probability ProbabilityProbability Probability Probability Probability Probability Probability Probability ProbabilityProbability Probability Probability Probability Probability Probability Probability Probability ProbabilityProbability Probability Probability Probability Probability Probability Probability Probability ProbabilityProbability Probability Probability Probability

WhatisthechancethatthePrimeMinisterwillwalkintoyourclassroominthe next5minutes?Youmightsaythatthereis‘notaverygoodchance’.Oryou mightratethechance‘notverylikely’onascalelikethis: impossible not very likely likely highly likely certain

Inmathematics,weusetheword probability todescribethechanceofanevent occurring(ortakingplace).

17A Probability

Writingprobabilities

Youcandrawa probabilityscale (liketheoneshownabove)usingnumbersinsteadof words.Theprobabilityofaneventhappeningisexpressedonascalefrom0to1.

Aneventthatisrarehasaprobabilitycloseto0,whileaneventthatisverycommon hasaprobabilitycloseto1.

Eventsthatwillnothappenhaveaprobabilityof0.Eventsthatarecertaintohappen haveaprobabilityof1.

Eventsthathappensometimeshaveaprobabilitybetween0and1,andareoften writtenasfractions.Thenumeratoristhenumberofwaystheeventcanhappen,and thedenominatoristhetotalnumberofpossibilities:

Probabilityofanevent = numberofwaystheeventmayhappen totalnumberofpossibilities

Forexample,theprobabilityofrollinga3onadieis‘oneinsix’or 1 6 .Thisisbecause thereisonly1waytorolla3,but6differentnumberscouldshowup.

Theprobabilityofrollinganoddnumberonadieis‘threeinsix’or 3 6,asthereare 3waystorollanoddnumberoutof6differentnumbersthatcouldcomeup.

Example1

Ifwerolla6-sideddienumberedfrom1to6,whatistheprobabilityofrolling:

a thenumber7?

b anumberfrom1to6?

c thenumber2?

Solution a numberofwayseventmayhappen totalnumberofpossibilities = 0 6 (7isnotonthedie) (6numbersonthedie)

Theprobabilityofa7appearing=0

b numberofwayseventmayhappen

totalnumberofpossibilities = 6 6 (1to6areallonthedie) (6numbersonthedie)

Theprobabilityofanumberfrom1to6appearing = 6 6 = 1

c numberofwayseventmayhappen

totalnumberofpossibilities = 1 6

Theprobabilityofa2beingrolled = 1 6 (1number2onthedie) (6numbersonthedie)

Thenextexampleshowsthatprobabilitiescanalsobewrittenasdecimalsor percentages.

Example2

Thereare10marblesinabag.Themarblesare differentcolours.Thereare4bluemarbles,3green marbles,2redmarblesand1whitemarble.Suppose thatyouselectoneofthemarbleswithoutlooking. Whatistheprobabilityofselecting:

a abluemarble?Writeyouranswerasadecimalor percentage.

b awhitemarble?Writeyouranswerasadecimalor percentage.

Solution

a numberofwayseventmayhappen

totalnumberofpossibilities = 4 10 (4bluemarbles) (10marblesinthebag) = 0.4 or 40%

Sotheprobabilityofpickingabluemarbleis0.4.

b numberofwayseventmayhappen

totalnumberofpossibilities = 1 10 (1whitemarble) (10marblesinthebag) = 0.1 or 10%

Sotheprobabilityofpickingawhitemarbleis0.1.

Expectedoutcomes

Sometimes,theprobabilitieswepredictarenotwhatweseewhenwedotheactual experiment.

Forexample,ifweweretoconductthemarbleexperimentasin Example2 (onthe previouspage)wemight expect toselectabluemarble4outof10times,butwe mightobserveadifferentresult.

17A Wholeclass LEARNINGTOGETHER

1 Tossingcoins

a Makeaprediction.Ifyoutossedtwocoins50times,howoftenmightyou throw2tails?

b Tosstwocoins50times.Keepatally,thenwritedownthetotalnumber oftimesyoutossedeachcombination.

2heads 1tail,1head 2tails

c Compareyourresultswithyourclassmates’results.

d Dothemaths:Howmanydifferentcombinationsarepossiblewhentossing twocoins?

Copythistwo-waytableanduseittohelpyou.

17A Individual APPLYYOURLEARNING

1 Rollingonenormaldie

Predictthefractionof30rollsofadiethateachnumber(1−6)ona6-sideddie willoccur.Writedownyourpredictedprobabilityasafractionbetween0and1. Nowcarryouttheexperiment,workingingroupsoffour.Rollthedie30times andrecordtheresultsinatable.Eachtimeanumberisrolled,recorditbydrawing atallymarkinthetable.After30rolls,recordthetotalnumberoftimeseach numberwasrolled.

Discussyourresults.Didyourpredictionmatchyourresults?Whathappensifyou rollthedie60times?

2 Thesumoftwodice

a Jenniistryingadiceexperiment.Sheisrollingtwodicethenfindingtheirsum. Shesaysthat6isthemostcommonsumfromrollingtwodice.Makea prediction.DoyouthinkJenniisright?Ifnot,whichsumdoyouthinkwillbe themostcommon?

b Dotheexperiment.Rolltwodice30times,recordingtheirsuminachartlike theoneonthefollowingpage.Colouraboxonthecharteachtimethesum isrolled.

Thischartshowsthatforthefirst10throws:2wasrolledonce,3wasrolled twice,7wasrolledsixtimes,and11wasrolledonce.

c Dothemaths:inhowmanydifferentwaysisitpossibletomakeeachnumber from2to12byaddingthenumbersonthefacesoftwo6-sideddice?Copy andcompletethischart.Thedifferentwaystomake2,6and10havebeen doneforyou.

d Howdoesthechartinpart c comparetoyourexperimentinpart b?Whydo youthinkthatsomepeoplebelievethat7isaluckynumber?

17B Challenge–Ready,set,explore!

Basketballchanceinvestigation

Inabasketballgame,aplayertakesshotsatthe basket.Theplayermightmaketheshotormissthe basket.Inthisinvestigation,wewillsimulatetaking 25freeshotsatthebaskettoseehowoftenthe playerscores.Wecanthencomparetheresultstoa real-worlddataset.

IntheNBA,theaveragegoalpercentageisaround 45–50%everytimeaplayerattemptsashot.

Ifaplayerhasafreethrow,theaveragegoal percentageisaround75–80%.

Duringabasketballgame,ateamusuallytakes around25freeshots.Wewilltake25freeshotsat thebaskettosimulateagame.

Materialsneeded

• paperandpencil

• randomnumbergenerator(a10-sideddieoracomputerprogramto randomlyselectnumbers)

Setuptheexperiment

1 Definetheoutcomes

• Numbers1to7representasuccessfulfreethrow

• Number8to10representamissedfreethrow

2 Calculatetheexpectedprobability

• Probabilityofmakingafreethrow 7 10 = 0.7(70%)

• Probabilityofmissingafreethrow 3 10 = 0 3(30%)

3 Createarandomnumbergenerator

• Usea10-sideddieoracomputertogeneraterandomnumbersbetween 1and10tosimulatetheoutcomesofeachfreethrow.

4 Collectingyourdata

• Discusswithpartnerhowtocollectthedatafromeachfreethrowe.g.atable, tallymarks.

5 Recordyourdataonatablesimilartothisone:

6 Convertingdata

• Countthetotalnumberoftallymarksforeachrandomnumber.Thisgivesthe frequencyofeachnumber.

• Combinethefrequenciesoftherandomnumbersthatrepresentamake (1 7) to findthetotalnumberofsuccessfulfreethrows.

• Combinethefrequenciesoftherandomnumbersthatrepresentamiss (8 10) to findthetotalnumberofmissedfreethrows.

Ourexampledata:

Totalmakes:15

Totalmisses:10

7 Calculateobservedprobabilityofmakingormissingthefreethrow

• Dividethefrequencyofeachoutcome(makeormiss)bythetotalnumberof shots.Thisisexpressedasafraction.

• Converttodecimalsandpercentages.

• Actualprobabilitytoexperimentalprobability.Supposeintheexperimentaldata set,theplayermade15shotsoutof25freeshots.

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Experimentalprobabilityofmakingafreethrowis 15 25 = 0.6or60%

Experimentalprobabilityofmissingafreethrowis 10 25 = 0 4or40%

Analysis

• Comparetheexperimentalprobabilitieswiththeactualprobabilities.

• Discussanydiscrepanciesandpotentialreasonse.g.randomchance,playerskill, gameconditionsthatmightaffecttheoutcome.

• Considerrunningadditionalsimulationstoseeifresultscovertheactualprobabilities withlargersamplesizes.

• Doestheexperimentdesignreflectthefreethrowaveragegoalpercentageinthe NBA?Ifnot,howcoulditbechangedtoimproveit?

CHAPTER

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Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating algorithmicthinking Incorporating

Definition:

An algorithm isafinite,unambiguoussequenceofinstructionsforperforming aspecifictask.

Thismaysoundlikeacomplicateddefinition,butitissomethingwithwhichyou areveryfamiliar.Forinstance,whenyoufollowarecipeformakingacake,you mustfollowtheinstructionsinorder(orsequence)andtheymustbeveryclear (thatis,unambiguous)oryoumaymisinterpretsomeofthem.Finally,itmustbea finitelist,oritwilltakeforevertomakethecake!

Therearemoremathematicalexamples,suchastheprocedureforaddingtwo fractions.Theremaybeseveralalgorithmsthatsolvethesameproblem,butifyou followanyoneofthemcarefully,youwillarriveatthecorrectresult.

Weareinterestedinalgorithmsbecausetheyarethebasisofprogramminga computer,orarobot,todowhatyouwishittodo.Inthiscase,thealgorithms needtobewritteninaparticularformwhichthecomputerorrobotwill understand,calledaprogramminglanguage.Inthischapter,wewillusethe language Scratch towriteourprograms.

Theaimsofthischapterareprimarilytodevelopyourabilityto:

• understandthepurposeofalgorithmsandtheiruseinsolvingproblems,and

• writeshortprogramstoimplementsomesimplealgorithmsusingScratch.

Weconcentratehereonusingalgorithmicthinkinginmathematicsproblems,but thiswayofthinkingcanbeusedinmanyothersubjects.Inparticular,whenever wethinkacomputermighthelpusinsolvingaproblem,weneedtouse algorithmicthinkingtodescribeourprobleminawaythatthecomputercan understand.

18A FunctionMachines

YouhaveprobablyplayedtheMachineGame,whereyouhaveamathematical machinewhichdoessomethingtothenumbersyouputinandgivesanoutput number.Yourtaskistoworkoutwhatthemachineisdoing.

Inthisactivity,themachineisperformingasimplealgorithmoneachinputnumber. Onceyouhavefiguredoutwhatthemachineisdoing,youcoulddescribethe algorithmusinga flowchart.Aflowchartisadiagramforsettingoutasequenceof instructionsintheorderinwhicheachoftheinstructionsistobeapplied.Hereisa flowchartforthemachineabove.

Soifwewanttofindthemissingoutput,weputthenumber5intoourflowchart

Theresultor output is16.Thenumberyouthoughtofisthe input.Youmaythinkthis isaverycomplicatedwayofdoingsomethingyoucanprobablydoinyourhead.but weneedtolearnhowtobreakanalgorithmdownintosmallstepssothatwecan designmorecomplicatedalgorithmsefficiently.Also,sometimeswearetoldtheoutput numberandaskedfortheinputnumberwhichgivesthisoutput.Whenwehave brokenthemachinedownintoseparatesteps,thiscaneasilybedownbyreversingthe flowchartandundoingeachofthestepsinturn.

Forinstance,areversalofthemachineabovewouldgiveaflowchartlikethis:

So,ifweknewtheoutputwas22,wecouldcalculatetheinputasfollows:

Example1

a Drawaflowchartforthefollowinginstructions.

Thinkofanumber • Multiplyby7 • Subtract4 •

b Givetheresult(output)whenthethenumberyouthinkofis

3 i 5 ii 2 iii

c Givetheinputnumberifyouknowtheoutputnumberis

3 i 24 ii 4 iii

Solution

a Start Think of a number Multiply by 7 Subtract 4 Stop

bi 3 ×7 ⟶ 21 4 ⟶ 17.Theouputis17.

ii 5 ×7 ⟶ 21 4 ⟶ 31.Theouputis31.

iii 2 ×7 ⟶ 14 4 ⟶ 18.Theouputis 18.

ci 1 ÷7 ⟵ 7 +4 ⟵ 3.Theinputis1.

ii 4 ÷7 ⟵ 28 +4 ⟵ 24.Theinputis4.

iii 0 ÷7 ⟵ 0 +4 ⟵ 4.Theinputis0.

18A Individual APPLYYOURLEARNING

1a Drawaflowchartforthefollowinginstructions.

Thinkofanumber • Add3 • Divideby2 •

b Givetheresult(output)whenthenumberyouthinkofis

3 a 5 b 1 c

c Givetheinputnumberifyouknowtheoutputnumberis 7 a 13 b 3 c

2a Drawaflowchartforthefollowinginstructions.

Thinkofanumber • Multiplyby5 • Add1 • Divideby3 •

b Givetheresult(output)whenthenumberyouthinkofis 4 i 13 ii 2 iii

c Givetheinputnumberifyouknowtheoutputnumberis 2 i 13 ii 3 iii

3a Drawaflowchartforthefollowinginstructions.

Thinkofanumber • Add2 • Multiplyby2 • Subtract2 • Divideby2 •

b Givetheresult(output)whenthenumberyouthinkofis 1 i 7 ii 2 iii

c Givetheinputnumberifyouknowtheoutputnumberis 4 i 24 ii 3 iii

4a Drawaflowchartforthefollowinginstructions.

• Thinkoftwopositivewholenumbers

• Multiplythenumberstogether

• Add2

b Givetheresultwhenthetwoinputnumbersare 2and3 i 4and5 ii 1and7 iii

c Findallpossiblepairsofinputnumberswhentheoutputis 13 i 12 ii 14 iii

d Explainwhyitisnotalwayspossibleinthisquestiontoknowpreciselywhatthe inputnumbersare.

5a Drawaflowchartforthefollowinginstructions.

• Thinkoftwopositivewholenumbers

• Multiplythefirstnumberby4

• Multiplythesecondnumberby3

• Addthetworesults

b Givetheresultwhenthetwoinputnumbersare 2and3 i 4and5 ii 1and7 iii

c Findallpossiblepairsofinputnumberswhentheoutputis 13 i 14 ii 19 iii

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d Explainwhyitisnotalwayspossibleinthisquestiontoknowpreciselywhatthe inputnumbersare.

18B Scratchprogramming

YoucanfindScratchonlineat https://scratch.mit.edu/.Youmaywellhave usedScratchtodrawpatternsorcreateanimations.However,wewillbefocusing,for themoment,onhowmathematicaloperationsareperformedinScratch.Whenyou firstopenScratchyoucanclickon Create inthetopmenutobeginwritinganew program.

Youwillseeascreenliketheoneabovedividedinto3sections.Theright-handsection iswheretheactiontakesplace-thatis,whereyourprogramwillactuallyrunwhen youstartit.Theleft-handsectioniswhereyouwillfindallofthecommandsand instructions(calledblocks)youcanusetowriteyourprogram.Notethattheyare grouped(andcolourcoded)intoblockcategories.Thesearethendraggedasrequired intothemiddlesectionwhereyouactuallyputtogetheryourprogram.

Wewillconstructasimpleprogramwhichperformsasthefirstmachineflowchartdid intheprevioussection.Todothis,weneedtobeabletodofourthings

• Inputanumber

• Performthemachineoperationontheinputnumber

• Outputtheresult

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• Understandhowtorunourprogram.

Firstly,wecanaskauserofourprogramtoinputanumberusingtheaskcommand whichisintheSensingcategory.

Thiscommandallowsyouto decidethequestionyouare goingtoaskbyeditingthe scriptinthewhitebox.

Edit‘What’syourname’to ‘What’syournumber’by typinginthewhitebox.

Whentheprogramisrun,itwillwaituntiltheuserhasansweredthequestionbefore placingtheinputinaboxcalled‘answer’.(Youwillaccessthislaterinyourprogram through answer whichisalsofoundintheSensingmenu.)

Next,weperformourmathematicaloperationwiththeinputanswer.Thisisdonewith theOperatorscategoryofcommands.Youwillseeinthiscategorycommandsto performallofthebasicoperationsaswellasvariousotherthings. Wefirstneedtomultiplyby 3andthentoadd1,sowe willneedamultiplyoperator andanaddoperator.We alsobringtheanswerblock tothatsectionofthescreen.

Wealsoneedtodefineavariablecalled‘output’whichwill beusedtostoreouranswer.ThisisdoneintheVariables category.FirstclickontheboxMakeavariable andtype inthenameofourvariable‘output’.

Wenowwanttocalculatethevalueofoutput.Weusethe ‘set’command(alsointheVariablescategory)todothis. Firstlyweusethemultiplyoperatortomultiplytheanswer by3andinsertthisintothe‘set’command.Thisputsthe inputfrom‘answer’into‘output’.

Wenextadd1to‘output’ whichhasthevaluefromthe previousstepandsetthisas thenewvalueof‘output’.

Wenowwishtodisplaytheoutputandwecandothisusingthe‘say’commandwhich isinthelooksmenu.Wecanjustputtheoutputbox(whichwillhaveour‘output’ value)intothe‘say’command,butthiswilljustgivethenumberwithnowords,sowe usea join operatorwhichallowsustoputsomewordsintothefirstpartandthe outputintothesecondpart.

Finally,wecontroltherunningoftheprogrambyputtingtheGreenFlagcommandat thebeginning.Thismeansthatwhenevertheuserclicksonthegreenflaginthe runningwindow,theprogramwillbegin.

YoucanseetheprogrambeingcreatedandrunningbyclickingtheGreenFlagicon.

Thenumberenteredis45and3 × 45 + 1 = 136.Thevalueofeachofthe variables,‘answer’and‘ouput’areshown.Youobtainthisbytickingtheboxnext tothem.

18B Individual APPLYYOURLEARNING

1 CopytheScratchprogramaboveandmodifyittoproducetheoutputof Question 1a inExercise18A.Checkthatyourprogramworkscorrectlybytesting withthedatagivenandseeingthatyouranswersarecorrect.

2 WriteaScratchprogram(ormodifyyourpreviousone)toproducetheoutputof Question 2a inExercise18A.Checkthatyourprogramworkscorrectlybytesting withthedatagivenandseeingthatyouranswersarecorrect.

3 Modifytheprograminthepreviousquestionto‘undo’theprocess.Thatis,ifyou weregiventheoutputnumberasaninput,itwouldproducetheinputnumberas anoutput.Checkthatyourprogramworkscorrectlybytestingwiththedatagiven andseeingthatyouranswersarecorrect.

4 WriteaScratchprogram(ormodifyapreviousone)toproducetheoutputof Question 3a inExercise18A.Checkthatyourprogramworkscorrectlybytesting withthedatagivenandseeingthatyouranswersarecorrect.

5 Modifytheprograminthepreviousquestionto‘undo’theprocess.Thatis,ifyou weregiventheoutputnumberasaninput,itwouldproducetheinputnumberas anoutput.Checkthatyourprogramworkscorrectlybytestingwiththedatagiven andseeingthatyouranswersarecorrect.

6 WriteaScratchprogramwhichinputstwopositivenumbers,addsthemtogether andaddstwo(asinQuestion 4a fromExercise18A).

7 InventyourownfunctionmachineandwriteaScratchprogramtoproducethe correctoutputforyourmachine.Workingwithapartner,trytofigureoutwhat eachother’smachinesaredoing(youcanhidetheinstructionsbyclickinginthe topright-handcorner).

18C Sequences

Amathematicalsequenceisalistofnumbersgeneratedbyarule.Therearetwoways ofwritingrules.Thefirstistousea recursive formula.Thisiswhereeachnumber(or term)ofthesequenceisgeneratedfromtheterm(orterms)whichcomebeforeit. Thisiswhathappenswhenyoudoskipcounting.Forexample,ifyouareaskedtoskip countin3sstartingat8andgoingupto,butnotbeyond30,yougeneratethe sequence

8, 11, 14, 17, 20, 23, 26, 29

Theruleis: Startingat8,eachtermis3morethanthepreviousterm,stopifthenext termismorethan30.

Towritethismoremathematically,wewillintroduceanotationforthetermsofa sequence.Wecallthefirstterm t1,thesecondterm t2 andsoon,withthe nthterm referredtoas tn.Nowwecanalsowritethedescriptionabovemathematicallyas

t1 = 8(thefirsttermis8)

tn+1 = tn + 3(Weadd3toanytermtogetthenextterm). ThiscanalsoberepresentedinaflowchartasshownExample3.

Example2

Givearecursivedefinition,thatis, tn+1 equaltoanexpressionwith tn,whichwould producethefollowingsequences. 1, 6, 11, 16,

Solution

a Thefirsttermis t1 = 1.Eachfollowingtermisobtainedbyadding5tothe previousterm. Thereforeapossiblerecursivedefinitionis:

tn+1 = tn + 5

b Thefirsttermis t1 = 2.Eachfollowingtermisobtainedbymultiplyingthe previoustermby3. Thereforeapossiblerecursivedefinitionis:

tn+1 = 3 × tn

Example3

Drawaflowchartwhichillustrateshowtoconstructthesequencewithfirstterm8 andfurthertermsobtainedbyadding3tothepreviousterm.

Solution

Thesecondtypeofruleisgivenbyan explicit formula.Thisisasingleformulawhich willgiveyouthe nthtermofasequencewhenyouput n intotheformula.Anexample isthesequenceofsquarenumberswhichcanbedefinedbytherule tn = n2.Inthis case,asbefore, tn standsforthe nthtermofthesequence,so,forinstance t3 isthe thirdtermand,bytheformula, t3 = 3 × 3 = 9.Thefirsttentermsofthesequenceare

Thiscanberepresentedinaflowchartliketheoneshownintheexamplebelow.

Example4

Drawaflowchartwhichillustrateshowtoconstructthesequenceofthefirstten squarenumbers.

Solution

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18C Individual APPLYYOURLEARNING

1 Givearecursivedefinition,thatis, tn+1 equaltoanexpressionwith tn,whichwould producethefollowingsequences.

2 ForeachofthesequencesinQuestion1,canyoufindanexplicitformulatodefine thesequence?

3 Drawaflowchart,thenwriteoutthesequencegivenbyeachofthefollowing rules:

a Startingat2,eachtermis10morethanthepreviousone,stopifthenextterm ismorethan100.

b Startingat1,eachtermis3timesthepreviousone,stopifthenexttermis morethan100.

c Startingat25,eachtermis3lessthanthepreviousone,stopifthenexttermis negative.

4 Giveanexplicitformulawhichwillgeneratethefollowingsequences.

5 ForeachofthesequencesdefinedinQuestion3,canyoufindarecursive definition?

WecancreateasequencelikethefirstexampleaboveinScratch.Todothis,weneed tounderstand lists and loops.

Lists

Alistisexactlywhatyouwouldexpectittobe.Itisasetofoutputswhicharewritten verticallyonthescreen.Wefirstneedtocreatealistvariableandthisisdoneinthe Variablescategory.Assoonasyounamealistvariableyouwillseeaboxforthelist appearintheStagescreen(youcantickorunticktoindicatewhetheryouwishthisto appearornot).

Loops

AloopinScratchallowsyourepeataspecificblockofcodemultipletimesuntila conditionismet.Therearenumberofdifferentwaysofcontrollingloops.Theyareall intheControlcategory.

Inthefollowingexample,wewillusethe ‘repeat … until’ commandwhichwillcarry ongoingroundtheloopuntilaconditionismet.ItisaprograminScratchwhich followsthestructureoftheflowchartinExample3.

Example5

WriteaScratchprogramtoproducethesequencebyrepeatedlyadding3:

Solution

Defineavariable,‘output’asintheearliersectionanddefinealistcalled‘Term’in asimilarwayasshownhere.

Youcanusewhatevernameyoulikeforyourvariablesbutitusuallymakessense togivethemnameswhichhelpyoutorememberwhattheydo. Ascreenforthelistimmediatelyappearsasshown

Wenowputtheprogramtogether.Westartbygivingouroutputthefirstvalue (inthiscase8).Thenweentertheloopwhichaddstheoutputtothelistand thenincreasesitsvalueby3.Itcarriesondoingthisuntilthevalueoftheoutput isgreaterthan30,thenitstops.Thedeletecommandatthebeginningofthe programistheretoclearthelisteachtimeyouruntheprogram.

Toproduceasequencedefinedexplicitlyiseveneasier.FortheflowchartinExample4, weusealoopwhichrepeats10times(aswewant10terms).Wefirstinitialisethe variablewearegoingtouse(inthiscase n)andinlcudeaninstructiontoclearthelist atthestart.Insidetheloopweadd n × n tothelist(calledtermintheexample)and increase n by1.

WriteaScratchprogramtoproducethefirst10squarenumbers.

Example6

18D Individual APPLYYOURLEARNING

1 CopytheScratchprogramtoreproducetherecursivesequenceaboveandmodify ittoproducetheoutputofeachofthefollowingsequences.

2 WriteaScratchprogram(ormodifyyourpreviousone)toperformtheoperation ofeachofthefollowing:

a Startingat2,eachtermis10morethanthepreviousone,stopifthenextterm ismorethan100.

b Startingat1,eachtermis3timesthepreviousone,stopifthenexttermis morethan100.

c Startingat25,eachtermis3lessthanthepreviousone,stopifthenexttermis negative.

3 WriteaScratchprogram(ormodifyyourpreviousone)toproducetheoutputof eachofthefollowingsequences,usingexplicitdefinitions. 2, 4, 6, 8, 10, 12, 14, 16

4 Inventyourownsequenceanddefineitrecursively.WriteaScratchprogramto producethecorrectoutputforyoursequence.Workingwithapartner,tryto figureoutwhattherulesareforeachother’ssequences.Nowdothesamefora sequencedefinedexplicitly.

Youmayhaveconductedsomeprobabilityexperimentslikethisone: Rolladie100timesandcounthowmanytimesyourolla6.WecanuseScratchto conducttheseexperimentsmuchmorequickly.First,let’slookataflowchartwhich describestheexperiementasanalgorithm.

Noticethatwehavesetuptwovariableshere.One‘Count’tokeeptrackofhow manytimeswehaverolledthedieandanother‘Sixes’tokeeptrackofhowmanysixes weroll.

InScratch,wehaveanoperatorcalled, pickrandom,whichwillgenerateanumber randomly(likerollingadie)between1andanywholenumberwechoose(inthiscase wewillchoose6).Wecanalsoavoidhavingacountvariablebysimplyusingarepeat loopwhichgoesround100times(oranyothernumberweputinthebox). Theothernewfeatureofthisprogramisthe ‘if’ statement.Thismeansthat instructionsinsideareonlycarriedoutifthestatementistrue(inthiscase,weonlyadd 1tothenumberofsixesifwerolla6).Notethatthefirstinstructionwhichsetsthe valueofsixesto0isnecessarysothateachtimeyouruntheprogramitstartscounting from0.Thisiscalled initialising

18E Individual APPLYYOURLEARNING

1 CopytheScratchprogramaboveandrunitseveraltimes.Whyistheoutputnot alwaysthesame?Modifyittocountthenumberoffivesrolledin6000rollsofa die,changingvariablenameswherenecessary.Howmanytimesdoyouexpectto rolla5?

2 WriteaScratchprogram(ormodifyyourpreviousone)tocountthenumberof tensrolledona10-sideddie,rolled500times.Istheanswerwhatyouexpect?

3 Nowintroduceaseconddieroll(youcouldcallit‘diceroll2’andwriteaprogram whichcountsthenumberoftimesadouble-sixisrolledin360rollsofthedie. [hint:youwillneedtousethe‘and’operatorinsideyourifstatement.Howmany timeswouldyouexpecttorolladouble-six?

4 WriteaScratchprogramtorolltwodice600timesandcounthowoftenthe numbersonthedicearethesame(thatis,howmanydoublesyouroll).Howmany wouldyouexpect?

5 WriteaScratchprogramtorolltwodice720timesandcounthowoftenthe numbersonthediceaddto2.Repeattheprogramfordifferentsums.Makea tableofyourresults.Areallsumsequallylikely?Describethepatternsyoufind.

18F

Stringart

Inthissection,wewillprovideanexampleofaloopstructure producingcreativeoutput.

Penmenu

Tounderstandthis,youneedtounderstandsomeofthepen commandsavailableinScratch.

Withthese,youcandrawonthescreenaccordingtoinstructions giveninyourprogram.The‘pendown’and‘penup’commands shouldbeself-explanatory.ToobtainthePenmenuclicktheblue ‘AddExtension’buttoninthebottom-leftcorneroftheScratch editorandselectPen.Thereareothercommandswhichallowyou tocontrolthesizeandcolourofthepenandyoucanexperiment withthese.

3rd

Motionmenuandgrid

Thefigure(orSprite)thatnormallyappearsintheStagescreen hasbeenhiddenheresoyoucanfocusjustonthedrawing.The positionofthecurserisstillvisible.Youcanmakeiteasiertosee wherethecurserisonthescreenbychoosingagridbackground (calledabackdrop).Todothis,clickBackdropbeneaththeStage screenandselectoneofthegridoptions.

Wecancontrolthecurserpositionusingthemotion commands.Thesecommandsletyou:

• movethespriteforwardagivendistance(move)

• turnthespritebyasetangle(turn)

Youcanexperimentwiththeseinstructionstoseehowthey behave.Inthisactivity,wewillonlyusethe’goto’command. Thissendsthemousetoaspecificcoordinate.

We’veincludedtwovariables, x and y,tohelpcontrolthecoordinateposition.Be carefultounderstanddifferencebetweenthevalueofthevariable x andthe x-coordinateonthegrid.

Tobegin,weset x to0and y to150.Next,westartaloop.Eachtimetheloopruns:

• y decreasesby5

• x increasesby5

Wedothisusingthechangecommand.Theprogramcontinuestoloopuntil y reaches0.

Ineachloop,theprogramdrawsalinefrom(0, y)to(x,0).As x and y change,the linesslide downthe y-axisandacrossthe x-axis.Thiscreatesacurvedpattern.

Weuse‘pendown’and‘penup’sothatthelinesareonlydrawninonedirection.You cancopythisprogramandexperimentwithdifferentchangevalues.

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1 CopytheScratchprogramaboveandrunit,makingsureyouunderstandwhat eachinstructiondoes.Modifythestartingvalueof y to 160andthechange valueof y to5.Whatdifferencedoesthismake?

2 Modifyyourprogrambychangingthe x changevalueto 5.Whathappens?Now combinesectionsofprogramtoproduceapieceofstringartinallfourcorner sectionsofthegrid.

3 SeeifyoucandeviseyourownartworkusingthePencommandsandaloop structure.

18G Algorithmicthinking problems

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

1 CircuitSolver Amathematicalcircuitconsistsofwiresjoininganumberofgates. Eachgatetakestwoinputnumbersandcreatesoneoutputnumber.Theaddition gateaddsthetwonumbers.Thesubtractiongatesubtractsthesmallernumber fromthelargerone,sotheanswerisalwayspositiveorzero.Themultiplication gatemultipliesthetwonumbers.Themaximumgateoutputsthelargerofthetwo numbers.Theminimumgateoutputsthesmallerofthetwonumbers.

Hereisanexampleofeachgatewiththeinputs2and5.Notethatthepositionof thetwoinputsdoesnotmatter.Theoutputwouldbethesameiftheinputswere 5and2.

Wirescanbesplittoprovidethesameinputtodifferentgates.Theycanalsocross eachotherwithoutaffectingthenumbers.

Theinputsinthefollowingcircuitare3and7.Thefinaloutputis20.Thegatesare labelledexceptforthepinkonewithaquestionmark.

Whattypeofgateisthepinkonewithaquestionmark?

2 GridRobot Arobotismovingaroundagrid.Everysquareonthegridisshaded oncetherobothasbeenonit.ItunderstandstheinstructionsF,R,XandYwhere:

• Ftellstherobottomoveforwardonesquareinthedirectionitisfacing.

• Rtellstherobottostayonthesquareandturn90°totherightfromthe directionitisfacing.

• XtellstherobottoFFRFRR.

• YtellstherobottoXXXX.

Forexample,FRFwouldshadethispatternofsquares:

Therobotisfacingright.WhatcouldthepatternlooklikeforY?