Essential Maths Core for the Australian Curriculum 2ed 8 - final pages by cambridge_aused

ESSENTIAL MATHEMATICS CORE

FOR THE AUSTRALIAN CURRICULUM

SECOND EDITION

David Greenwood

Bryn Humberstone

Justin Robinson

Jenny Goodman

Jennifer Vaughan

Stuart Palmer

TEXTBOOK ADVANCE COPY

March 2025

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3 Fractions,decimalsandpercentages

3G Convertingfractions,decimalsandpercentages

4 Measurement

Measurement,Number

8 Statisticsandprobability

8A Interpretinggraphsandtables

8B Frequencytablesandtallies

8C Graphsoffrequencytables

8D Rangeandmeasuresofcentre

8E Surveyingandsampling

8G Two-stepexperiments

8H Treediagrams

8I Venndiagrams

8J Two-waytables 527

8K Experimentalprobability

9A TheCartesianplane

9B Usingrulesandtablestoexplorelinearrelationships

9C Plottingstraightlinegraphs

9E Usinggraphstosolvelinearequations

9F Usinggraphstosolvelinearinequalities 591

9G Gradient

9H Applicationsoflineargraphs

9I Non-lineargraphs

10C Rotation

10D Congruentﬁgures

10E Congruenttriangles

10F Tessellations

10G Congruenceandquadrilaterals 678

10H Similarﬁgures

10I Similartriangles

Maths@Work:Fashiondesigner

AbouttheAuthors

DavidGreenwood istheHeadofMathematicsatTrinityGrammarSchoolinMelbourneand has30+ years’experiencemathematicsfromYear7to12.Heistheleadauthorfor theCambridgeEssentialseriesandhasauthoredmorethan80titlesfortheAustralian Curriculumandforthesyllabusesofthestatesandterritories.Hespecialisesinanalysing curriculumandthesequencingofcoursecontentforschoolmathematicscourses. Healsohasaninterestintheuseoftechnologyfortheteachingofmathematics.

BrynHumberstone graduatedfromtheUniversityofMelbournewithanHonoursdegreein PureMathematics,andhas20+ years’experienceteachingsecondaryschoolmathematics. HewasaHeadofMathematicsfrom2014–2024attwoindependentschoolsinVictoria. Brynispassionateaboutapplyingthescienceoflearningtoteachingandcurriculum design,tomaximisethechancesofstudentsuccess.

JustinRobinson isco-founderof TheWellbeingDistillery,anorganisationcommittedtoenergising, equippingandempoweringteachers.Heconsultswithschoolsgloballyontheﬁeldsof studentandeducatorwellbeing.Priortothis,Justinspent25yearsteachingmathematics, coveringalllevelsofsecondaryeducationincludingteachingVCE,IBandA-Levels.His drivingpassionwasengagingandchallengingstudentswithinasafelearningenvironment. JustinisanHonoraryFellowoftheUniversityofMelbourne’sGraduateSchoolofEducation.

JennyGoodman hastaughtinschoolsforover28yearsandiscurrentlyteachingataselective highschoolinSydney.Jennyhasaninterestintheimportanceofliteracyinmathematics education,andinteachingstudentsofdifferingabilitylevels.ShewasawardedtheJones MedalforeducationatSydneyUniversityandtheBourkePrizeforMathematics.Shehas writtenforCambridgeMATHSNSWandwasinvolvedintheSpectrumandSpectrumGoldseries.

JenniferVaughan hastaughtsecondarymathematicsforover30yearsinNewSouthWales, WesternAustralia,QueenslandandNewZealandandhastutoredandlecturedinmathematics atQueenslandUniversityofTechnology.Sheispassionateaboutprovidingstudentsofallability levelswithopportunitiestounderstandandtohavesuccessinusingmathematics.Shehas hadextensiveexperienceindevelopingresourcesthatmakemathematicalconceptsmore accessible;hence,facilitatingstudentconﬁdence,achievementandanenjoymentofmaths.

StuartPalmer wasbornandeducatedinNSW.Heisafullyqualiﬁedhighschoolmathematics teacherwithmorethan25years’experienceteachingstudentsfromallwalksoflifein avarietyofschools.HehasbeenHeadofMathematicsintwoschools.Heisverywell knownbyteachersthroughoutthestatefortheprofessionallearningworkshopshe delivers.StuartalsoassiststhousandsofYear12studentseveryyearastheyprepare fortheirHSCExaminations.AttheUniversityofSydney,Stuartspentmorethana decaderunningtutorialsforpre-servicemathematicsteachers.

Acknowledgements

Theauthorandpublisherwishtothankthefollowingsourcesforpermissiontoreproducematerial: Cover: © GettyImages/Westend61.

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Introduction

Thesecondeditionof EssentialMathematicsCOREfortheAustralianCurriculum hasbeensigniﬁcantlyrevisedandupdatedtosuit theteachingandlearningofVersion9.0oftheAustralianCurriculum.Manyoftheestablishedfeaturesoftheserieshavebeen retained,buttherehavebeensomesubstantialrevisions,improvementsandnewelementsintroducedforthiseditionacrossthe print,digitalandteacherresources.

Newcontentandsomerestructuring

Newcontenthasbeenaddedatallyearlevels.In Year7,thereisnewcontentonratiosandproportions,volumeoftriangular prisms,netsofsolidsandmeasurementrelatingtocircles.Allgeometrytopicsarenowcontainedinasinglechapter(Chapter7).In Year8,thereisnewcontentonorderofoperations,3D-coordinates,operationswithnegativefractions,areasofsectorsand compositeshapes,Pythagoras’theorem,inequalities,similarﬁgures,two-stepexperimentsandtreediagrams.For Year9,thereis newcontentonerrorsinmeasurement,inequalities,factorisation,samplingandproportion,quadraticsexpressionsandparabolas. In Year10,thereisnewcontentoncompositesolids,errorsinmeasurement,networksandlogarithmicscales.

Version9.0placesincreasedemphasison investigations and modelling,andthisiscoveredwithrevisedModellingactivitiesat theendofchaptersanddownloadableInvestigations.Therearealsomanynewelaborationscovering FirstNationsPeoples’ perspectives onmathematics,rangingacrossallsixcontentstrandsofthecurriculum.Thesearecoveredinasuiteofspecialised investigationsprovidedintheOnlineTeachingSuite.

Othernewfeatures

• Technologyandcomputationalthinking activitieshavebeenaddedtotheendofeverychaptertoaddressthecurriculum’s increasedfocusontheuseoftechnologyandtheunderstandingandapplicationofalgorithms.

• Targetedskillsheets –downloadableandprintable–havebeenwrittenforeverylessonintheseries,withtheintentionof providingadditionalpracticeforstudentswhoneedsupportatthebasicskillscoveredinthelesson,withquestionslinkedto workedexamplesinthebook.

• EditablePowerPointlessonsummaries arealsoprovidedforeachlessonintheseries,withtheintentionofsavingthetime ofteacherswhowerepreviouslycreatingthesethemselves.

Diagnosticassessmenttool

Alsonewforthiseditionisaﬂexible,comprehensivediagnosticassessmenttool,availablethroughtheOnlineTeachingSuite.This tool,featuringaround10,000newquestions,allowsteacherstosetdiagnosticteststhatarecloselyalignedwiththetextbook content,viewstudentperformanceandgrowthviaarangeofreports,setfollow-upworkwithaviewtohelpingstudents improve,andexportdataasneeded.

Guidetotheworkingprograms

EssentialMathematicsCOREfortheAustralianCurriculum9.0 containsworkingprogramsthataresubtlyembeddedinthe exercises.ThesuggestedworkingprogramsprovidetwopathwaysthroughthebooktoallowdifferentiationforBuildingand Progressingstudents.

EachexerciseisstructuredinsubsectionsthatmatchtheAustralianCurriculum9.0proﬁciencystrands(withProblem-solving andReasoningcombinedintoonesectiontoreduceexerciselength),aswellas‘Goldstar’( ).Thequestions* suggestedforeach pathwayarelistedintwocolumnsatthetopofeachsubsection.

• Theleftcolumn(lightestshade)showsthequestionsintheBuildingworkingprogram.

• Therightcolumn(darkestshade)showsthequestionsintheProgressingworkingprogram.

Gradientswithinexercisesandproﬁciencystrands

Theworkingprogramsmakeuseoftwo gradientsthathavebeencarefullyintegrated intotheexercises.Agradientrunsthrough theoverallstructureofeachexercise–where there’sanincreasinglevelofsophistication requiredasastudentprogressesthrough theproficiencystrandsandthenontothe ‘GoldStar’question(s)–butalsowithineach proficiencystrand;thefirstfewquestions inFluencyareeasierthanthelastfew,for example,andthefirstfewProblem-solvingand reasoningquestionsareeasierthanthelastfew.

Therightmixofquestions

Questionsintheworkingprogramshavebeenselectedtogivethemostappropriatemixoftypesofquestionsforeachlearning pathway.StudentsgoingthroughtheBuildingpathwayaregivenextrapracticeattheUnderstandingandbasicFluencyquestions andonlytheeasiestProblem-solvingandreasoningquestions.TheProgressingpathway,whilenotchallenging,spendsalittleless timeonbasicUnderstandingquestionsandalittlemoreonFluencyandProblem-solvingandreasoningquestions.TheProgressing pathwayalsoincludesthe‘Goldstar’question(s).

Choosingapathway

Thereareavarietyofwaysofdeterminingtheappropriatepathwayforstudentsthroughthecourse.Schoolsandindividual teachersshouldfollowthemethodthatworksbestforthem.Ifrequired,theWarm-upquizatthestartofeachchaptercanbe usedasadiagnostictool.Thefollowingarerecommendedguidelines:

• Astudentwhogets 40% orlowershouldheavilyrevisecoreconceptsbeforedoingtheBuildingquestions,andmayrequire furtherassistance.

• Astudentwhogetsbetween 40% and 75% shoulddotheBuildingquestions.

• Astudentwhogets 75% andhighershoulddotheProgressingquestions.

Forschoolsthathaveclassesgroupedaccordingtoability,teachersmaywishtoseteithertheBuildingorProgressingpathwaysas thedefaultpathwayforanentireclassandthenmakeindividualalterationsdependingonstudentneed.Forschoolsthathave mixed-abilityclasses,teachersmaywishtosetanumberofpathwayswithintheoneclass,dependingonpreviousperformance andotherfactors.

* Thenomenclatureusedtolistquestionsisasfollows:

3,4:completeallpartsofquestions3and4

• 1–4:completeallpartsofquestions1,2,3and4

• 10(½):completehalfofthepartsfromquestion 10(a,c,e,.....orb,d,f,.....)

• 2–4(½):completehalfofthepartsofquestions2,3and4

• –:completenoneofthequestionsinthissection.

• 4(½),5:completehalfofthepartsofquestion4 andallpartsofquestion5

•

Guidetothisresource

PRINTTEXTBOOKFEATURES

1 NEW Newlessons: authoritativecoverageofnewtopicsintheAustralianCurriculum9.0intheformofnew,road-tested lessonsthroughouteachbook.

2 AustralianCurriculum9.0: contentstrands,sub-strandsandcontentdescriptionsarelistedatthebeginningofthechapter (seetheteachingprogramformoredetailedcurriculumdocuments)

3 Inthischapter: anoverviewofthechaptercontents

4 NEW Quickreference: Multiplication,primenumber,fractionwallanddivisibilityrulestablesatthebackofthebook

5 Chapterintroduction: setscontextforstudentsabouthowthetopicconnectswiththerealworldandthehistoryof mathematics

6 Warm-upquiz: aquizforstudentsonthepriorknowledgeandessentialskillsrequiredbeforebeginningeachchapter

7 Sectionslabelledtoaidplanning: Allnon-coresectionsarelabelledas‘Consolidating’(indicatingarevisionsection)or withagoldstar(indicatingatopicthatcouldbeconsideredchallenging)tohelpteachersdecideonthemostsuitablewayof approachingthecoursefortheirclassorforindividualstudents.

8 Learningintentions: setsoutwhatastudentwillbeexpectedtolearninthelesson

9 Lessonstarter: anactivity,whichcanoftenbedoneingroups,tostartthelesson

10 Keyideas: summarisestheknowledgeandskillsforthesection

11 Workedexamples: solutionsandexplanationsofeachlineofworking,alongwithadescriptionthatclearlydescribesthe mathematicscoveredbytheexample.Workedexamplesareplacedwithintheexercisesotheycanbereferencedquickly, witheachexamplefollowedbythequestionsthatdirectlyrelatetoit.

12 Nowyoutry: try-it-yourselfquestionsprovidedaftereveryworkedexampleinexactlythesamestyleastheworkedexample togivestudentsimmediatepractice

differentiatedquestionsetsfortwoabilitylevelsinexercises

14 Puzzlesandgames: ineachchapterprovideproblem-solvingpracticeinthecontextofpuzzlesandgamesconnectedwith thetopic

15 Gentlestarttoexercises: theexercisebeginsatUnderstandingandthenFluency,withtheﬁrstquestionalwayslinkedto theﬁrstworkedexampleinthelesson

16 Chapterchecklist: achecklistofthelearningintentionsforthechapter,withexamplequestions

17 Chapterreviews: withshort-answer,multiple-choiceandextended-responsequestions;questionsthatare‘GoldStar’are clearlysignposted

18 Maths@Work: asetofextendedquestionsacrosstwopagesthatgivepracticeatapplyingthemathematicsofthechapter toreal-lifecontexts

19 NEW Technologyandcomputationalthinking activityineachchapteraddressesthecurriculum’sincreasedfocusonthe useofdifferentformsoftechnology,andtheunderstandingandimplementationofalgorithms

20 Modellingactivities: anactivityineachchaptergivesstudentstheopportunitytolearnandapplythemathematicalmodelling processtosolverealisticproblems

INTERACTIVETEXTBOOKFEATURES

21 NEW TargetedSkillsheets,oneforeachlesson,focusonasmallsetofrelatedFluency-styleskillsforstudentswhoneed extrasupport,withquestionslinkedtoworkedexamples

22 Workspaces: almosteverytextbookquestion–including allworking-out–canbecompletedinsidetheInteractive Textbookbyusingeitherastylus,akeyboardandsymbol palette,oruploadinganimageofthework

23 Self-assessment: studentscanthenself-assesstheir ownworkandsendalertstotheteacher.Seethe Introductiononpagexformoreinformation

24 Interactivequestiontabs canbeclickedonsothat onlyquestionsincludedinthatworkingprogramare shownonthescreen

25 HOTmathsresources: ahugecateredlibraryofwidgets, HOTsheetsandwalkthroughsseamlesslyblendedwith thedigitaltextbook

26 Desmosgraphingcalculator,scientiﬁccalculatorand geometrytoolarealwaysavailabletoopenwithinevery lesson

27 Scorcher: thepopularcompetitivegame

28 Workedexamplevideos: everyworkedexampleis linkedtoahigh-qualityvideodemonstration,supporting bothin-classlearningandtheﬂippedclassroom

29 Arevisedsetof differentiatedauto-marked practicequizzes perlessonwithsavedscores

30 Auto-markedmaths literacyactivitiesteststudents ontheirabilitytounderstandandusethekey mathematicallanguageusedinthechapter 29

31 Auto-markedpriorknowledgepre-test (the‘Warm-upquiz’oftheprintbook)fortestingtheknowledgethatstudents willneedbeforestartingthechapter

32 Auto-markedprogressquizzesandchapterreviewquestions inthechapterreviewscanbecompletedonline

DOWNLOADABLEPDFTEXTBOOK

33 InadditiontotheInteractiveTextbook,a PDFversionofthetextbook hasbeenretainedfortimeswhenuserscannotgo online.PDFsearchandcommentingtoolsareenabled.

ONLINETEACHINGSUITE

34 NEW DiagnosticAssessmentTool included withtheOnlineTeachingSuiteallowsforﬂexible diagnostictesting,reportingandrecommendations forfollow-upworktoassistyoutohelpyour studentstoimprove

35 NEW PowerPointlesson summariescontainthe mainelementsofeachlessoninaformthatcanbe annotatedandprojectedinfrontofclass

36 LearningManagementSystem withclass andstudentanalytics,includingreportsand communicationtools

37 Teacherviewofstudents’workand self-assessment allowstheteachertoseetheir class’sworkout,howstudentsintheclassassessed theirownwork,andany‘redﬂags’thattheclass hassubmittedtotheteacher

38 Powerfultestgenerator withahugebankof levelledquestionsaswellasready-madetests

39 Revampedtaskmanager allowsteachersto incorporatemanyoftheactivitiesandtoolslisted aboveintoteacher-controlledlearningpathways thatcanbebuiltforindividualstudents,groupsof studentsandwholeclasses

40 Worksheets,Skillanddrill,mathsliteracy worksheets,and twodifferentiatedchapter testsineverychapter,providedineditableWord documents

41 Moreprintableresources: allPre-testsand Progressquizzesareprovidedinprintableworksheet versions

1 Computationwith integers

Essentialmathematics:whybeingabletocomputewith integersisimportant

Inanemergency,ambulanceparamedicsoftenadministerintravenousﬂuidstoresuscitatea patient.Theyusemultiplicationanddivisiontocalculatedriprates,volumesandtimes.

Electronicsengineersandtechnicianscomputesquaresandsquarerootswhendesigningaudio ampliﬁerstoproduceastrongenoughmusicalsignaltorunloudspeakers.

Boardgamedesignersusedivisibilityrulesinvariouswaystomakegamesmoreengagingand complex.Scoringsystemsandplayerprogressionrulesbasedonwhetheraplayer’sscoreis divisiblebyacertainnumber,encouragestrategicplay.Also,distributingskillpointsevenlyacross charactersensuresequalopportunityforall.

Skiers,snowboardersandmountainclimbersselecttheirgearaccordingtotemperatureforecasts. OnewintermorningatAustralia’sThredboTopStation,thetemperaturewas −6°Cbutwindsof 72 km/hhelpedtocauseafreezingcold‘feelslike’temperatureof −21°C.

Inthischapter

1AAddingandsubtracting positiveintegers (Consolidating)

1BMultiplyinganddividing positiveintegers (Consolidating)

1CSquares,cubesandother powers

1DNumberproperties

1EDivisibilityandprime factorisation

1FNegativeintegers (Consolidating)

1GAddingandsubtracting negativeintegers

1HMultiplyinganddividing negativeintegers

1IOrderofoperationsand substitution

AustralianCurriculum9.0

NUMBER

Usethe4operationswithintegersand withrationalnumbers,choosingand usingefﬁcientstrategiesanddigital toolswhereappropriate(AC9M8N04)

Usemathematicalmodellingtosolve practicalproblemsinvolvingrational numbersandpercentages,including ﬁnancialcontexts;formulateproblems, choosingefﬁcientcalculationstrategies andusingdigitaltoolswhere appropriate;interpretand communicatesolutionsintermsofthe situation,reviewingtheappropriateness ofthemodel(AC9M8N05)

©ACARA

Onlineresources

Ahostofadditionalonlineresources areincludedaspartofyourInteractive Textbook,includingHOTmathscontent, videodemonstrationsofallworked examples,auto-markedquizzesand muchmore.

1 Decideifthefollowingexpressionsrelateto A addition (+) S subtraction ( ) M multiplication (×) or D division (÷) Totalsum a Difference b 10 morethan 7 c 10 lessthan 13 d 5 groupsof 3 e

3sin 18 f

2 Completetheseadditions.

3 Completethesesubtractions.

4 Completethesemultiplications.

5 Completethesedivisions.

Listtheﬁrst 5 multiplesof 6. a

Listtheﬁrst 4 multiplesof 9 b

Whatisthelowestcommonmultiple(LCM)of 6 and 9? c

7 Listallthefactorsof 12. a

Listallthefactorsof 15 b

Whatisthehighestcommonfactor(HCF)of 12 and 15? c

8 Primenumbershaveexactlytwofactors.Oftheﬁrst 15 positiveintegers(listedbelow),listthe numberswhichareprime.Writeyourprimesinascending(increasing)order.Theﬁrstprime iscircled.

1 2 3456789101112131415

9 Answerthefollowingastrue(T)orfalse(F). 2 + 3 × 4 = 2 + 12 a 10−8 ÷ 2 = 10−4 b (5−2) × 7 = 3 × 7 c 9 × 3 + 5 = 9 × 8 d 9 × (3 + 5) = 9 × 8 e 12 ÷ 3 × 4 = 1 f

10 Statethemissingnumbersforeachpartinthistable.

a 2 × 2 = √4 = 2

b 3 × 3 = √9 =

c 4 × 4 = √16 =

d 6 × 6 = √36 =

e 9 × 9 = √ = 9

f 10 × 10 = √ = 10

g × = 49 √49 = h × = 144 √144 =

11 Whatarethenexttwonumbersineachofthesepatterns?

3, 2, 1, , a

2, 0, −2, , b −9, −10, −11, , c

12 Usethisnumberlinetohelpﬁndtheanswer.

2−5 a 0−3 b

−4 + 6 c

−2 + 7 d

CONSOLIDATING

1A 1A Addingandsubtractingpositive integers

Learningintentions

• Tounderstandthecommutativelawforaddition.

• Tobeabletousethementalstrategiesofpartitioning,compensatinganddoublingtocalculateasumordifference ofwholenumbersmentally.

• Tobeabletousetheadditionandsubtractionalgorithmsto ndthesumanddifferenceofwholenumbers.

Keyvocabulary: sum,difference,algorithm,commutativelaw,compensating,doubling,countingon

ThenumbersystemthatweusetodayiscalledtheHindu–Arabicordecimalsystem.Itusesthedigits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

Thevalueofeachdigitdependsonitsplaceinthenumber,so,forexample,the 4 in 3407 hasaplace valueof 400.Wholenumbersinclude 0 (zero)andthecounting(natural)numbers 1, 2, 3, 4, … TheCounting numbers 1, 2, 3, 4, … arecalledpositiveintegers.Wecanaddorsubtractwholenumberstoﬁndsumsand differences.

Lessonstarter:Sumanddifference

Useaguess-and-checkmethodtotrytoﬁndapairofnumbersdescribedbythesesentences.

• Thesumoftwonumbersis 41 andtheirdifferenceis 11

• Thesumoftwonumbersis 41 andtheirdifferenceis 1

Describethemeaningofthewords‘sum’and‘difference’.Discusshowyoufoundthepairofnumbers ineachcase.

Keyideas

Youcanaddinanyorder.

e.g. 7 + 5 = 5 + 7 9 + 3 + 1 = 9 + 1 + 3

• Thisiscalledthe commutativelaw foraddition.

Youcannotsubtractinanyorder.

e.g. 7−5 ¢ 5−7

Ifthenumbersarelarge,writenumbersincolumnsanduseknown algorithms tocalculatethe answer.

1 Matcheachofthequestionsintheleft-handcolumn(a, b, c and d)totheworkingoutintheright-hand column(I, II, III, IV).

Thetotalof 156, 94 and 6 a 2491 + 945 I Take 856 awayfrom 2491 b 2491−856 II 945 morethan 2491 c

less 863 d

2 Writeeachofthefollowingusinganaddition (+) orasubtraction ( ) signinsteadofthewords.Donot workouttheanswer.

26 plus 17 a 43 takeaway 9 b

134 minus 23 c

Thesumof 19 and 29 e

Thedifferencebetween 59 and 43 g

36 morethan 8 i

32 lessthan 49 k

451 add 50 d

Thesumof 111 and 236 f

Thedifferencebetween 339 and 298 h

142 morethan 421 j

120 lessthan 251 l

3 Describethesesumsanddifferencesastrue(T)orfalse(F)?

Fluency

Example1Usingmentalarithmetic

Evaluatethisdifferenceandthesesumsmentally.

Nowyoutry

Evaluatethisdifferenceandthesesumsmentally.

Thismethodiscalledcompensating.

Thismethodiscalleddoubling.

Thismethodiscalledcountingon.

5 Completethesedifferences.

4 Completethesesums. HintforQ4andQ5: Dothesewithoutacalculator oralgorithm.

6 Evaluatethesesumsanddifferencesmentally.

+ 25 e

Example2Usinganalgorithm

Useanalgorithmtoﬁndthissumanddifference.

a 9138 + 217

b ✁ 113 ✁ 411

Nowyoutry

Useanalgorithmtoﬁndthissumanddifference.

8 + 7 = 15 (carrythe 1 tothetenscolumn) 1 + 3 + 1 = 5 9 + 2 = 11

Borrowfromthetenscolumnthensubtract 6 from 11.Nowborrowfromthehundreds columnandthensubtract 8 from 13

7 Useanalgorithmtoﬁndthesesumsanddifferences.

HintforQ7:Carrythe 1 for sumslargerthan 9 and borrow‘ten’forsubtraction.

Problem-solving and reasoning

8 Aracingbike’sodometershows 21432 kmatthestartof araceand 22110 kmattheendoftherace.Howfarwas therace?

9 Kristianhas $246 morethanSally.Davidhas $56 lessthanSally.IfSallyhas $492,howmuchdoKristian andDavideachhave?

10 Callumwalks 15 kmonMondayand 3 kmmoreeachday.HowmanykilometresdoesCallumwalkon Thursday?

11 Thesumoftwonumbersis 39 andtheirdifferenceis 5.Whatisthelargernumber?

Fillingthegapandmagictriangles

12a Writethedigitmissingfromthesesumsanddifferences.

b Findthemissingdigitsinthesesumsanddifferences.

c Thesidesofamagictriangleallsumtothesametotal.

Showhowitispossibletoarrangeallthedigitsfrom 1 to 9 sothateachsideaddsto 17. i

Showhowitispossibletoarrangethesamedigitstoadifferenttotal.Howmanydifferent totalscanyouﬁnd?

CONSOLIDATING

1B 1B Multiplyinganddividingpositive integers

Learningintentions

• Tounderstandthecommutativeanddistributivelawformultiplication.

• Tobeabletousementalstrategiestocalculatesimpleproductsandquotients.

• Tobeabletousethemultiplicationanddivisionalgorithmsto ndtheproductandquotientofwholenumbers.

Keyvocabulary: product,quotient,remainder,distributivelaw,commutativelaw

Multiplyinganddividingaretwokeyoperationsinmathematicsandareusefulinmanypracticalsituations suchasﬁndingthecostof 9 ticketsat $109 eachorthenumberoftrucksneededtocarry 280 tonnes ofcoal.

Lessonstarter:Multiplicationordivision?

Insolvingmanyproblemsitisimportanttoknowwhethermultiplicationordivisionshouldbeused. Decideifthefollowingsituationsrequiretheuseofmultiplicationordivision.Discusstheminagroup orwithapartner.

• Thenumberofcookies 4 peoplecangetifapacketof 32 cookiesissharedequallybetweenthem.

• Thecostofpaving 30 squaremetresofcourtyardatacostof $41 persquaremetre.

• Thenumberofsheetsofpaperin 4000 boxesof 5 reamseach(1 reamis 500 sheets).

• ThenumberofhoursIcanaffordaplumberat $75 perhourifIhaveaﬁxedbudgetof $1650 Makeupyourownsituationthatrequirestheuseofmultiplicationandanotherfordivision.

Keyideas

A product istheresultofmultiplication.

Multiplicationcanbedone: mentally

e.g. 6 × 5 = 30

• using an algorithm e.g. 217 26 1302 2176 4340 21720 5642 13024340 × × × + •

Youcanmultiplynumbersinanyorder.

e.g. 6 × 5 = 30 and 5 × 6 = 30

• Thisisthe commutativelaw formultiplication.

The distributivelaw ishelpfulwhenmultiplying.

e.g. 5 × 34 = 5 × (30 + 4)

= 5 × 30 + 5 × 4

= 150 + 20 = 170

Usingdivisionresultsinﬁndinga quotient anda remainder

e.g. 38113 ÷ = and 5 remainder

dividend divisor quotient or 38 ÷ 11 = 3 5 11

Divisioncanbedone: mentally e.g. 56 ÷ 8 = 7

Exercise1B

usinganalgorithm 7 732 ) 512214

Und er stand ing 1–3 3

1 Matcheachofthequestionstotheworkingoutontheright.

Theproductof 9 and 6 a 15 × 12 I

36 dividedby 12 b 15 ÷ 5 II 15 lotsof 12 c 9 × 6 III

Thequotientwhen 15 isdividedby 5 d 15 ÷ 12 IV

Divide 12 into 15 e 36 ÷ 12 V

2 Useyourknowledgeofthemultiplicationtablestoanswerthefollowing.

5 × 8 a 11 × 9 b 6 × 7 c 9 × 8 d 11 × 6 e 12 × 11 f 8 × 4 g 7 × 9 h

3 Arethesesimpleequationstrue(T)orfalse(F)?

4 × 13 = 13 × 4 a 2 × 7 × 9 = 7 × 9 × 2 b 6 ÷ 3 = 3 ÷ 6 c 60 ÷

Fluency

Example3Usingmentalstrategiesformultiplication

Useamentalstrategytoevaluatethefollowing.

5 × 160 a

Solution

HintforQ2:You shouldknowmostof theseoffbyheart.

Explanation

a 5 × 160 = 800 Tomultiplyby 5 youcanmultiplyby 10 thenhalvetheresult. 160 × 10 = 1600, 1600 ÷ 2 = 800

b 7 × 89 = 623 89 = 90−1 Â 7 × 89 = 7 × 90−7 × 1 = 630−7 = 623 (thisisthedistributivelaw)

c 5 × 43 × 2 = 430 5 43 25243 1043 430 ××=×× =× = look for easy pairs

Nowyoutry

Useamentalstrategytoevaluatethefollowing.

4 Useamentalstrategytoevaluatethefollowing.

HintforQ4:Dothese mentally.

Example4Usingmentalstrategiesfordivision

Useamentalstrategytoevaluatethefollowing.

Solution

Explanation

a 464 ÷ 4 = 116 Todivideby 4 youcandivideby 2 twice. 464 ÷

halvingthenumber)

b 480 ÷ 5 ÷ 2 =

Nowyoutry

Useamentalstrategytoevaluatethefollowing.

5 Useamentalstrategytoevaluatethefollowing.

Dividingby 5 andthenby 2 isthesameas dividingby 10 480 ÷ 10 = 48

HintforQ5:Chooseoneofthe mentalstrategiesdescribed above.

Example5Usinganalgorithmformultiplicationanddivision Useanalgorithmtoevaluatethefollowing.

412 × 25 a 938 ÷ 13 b

Solution

Explanation a 412 × 25 2060 8240 10300

b 13 72 ) 9328 rem 2

So 938 ÷ 13 = 72 and 2 remainder.

938 ÷ 13 = 72 2 13

412 × 5 = 2060 and 412 × 20 = 8240 Addthesetwoproductstogettheﬁnal answer.

93 ÷ 13 = 7 and 2 remainder

28 ÷ 13 = 2 and 2 remainder

Wewriteremaindersasfractions 72 2 13

Nowyoutry Useanalgorithmtoevaluatethefollowing. 137 × 12 a 354 ÷ 7 b

6 Useanalgorithmtoevaluatethefollowing.

HintforQ6:Usethesettingout describedin Example5

7 Usetheshortdivisionalgorithmtoevaluatethefollowing.Writeyouranswerusingfractionsifthereis aremainder.

Problem-solving and reasoning

8 Auniversitystudentearns $550 for 20 hoursofwork.Whatisthestudent’spayrateperhour?

9 Packetsofbiscuitsarepurchasedbyasupermarket inboxesof 12.Thesupermarketorders 220 boxes andsells 89 boxesinoneday.Howmanyboxesare left?Howmanypacketsofbiscuitsremaininthe supermarket?

10 Rileybuysafridgewhichhecanpayforbythe followingoptions.

9 paymentsof $183 A $1559 upfront B Whichoptionischeaperandbyhowmuch?

11 Theshovelofagiantexcavatorcanmove 6 tonnesofrockineachload.Howmanyloadsare neededtoshift 750 tonnesofrock?

12 Tomsaves $362 aweek.Howmuchwillhesavein 52 weeks? Maximumtickets 13

13 Achildtickettoatheatreis $7 andanadultticketis $12. Findthecostof 2 adultand 3 childtickets. a Findthecostof 1 adultand 5 childtickets. b Genspendsexactly $90 tobuychildticketsandadulttickets.Findthemaximumnumberoftickets thatGencouldpurchase.

1C 1C Squares,cubesandotherpowers

Learningintentions

• Tounderstandthemeaningofanexpressionwrittenintheform an intermsofrepeatedmultiplicationof a.

• Tobeableto ndthesquare,squareroot,cubeandcuberootofcertainsmallwholenumbers.

Keyvocabulary: base,index,power,indexnotation,expandedform,product,square,squareroot, cube,cuberoot

Inmathematicstherearemanywaystoabbreviateexpressions.

Usingrepeatedaddition, 4 + 4 + 4 + 4 + 4 canbewrittenas 5 × 4 usingmultiplication.

Usingrepeatedmultiplication, 3 × 3 × 3 × 3 canbewrittenas 34 usingindexnotation. Weread 34 as“3 tothepowerof 4”.

Lessonstarter:Squarenumbers

Canyouexplainwhywecallthenumbers 1, 4, 9 and 16 squarenumbers?

Drawdiagramsforthenexttwosquarenumbers.

Usecenticubestobuildtheﬁrstthreecubenumbers.Write downthenextcubenumber.

Keyideas

Indexnotation

expanded form base index or power

The base of 3 showsthefactorthatisrepeatinginmultiplication,andthe power or index isthe numberoftimesitappears.

The square ofanumberiswritten a2 anditmeans a × a

e.g. 52 means 5 × 5 (wesay 5 squared,thesquareof 5,or 5 tothepowerof 2)

Theoppositeofsquaringisﬁndingthe squareroot ofanumber.Thesymbol √ means squareroot.

e.g. √9 = 3 as 32 = 9

• Thesquarerootofanumberisalwayspositiveorzero.

The cube ofanumber a is a3 = a × a × a

e.g. 53 = 5 × 5 × 5 (wesay 5 cubed,or, 5 tothepowerof 3)

Theoppositeofcubingistakingthe cuberoot ofanumber.Thesymbolforcuberootis 3 √

e.g. 3 √8 = 2 as 23 = 2 × 2 × 2 = 8

Exercise1C

Und er stand ing

1 Writeeachofthefollowingusingindexnotation. 2 × 2

2 Matcheachexpressioninwordstoanexpressioninsymbols,givenontheright.

Thesquareof 10 a

3 Copyandcomplete.

32 = 3 × 3 = a

4 Copyandcomplete.

Fluency

Example6Usingindexnotation

Writeeachproductusingindexnotation.

Solution

a 8 × 8 × 8 = 83

b 7 × 7 × 7 × 7 × 7 × 7 = 76

Nowyoutry

Writeeachproductusingindexnotation.

Explanation

Thenumber 8 appears 3 times.Wewrite 8 to thepowerof 3

The 7 appears 6 times.Wewrite 7 tothe powerof 6

5 Writeeachofthefollowingproductsusingindexnotation.

Example7Usingexpandednotation

Write 54 inexpandedform. a

Solution

a 54 = 5 × 5 × 5 × 5

b 54 = 625

Nowyoutry

Write 25 inexpandedform. a

6 Writeeachindexnotationinexpandedform.

Findthevalueof 54 b

Explanation

Thepowerof 4 tellsusthatthenumber 5 appears 4 times. 54 = 5 × 5 × 5 × 5

Findthevalueof 25 b

7 Findthevalueofthefollowingbyﬁrstwritingtheminexpandedform.

Example8Findingsquares,cubes,squarerootsandcuberoots

Evaluatethefollowing.

a 62 = 6 × 6 = 36

23 = 2 × 2 × 2 = 8

Nowyoutry Evaluatethefollowing.

Explanation

Findtheproductof 6 withitself.

8 Evaluatethesesquaresandsquareroots.

2 d

HintforQ8: 32 = 9 and √9 = 3 42 a 102 b 132 c

25 g

9 Evaluatethesecubesandcuberoots.

Problem-solving and reasoning

10 Decidewhichofthefollowingislarger.

23 or 32 a 24 or 32 b 25 or 52 c

11 Copyandcomplete.

If 132 = 169,then √169 = a If 152 =

,then √225 = b If √625 = 25,then 252 = c If 93 = 729,then 3 √729 = d

If 3 √1331 = 11,then 113 = e

12 Given 5 × 5 × 5 × 4 × 4 iswrittenas 53 × 42 (thedifferentbasesof 5 and 4 arekeptseparate),write eachofthefollowinginindexform.

× 12 × 4 × 4 ×

Algebraicindices

13 Writeeachofthefollowinginindexform.Remember,differentbasesarekeptseparate. HintforQ13:

p × p × p × q × q e

a × a × a × a × b × b f

a × a × b × b × b × b g

x × x × x × x × y h

1D 1D Numberproperties

Learningintentions

• Tounderstandthataprimenumberhasexactlytwofactorsandacompositenumberhasmorethantwofactors.

• Tobeableto ndthelowestcommonmultiple(LCM)oftwonumbers.

• Tobeableto ndthehighestcommonfactor(HCF)oftwonumbers.

Keyvocabulary: countingnumbers,multiple,factor,lowestcommonmultiple(LCM),highestcommonfactor(HCF), primenumbers,compositenumbers

Simplepropertiesofnumbersareattheheartofmorecomplexmathematicsandassociatedproblems.Prime numbersforexampleformthebasisofouronlinebankingencryptioncodesasitisverydifﬁculttoﬁndthe primefactorsoflargenumbers.

Lessonstarter:Howmanyin 60 seconds?

In 60 seconds,writedownasmanynumbersasyoucanthatﬁteachdescription.

• Multiplesof 7

• Factorsof 144

• Primenumbers

Compareyourlistswiththeresultsoftheclass.Whatisthelargestprimenumberthattheclass cameupwith?

Keyideas

A multiple ofanumberisobtainedbymultiplyingthenumberbythe countingnumbers 1, 2, 3, …

e.g.Multiplesof 9 include 9, 18, 27, 36, 45, … (thinkofyourmultiplicationtables).

The lowestcommonmultiple(LCM) isthesmallestmultipleoftwoormorenumbersthatis common.

e.g.Multiplesof 3 are 3, 6, 9, 12, 15 , 18, …

Multiplesof 5 are 5, 10, 15 , 20, 25, …

TheLCMof 3 and 5 istherefore 15

A factor ofanumberhasaremainderofzerowhendividedintothegivennumber.

e.g. 11 isafactorof 77 since 77 ÷ 11 = 7 with 0 remainder.

The highestcommonfactor(HCF) isthelargestfactoroftwoormorenumbersthatiscommon.

• Factorsof 24 are 1, 2, 3, 4, 6, 8, 12 , 24

• Factorsof 36 are 1, 2, 3, 4, 6, 9, 12 , 18, 36

• TheHCFof 24 and 36 istherefore 12

Primenumbers haveonlytwofactors:thenumberitselfand 1

• 2, 13 and 61 areexamplesofprimenumbers.

• 1 isnotconsideredtobeaprimenumber.(Ithasonlyonefactor.)

Compositenumbers havemorethantwofactors.

• 6, 20 and 57 areexamplesofcompositenumbers.

Exercise1D

Und er stand ing

1 Writedownthefactorsofeachnumber. 4 a 6 b 12 c

2 Writedownthenextterm(multiple)ineachofthesepatterns.

2, 4, 6, 8, a 3, 6, 9, 12, b

7, 14, 21, d 6,

Thefactorsof 16 are 1, 2, 4, 8, 16

Thefactorsof 24 are 1, 2, 3, 4, 6, 8, 12, 24

Thefactorsof 18 are 1, 2, 3, 6, 9, 18

Thefactorsof 30 are 1, 2, 3, 5, 6, 10, 15, 30.

Thefactorsof 8 are 1, 2, 4, 8

Usingtheinformationgiveninthetable,writedowntheHCFofeach pairofnumbers.

HintforQ3:HCF istheHighest CommonFactor. 16 and 24 a 24 and 30 b 18 and 30 c 16 and 8 d 24 and 18 e 8 and 24 f 16 and 18 g 18 and 8 h

4 UsetheﬁrstsixmultiplesofthenumbersgiventoﬁndtheLCMofeach pairofnumbers.

Number Multiples

2 2, 4, 6, 8, 10, 12

4 4, 8, 12, 16, 20, 24

3 3, 6, 9, 12, 15, 18

5 5, 10, 15, 20, 25, 30

6 6, 12, 18, 24, 30, 36

2 and 4 a 4 and 3 b 3 and 6 c 4 and 6 d 4 and 5 e 5 and 6 f

Fluency

Example9Workingwithprimesandcomposites

HintforQ4:LCM istheLowest CommonMultiple.

5,6,7–8(½)

Decidewhethereachofthefollowingisaprimenumberoracompositenumber.

29 a 63 b

Solution

5,6,7–8(½)

Explanation

a 29 isaprimenumber 29 hasonly 2 factors— 1 and 29.Itisaprime number.

b 63 isacompositenumber 63 hasfactors 1, 3, 7, 9, 21, 63

Nowyoutry

Decidewhethereachofthefollowingisaprimenumberoracompositenumber.

39 a 53 b

5 Decidewhethereachofthefollowingnumbersisprime orcomposite.

7 a 12 b 27 c 69 d 105 e 28 f 15 g 11 h 31 i 37 j 49 k 99 l

6 Choosetheprimenumbersfromthefollowinglist:

Example10FindingtheLCM

FindtheLCMof 6 and 8

Solution

Multiplesof 6 are:

6, 12, 18, 24, 30, …

Multiplesof 8 are:

8, 16, 24, 32, 40,

TheLCMis 24.

Nowyoutry

FindtheLCMof 4 and 10

7 FindtheLCMofthesepairsofnumbers.

2, 3 a 5, 9 b 8, 12 c 4, 8 d 25, 50 e

Example11FindingtheHCF

FindtheHCFof 36 and 48.

Solution

Factorsof 36 are:

1, 2, 3, 4, 6, 9, 12, 18, 36

Factorsof 48 are:

1, 2, 3, 4, 6, 8, 12, 16, 24, 48

TheHCFis 12

Nowyoutry

FindtheHCFof 24 and 32

8 FindtheHCFofthesepairsofnumbers.

6, 8 a 18, 9 b

HintforQ5:Primeshaveexactlytwo factors,compositeshavemorethan twofactors.

, 25, 26, 27, 28, 29, 30.

Explanation

First,listsomemultiplesof 6 and 8

Continuethelistsuntilthereisatleastonein common.

Choosethesmallestnumberthatiscommonto bothlists.

Explanation

First,listfactorsof 36 and 48

Choosethelargestnumberthatiscommonto bothlists.

, 24 c 24, 30 d

, 13 e 19, 31 f 72, 36 g 108, 64 h 6, 4 i 6, 12 j 8, 24 k 15, 25 l

Problem-solving and reasoning

9 Find:

theLCMof 8, 12 and 6 a

theLCMof 7, 3 and 5 b

theHCFof 20, 15 and 10 c

theHCFof 32, 60 and 48 d

9–119,11,12

10 Ateacherhas 64 studentstodivideintoequalgroupsofgreaterthan 2 withnoremainder.Inhow manywayscanthisbedone?

11 Belowarethenumbers 1 to 100 Listalltheprimenumbers.Howmanynumbersareprimenumbers?

12 Threesetsoftrafﬁclights(A,BandC)allturnredat9:00amexactly.LightsetAturnsredevery 2 minutes,lightsetBturnsredevery 3 minutesandlightsetCturnsredevery 5 minutes. Howlongdoesittakeforallthreelightstoturnredagainatthesametime?

13 Goldbach’sconjecture isafamousmathematicalstatementthatsaysthateveryevennumbergreater thantwocanbewrittenasthesumoftwoprimenumbers.

Theevennumbers 4, 6 and 8 havebeenwrittenasthesumoftwoprimes. Showhowtheevennumbers 10 to 30 canbewrittenasthesumoftwoprimes.Somecanbedonein morethanoneway.

4 = 2 + 2

6 = 3 + 3

8 = 3 + 5 10 = 12 = 14 = 16 = 18 = 20 = 22 = 24 = 26 = 28 = 30 = Theﬁrsttenprimenumbers.

AgraphillustratingGoldbach’sconjectureuptoandincluding 50 isobtainedbyplotting thenumberofwaysofexpressingevennumbersgreaterthan 4 asthesumoftwoprimes.

14 Twinprimesarepairsofprimenumbersthatdifferby 2.Ithasbeensuggestedthatthereareinﬁnitely manytwinprimes.UsethetableofprimesyoucreatedinQuestion 11 ofthisexerciseandlistthepairs oftwinprimeslessthan 100

1E 1E Divisibilityandprimefactorisation

Learningintentions

• Tobeabletowriteanumberasaproductofprimefactors.

• Tobeabletoconstructafactortree.

• Tobeabletousethedivisibilitytestsforsingle-digitfactorsotherthan 7.

• Tounderstandhowthelowestcommonmultipleandhighestcommonfactoroftwonumberscanbefoundusing theirprimefactorform.

Keyvocabulary: primenumber,factortree,highestcommonfactor(HCF),lowestcommonmultiple(LCM),divisibility tests,primefactorisation

Everywholenumbergreaterthan 1 canbe writtenasaproductofprimenumbers,e.g. 6 = 3 × 2 and 20 = 2 × 2 × 5

Writingnumbersasaproductofprimenumbers canhelptosimplifyexpressionsanddetermine otherpropertiesofnumbersorpairsofnumbers.

Lessonstarter:Remembering divisibilitytests

Totestifanumberisdivisibleby 2,wesimply needtoseeifthenumberisevenorodd.All evennumbersaredivisibleby 2.Asaclass, canyoudescribedivisibilitytestsforanyofthe following?

• Divisibleby 3

• Divisibleby 4

• Divisibleby 5

• Divisibleby 6

• Divisibleby 8

• Divisibleby 9

• Divisibleby 10

Keyideas

Afactortreecanbeusedtoseehowanumbercanbebroken downintoitsprimefactors.

A factortree isanillustratedbreakdownofanumberintoitsprimefactors.

Primefactorisation usesafactortree,orsimilar,towriteanumberasaproduct ofitsprimefactors.

e.g. 12 = 2 × 2 × 3 or 22 × 3 (usingindices)

The highestcommonfactor(HCF) canbefoundusingprimefactors.

TheHCF=Allcommonprimesraisedtothesmallestpower.

e.g. 12 = 22 × 320 = 22 × 5 Â HCF = 22 or 4

The lowestcommonmultiple(LCM) canbefoundusingprimefactors.

TheLCM = Alldifferentprimesraisedtothehighestpower.

e.g. 12 = 22 × 320 = 22 × 5 Â LCM = 22 × 3 × 5 = 60

Divisibilitytests

Anumberis:

• divisibleby 2 ifitiseven(endswiththedigit 0, 2, 4, 6 or 8),e.g. 24

• divisibleby 3 ifthesumofallthedigitsisdivisibleby 3

e.g. 162 where 1 + 6 + 2 = 9,whichisdivisibleby 3

• divisibleby 4 ifthenumberformedbythelasttwodigitsisdivisibleby 4

e.g. 148 where 48 isdivisibleby 4

• divisibleby 5 ifthelastdigitisa 0 or 5

e.g. 145 or 2090

• divisibleby 6 ifitisdivisiblebyboth 2 and 3

e.g. 456 where 6 isevenand 4 + 5 + 6 = 15,whichisdivisibleby 3

• divisibleby 8 ifthenumberformedfromthelast 3 digitsisdivisibleby 8,orifthelastthree digitsare 000

e.g. 2112 where 112 isdivisibleby 8 and 2000 whichendsin 000

• divisibleby 9 ifthesumofallthedigitsaredivisibleby 9

e.g. 3843 where 3 + 8 + 4 + 3 = 18 whichisdivisibleby 9

• divisibleby 10 ifthelastdigitisa 0

e.g. 4230

• Thereisnosimpletestfor 7

1 Givethemissingnumbersinthesefactortrees.

2 Givethemissingwordornumber.Anumberisdivisibleby:

5 ifthelastdigitis 0 or a

2 ifthelastdigitis b

10 ifthelastdigitis c

8 ifthenumberformedbythelast digitsisdivisibleby 8 d

6 ifitisdivisiblebyboth 2 and . e

3 ifthesumofthedigitsisdivisibleby f

9 ifthe ofthedigitsisdivisibleby 9 g

4 ifthenumberformedbythelast digitsisdivisibleby 4 h

Example12Findingprimefactorform

Useafactortreetowrite 300 asaproductofprimefactors.

Solution

Explanation

First,divide 300 intotheproductof any two factors. Choosetheeasiestpair.

300 = 30 × 10

Continuedividingnumbersintotwofactors untilthefactorsareprime.

Circletheprimefactors.

Writethefactorsinascendingorder.

Useindexnotation(powers)toabbreviateyour answer.

Nowyoutry

Useafactortreetowrite 224 asaproductofprimefactors.

3 Copyandcompletethesefactortreestohelpwritetheprimefactorformofthegivennumbers.

4 Useafactortreetoﬁndtheprimefactorformofthesenumbers.

Example13Testingfordivisibility

Usedivisibilityteststodecideifthenumber 627 isdivisibleby 2, 3, 4, 5, 6, 8 or 9

Solution

Explanation

Notdivisibleby 2 since 7 isodd.Thelastdigitneedstobeeven.

Divisibleby 3 since 6 + 2 + 7 = 15 and thisisdivisibleby 3

Notdivisibleby 4 as 27 isnotdivisible by 4

Notdivisibleby 5 asthelastdigitisnot a 0 or 5

Notdivisibleby 6 asitisnotdivisible by 2

Notdivisibleby 8 asthelast 3 digits togetherarenotdivisibleby 8

Notdivisibleby 9 as 6 + 2 + 7 = 15 whichisnotdivisibleby 9

Nowyoutry

Thesumofallthedigitsneedstobedivisibleby 3

Thenumberformedfromthelasttwodigitsneedsto bedivisibleby 4

Thelastdigitneedstobea 0 or 5

Thenumberneedstobedivisiblebyboth 2 and 3

Thenumberformedfromthelastthreedigitsneedsto bedivisibleby 8

Thesumofallthedigitsneedstobedivisibleby 9

Usedivisibilityteststodecideifthenumber 342 isdivisibleby 2, 3, 4, 5, 6,

5 Usedivisibilityteststodecideifthesenumbersaredivisibleby 2,

or 9.

HintforQ5:Dotheseventests oneachnumber.

Example14FindingtheLCMandHCF

FindtheLCMandHCFof 105 and 90,usingprimefactorisation.

Solution

105 = 3 × 5 × 7 90 = 2 × 32 × 5

LCM = 2 × 32 × 5 × 7 = 630

HCF = 3 × 5 = 15

Nowyoutry

Explanation

First,expresseachnumberinprimefactorform. Notethat 3 and 5 arecommonprimes.

FortheLCMincludeallthedifferentprimes, raisingthecommonprimestotheirhighest power.

FortheHCFincludeonlythecommonprimes raisedtothesmallestpower. 105 and 90 both haveone 3 andone 5.

FindtheLCMandHCFof 18 and 42,usingprimefactorisation.

6 CopyandcompletethistableofLCMsandHCFs.

7 Findthehighestcommonprimefactorsofthesepairsofnumbers.

8 FindtheLCMandtheHCFofthesepairsofnumbers,usingprimefactorisation.

Problem-solving and reasoning 9 9,10

9 Whatisthesmallestnumberthatcanbedivided,withoutgivingaremainder,byallofthefollowing fournumbers? 2, 3,

10 NanaMagoo’stwograndchildrenlovetovisither.Lachlanvisitsherevery 8 dayswhileBrycevisitsevery 18 days.TheybothvisitedherlastMonday. Howmanydayswillitbefromthatvisitbeforetheybothvisitheronthe samedayagain?

HintforQ10:You mightliketomakea listtohelpyouhere!

Findthemissingdigit 11

11 Usethedivisibilityrulesgiventoyouatthestartofthissectiontoﬁndthemissingdigitforeachof thefollowing.Insomecasestheremightbemorethanonedigitthatworks.Inthesecases,listall thepossibleanswers.

2 6 ifthenumberisdivisibleby 3 (remembertolistallpossibleanswers). a

1 35 ifthenumberisdivisibleby 9 b

4 3 ifthenumberisdivisibleby 3. c

4 3 ifthenumberisdivisibleby 3 and 9. d

276 ifthenumberisdivisibleby 2. e

276 ifthenumberisdivisibleby 2 and 5. f

1 1A Evaluatethesesumsandthesedifferencesmentally.

2 1A Useanalgorithmtoﬁndthesesumsandthesedifferences.

3 1B Useamentalstrategytoevaluatethefollowing.

4 1B Useanalgorithmtoevaluatethefollowing.

5 1C Writeeachproductinindexnotation.

7 × 7 × 7 × 7 a 5 × 5 × 2 × 2 × 2 b 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 c

6 1C Evaluatethefollowing.

7 1D FindtheLCMofthesepairsofnumbers.

8 1D FindtheHCFofthesepairsofnumbers.

9 1E Useafactortreetowrite 240 asaproductofprimefactors.

10 1E Usedivisibilityteststodecideif 72 isdivisibleby 2, 3, 4, 5, 6, 8,or 9

11 1E FindtheLCMandHCFof 40 and 110,usingprimefactorisation.

1F 1F Negativeintegers CONSOLIDATING

Learningintentions

• Tounderstandthatintegerscanbenegative,zeroorpositive.

• Tounderstandhowtouseanumberlinetoaddorsubtractpositiveintegers.

• Tobeabletoaddapositiveintegertoanegativeinteger.

• Tobeabletosubtractapositiveintegerfromapositiveornegativeinteger.

Keyvocabulary: integer,positivenumber,negativenumber,numberline

TheIndianmathematicianBrahmaguptasetoutrulesfornegativenumbersinthe 7thcentury. Today,negativenumbersareusedinscience,engineeringandbusiness.Theyhelpusdescribeopposites suchasleftandright,upanddown,proﬁtandloss,andtemperaturesaboveandbelowfreezing.

Negativenumberare usedtodescribe temperaturesbelow freezing(i.e.below 0° C).

Lessonstarter:Anegativeworld

Describehowtousenegativenumbersinthesesituations.

• 6°Cbelowzero

• Alossof $4200

• 150 mbelowsealevel

• Aturnof 90° anticlockwise

• Thesolutiontotheequation x + 5 = 3

Canyoudescribeanothersituationinwhichyoumightmakeuseof negativenumbers?

Keyideas

Negativenumbers arenumberslessthanzero.

The integers are , −4, −3, −2, −1, 0, 1, 2, 3, 4

• Theseinclude positive integers(naturalnumbers),zeroandnegativeintegers.

• Theseareillustratedclearlyona numberline 110 2 2 3 4 3 4

Zero

negative numbers

Valuesdecreaseasyoumovetothe leftalongthenumberline

positive numbers

Valuesincreaseasyoumovetotheright alongthenumberline

Addingorsubtractingapositiveintegercanresultinapositiveornegativenumber.

• Addingapositiveinteger

2 + 3 = 5

• Subtractingapositiveinteger

Exercise1F

Und er stand ing

1 Writedownthenumbersuggestedby:

2 Copythenumberlinebelowandmark(withadot)theintegers −3, −1, 1, 3 and 5

3 Writethesymbol < (lessthan)or > (greaterthan)tomakethesestatementstrue.

4 Whatistheﬁnaltemperature?

Example15Addingapositiveinteger

Evaluatethefollowing.

Nowyoutry

Evaluatethefollowing.

5 Evaluatethefollowing.

−1 + 2 a

−10 + 11 c

+ 7 b

+ 12 d

−20 + 35 e −6 + 4 f

−7 + 2 g −15 + 8 h

−26 + 19 i −38 + 24 j

−10 + 15 k −2 + 9 l

−7 + 3 m

−6 + 9 o

Example16Subtractingapositiveinteger

Evaluatethefollowing. 3−7 a

+ 7 n

+ 1 p

3−7 = −4

−2−3 = −5

Nowyoutry

Evaluatethefollowing.

13−20 a −8−17 b

6 Evaluatethefollowing.

4−5 a 10−15 b

0−26 c 14−31 d

6−8 e 10−9 f

−4−7 g −11−20 h

−14−15 i −10−100 j

−11−6 k 0−12 l

−15−5 m 3−12 n

8−4 o −8−4 p

7 Evaluatethefollowing.

−9 + 6 a −9−6 b

HintforQ5:Start withtheleftnumber andmoverighton thenumberline.

HintforQ6:Start withtheleftnumber andmovelefton thenumberline.

+ 12 c

d −7−7 e −7 + 0 f 15−14 g 15−16 h −9−10 i

+ 10 j

+ 10 l 100−101 m

8 Workfromlefttorighttoevaluatethefollowing.

+ 25 o

+ 40 p

−3 + 4−8 + 6 a 0−10 + 19−1 b 26−38 + 14−9 c 9−18 + 61−53 d

Problem-solving and reasoning

9 Writethesum(e.g. −3 + 4 = 1)ordifference(e.g. 1−5 = −4)tomatchthesenumberlines.

10 Writethemissingnumber.

11 Inahigh-risebuildingthereare 8 floorsabove groundleveland 6 floorsbelowgroundlevel. Aliftstartsatthe 2ndfloorandmoves 4 floorsup, then 7 floorsdownbeforemovingdowna further 3 floors.

Atwhatﬂoordoestheliftﬁnish?

12 OnMondayMillyborrows $35 fromafriend. OnTuesdayshepaysherfriend $40.OnFriday sheborrows $42 andpaysback $30 thatnight. HowmuchdoesMillyoweherfriendthen?

CompleteSuzanne’saccountfortheweekshown.Acreditisanaddition (+) andadebitisa subtraction ( )

Spendingandearning

openingbalance

pays 1 week’srentof $375 375 earns $80 babysitting receives $100 fromherparentsforherbirthday buysapairofjeansfor $90 buysatopfor $45 payshermonthlymobilephonebill $49 gives $25 tocharity

HowmuchwouldSuzanneneedtodeposit(credit)intoheraccountsothatshecanpaytherentfor thenextweek?

14 Findwhatintegerneedstobeaddedorsubtractedtoeachsothattheendresultisalwayszero.

1G 1G Addingandsubtractingnegativeintegers

Learningintentions

• Tounderstandthataddinganegativenumberisthesameassubtractingitsopposite.

• Tounderstandthatsubtractinganegativenumberisthesameasaddingitsopposite.

• Tobeabletoaddorsubtractnegativeintegers.

Keyvocabulary: integer,positivenumber,negativenumber,opposite

If + represents +1 and represents −1 then + addedtogetherhasavalueofzero.

Usingthesesymbols, 5 + (−2) = 3 couldbeillustratedastheadditionof 2 ,leavingabalanceof 3

So 5 + (−2) isthesameas 5−2.

Also 5− (−2) = 7 couldbeillustratedﬁrstas 5 + and 2 togetherthensubtractingthe 2

So 5− (−2) isthesameas 5 + 2

Whenaddingorsubtractingnegativeintegerswefollowtherulessetoutbytheabovetwoillustrations, aswellasthepatternsbelow.

Lessonstarter:Lookingatpatternsforaddingandsubtracting negativenumbers

Copyandcomplete.

64 + 10

63 + 9

6 2 + 8

61 + 60 +

6(1)+−→ same as 615 −=

6(2) +− → same as 62 =

6 ( 3)+− → same as 63 =

6 ( 4)+− → same as 64 =

6(1) → same as 61+=

6(2) → same as

6(3) → same as 6(4) → same as

Keyideas

The opposite ofanumberdiffersbyafactorof −1

e.g.Theoppositeof 7 is −7 andtheoppositeof −12 is 12

Addinga negativenumber isthesameassubtractingitsopposite.

e.g. 2 + (−3) = 2−3 = −1 twooppositesignsgiveasubtraction/minus −4 + (−7) = −4−7 = −11

Subtractinganegativenumberisthesameasaddingitsopposite.

e.g. 2− (−5) = 2 + 5 = 7 twolikesignsgiveanaddition/plus −6− (−4) = −6 + 4 = −2

Exercise1G

Und er stand ing 1–3 3

1 −3 and 3 areopposites.Writedowntheoppositesofthesenumbers.

2 Writethewords‘add’or‘subtract’tosuiteachsentence. Toaddanegativenumber, itsopposite. a Tosubtractanegativenumber, itsopposite. b

3 Arethefollowingstatementstrue(T)orfalse(F)? 5 + (−2) = 5 + 2 a 3 + (−4) = 3−4 b

+ (−4) = −6−4 c −1 + (−3) = 1−3 d 8− (−3

(

Fluency

Example17Addingnegativenumbers

Evaluatethefollowing. 10 + (−3) a −3 + (−5) b

Solution

a 10 + (−3) = 10−3 = 7

b −3 + (−5) = −3−5 = −8

Explanation

Adding −3 isthesameassubtracting 3

1011987 6 + and =

Adding −5 isthesameassubtracting 5

+ and =

Nowyoutry

Evaluatethefollowing. 24 + (−7) a −13 + (−5) b

4 Evaluatethefollowing.

6 + (−2) a 4 + (−1) b

+ (−12) c

+ (−5) d 2 + (−4) e

+ (−20) i

+ (−40) f

+ (−50) j

+ (−6) g

+ (−22) k

+ (−5) h

+ (−10) l 7 + (−8) m

+ (−12) n 6 + (−12) o

+ (−12) p −8 + (−8) q 5 + (−5) r

+ (−15) s

Example18Subtractingnegativenumbers

Evaluatethefollowing.

4− (−2) a

Solution

a 4− (−2) = 4 + 2 = 6

b −11− (−6) = −11 + 6 = −5

Nowyoutry

Evaluatethefollowing.

(−12) a

5 Evaluatethefollowing.

(−14) d

HintforQ4:Toadda negative,subtract itsopposite.

+ (−6) t

(−6) b

Explanation

Subtracting −2 isthesameasadding 2 7654 3 and =+

Subtracting −6 isthesameasadding 6 7 8 9 10 11 12

(−4) b

(−13) e

(−320) f −5− (−3) g

(−10) h

HintforQ5:To subtractanegative, additsopposite. 2− (−3) a

(−16) i −10− (−42) j

(−5) m

(−18) p

(−8) s

(−31) k

(−5) n

(−41) q

(−12) t

6 Evaluatethefollowingmixedproblems.

+ (−50) b

+ (−12) d

(−5) l

(−12) o

(−52) r

+ 6 e

+ (−15) h

(−6) f

+ (−7) i

Problem-solving and reasoning

7 Anicecubeisremovedfromafreezerat −25°Candplacedinto aglassofjuiceat 7°C.Whatisthedifferencebetweenthetwo temperatures?

8 Kelvinowesthebank $450000.Whatmusthedepositintohisaccounttoonlyowe $270000?

9 Statethemissingnumberineachofthefollowing.

10 Anadditionfactlike 2 + 3 = 5 canbeusedtogeneratetwosubtractionfacts: 5−2 = 3 and 5−3 = 2

Writetwosubtractionfactsthatcanbegeneratedfrom 4 + 6 = 10. a

Writetwosubtractionfactsthatcanbegeneratedfrom 7 + (−2) = 5 b

Explainwhy 5− (−6) = 11 byusinganadditionfactinvolving 5, −6 and 11. c

11 If a = −5 and b = −3,ﬁndthevalueof:

12 Writedownthemissingnumber. 4 + = 1 a 6 + = 0 b −2 + = −1 c + (−8) = 2 d + (−5) = −3 e + (−3) = −17 f 12− = 14 g 8− = 12 h −1− = 29 i (−7) =

Puzzleswithnegatives

HintforQ11:Replacethe pronumeralinthestatement withthenumberitrepresents. e.g. a = −2 then a + (−5) = −2 + (−5) = −2−5 = −7

13 Placetheintegersfrom −3 to 2 inthismagictrianglesothateachsideaddsto thegivennumber.

−3 a 0 b

14 Amagicsquarehaseachrow,columnanddiagonalsaddingtothesamemagicsum. Completethesemagicsquares.

1H 1H Multiplyinganddividingnegative integers

Learningintentions

• Tounderstandthattheproductorquotientoftwointegerswillbepositiveifthetwointegershavethesamesign.

• Tounderstandthattheproductorquotientoftwointegerswillbenegativeifthetwointegershaveoppositesigns.

• Tobeabletouseorderofoperationswithintegers.

Keyvocabulary: sign,integer,positiveinteger,negativeinteger,product,quotient,orderofoperations

Asarepeatedaddition,theproduct 3 × (−2) canbewrittenas −2 + (−2) + (−2) = −6.So 3 × (−2) = −6 and, since a × b = b × a forallnumbers a and b,then −2 × 3 isalsoequalto −6

Fordivisionwecanwritetheproduct 3 × 2 = 6 asaquotient 6 ÷ 2 = 3

So,if 3 × (−2) = −6 then −6 ÷ (−2) = 3

Alsoif −2 × 3 = −6 then −6 ÷ 3 = −2

Thequotientoftwonegativenumbersresultsinapositivenumber,andtheproductorquotientoftwo numbersofoppositesignisanegativenumber.

Also, 6 ÷ (−2) = −3 canalsoberearrangedto −3 × (−2) = 6.Sotheproductoftwonegativenumbersis apositivenumber.

Lessonstarter:Seeingthepattern

• Writethemissingnumbersinthesetables.Youshouldcreateapatterninthethirdcolumn.

• Writethemissingnumbersinthesesentences.Usethetablesabovetohelp.

3 × 5 = so 15 ÷ 5 = a −3 × 5 = so −15 ÷ 5 = b

3 × (−5) = so −15 ÷ (−5) = c −3 × (−5) = so 15 ÷ (−5) = d

Keyideas

The product or quotient oftwo integers ofthesame sign isa positiveinteger.

• Positive × Positive = Positive

• Positive ÷ Positive = Positive

• Negative × Negative = Positive

• Negative ÷ Negative = Positive

Theproductorquotientoftwointegersofoppositesignsisa negativeinteger

• Positive × Negative = Negative

• Positive ÷ Negative = Negative

• Negative × Positive = Negative

• Negative ÷ Positive = Negative

Exercise1H

Und er stand ing 1–3 3

1 Choosethecorrectwordstocompleteeachsentence.

Theproduct (×) orquotient (÷) oftwonumbersofthe signisa integer. a

Theproduct (×) orquotient (÷) oftwonumbersofthe signisa integer. b

2 Withoutﬁndingtheanswertotheseproducts,decideiftheanswerwouldbepositiveornegative.

3 Withoutﬁndingtheanswertothesequotients,decideiftheanswerwouldbepositiveornegative.

Fluency

Example19Findingproductsofintegers

Evaluatethefollowing. 3 × (−7) a

Solution

× (

Explanation

a 3 × (−7) = −21 Theproductoftwonumbersofoppositesignisnegative. + × =

b −4 × (−12) = 48−4 and −12 arebothnegativeandsotheproductwillbepositive. × =+

Nowyoutry

Evaluatethefollowing. −4 × 6 a −7 × (−11) b

4 Evaluatethefollowing.

Example20Findingquotientsofintegers

Evaluatethefollowing.

−63 ÷ 7 a −121 ÷ (−11) b

Solution

Explanation

a −63 ÷ 7 = −9 Thetwonumbersareofoppositesignssotheanswerwillbenegative. ÷+ =

b −121 ÷ (−11) = 11−121 and −11 arebothnegativesothequotientwillbepositive. ÷ =+

Nowyoutry

Evaluatethefollowing.

÷ (−8) a

5 Evaluatethefollowing. −10 ÷ 2 a

÷ (−9) b

÷ 19 b

÷ 15 c −120 ÷ 4 d 32 ÷ (−16) e

6 ÷ (−2) g

÷ (−3) j

÷ (−6) m

÷ 2 f

÷ (−2) h −12 ÷ 6 i

÷ 5 k

÷ (−5) n

÷ (−9) l

÷ 1 o −8 ÷ (−1) p

6 Decideiftheanswertothefollowingis −2

8 ÷ (−2) a −1 × (−2) b

−10 ÷ 5 c −16 ÷ 8 d −2 × 1 e −2 × 0 f

7 If (−2)2 = −2 × −2 = 4,ﬁndthevalueofthefollowing.

(−5)2 a (−6)2 b (−7)2 c (−8)2 d (−9)2 e (−10)2 f

Problem-solving and reasoning

8 Writethemissingnumber. × 3 = −9 a × (−7) = 35 b × (−4) = −28 c −3 × = −18 d

−19 × = 57 e ÷ (−9) = 8 f ÷ 6 = −42 g 85 ÷ = −17 h

−150 ÷ = 5 i

9 Will (−2)3 giveapositiveornegativeanswer?Explainwhy.

10 Insert × and/or ÷ signstomaketheseequationstrue.

−2 3 (−6) = 1 a

10 (−5) (−2) = 25 b

6 (−6) 20 = −20 c

−14 (−7) (−2) = −1 d

11 Theproductoftwonumbersis −24 andtheirsumis −5.Whatarethetwonumbers?

Bracketplacements

12 Insertbracketsinthesestatementstomakethemtrue.

−2 + 1 × 3 = −3 a −10 ÷ 3− (−2) = −2 b

−8 ÷ (−1) + 5 = −2 c −1−4 × 2 + (−3) = 5 d

−4 + (−2) ÷ 10 + (−7) = −2 e 20 + 2−8 × (−3) = 38 f 1− (−7) × 3 × 2 = 44 g 4 + (−5) ÷ 5 × (−2) = −6 h

(½)

1I 1I Orderofoperationsandsubstitution

Learningintentions

• Tounderstandtherulesfororderofoperations

• Tobeabletoevaluatenumericalexpressionsusingtheorderofoperations

• Tobeabletosubstituteintegersforpronumeralsinordertoevaluateexpressions

Keyvocabulary: Orderofoperations,brackets,evaluate,substitute

Anexpressionsuchas a + 2 × b canbeevaluated ifweknowthevaluesof a and b.Theexpression includestheoperationsaddition(listedﬁrst)and multiplication;however,byconvention,weknow thatmultiplicationisdonebeforetheaddition.In thissectionwewilldealwithorderofoperations, usingbothpositiveandnegativeintegers.

Expansionjointspreventbridgesfrombuckinginhotweather. Engineersapplytheorderofoperationsaftersubstitutingvalues forthebridgelength, Lm,temperatures, t°Cto T °C,andagiven a valueintotheexpansionlengthformula: l = aL(T t)

Lessonstarter:Bracketplacement

Isitpossible,byinsertingbrackets,tomake 3 × 5−2 + 6 = 15 true?

Insertapairofbracketstomaketheequationcorrect. Trymakingupyourownsimilarproblem.

Keyideas

Therulesfororderofoperationsare:

• Dealwithoperationsinsidebracketsﬁrst.

• Dealwithpowers.

• Domultiplicationanddivisionnext,workingfromlefttoright.

• Doadditionandsubtractionlast,workingfromlefttoright.

Expressionscanbeevaluatedbysubstitutingnumbersforthegivenpronumerals.

Forexample:If a = −2 and b = −3,then a + 5b = −2 + 5 × (−3) = −2 + (−15) = −17

• Remember,forexample,that 5b means 5 × b and a 3 means a ÷ 3

10(74)2

Exercise1I

Und er stand ing 1–3 2,3

1 Byfollowingtheorderofoperations,describetheoperationthatneedstobedoneﬁrst.

2 + 3 × 9 a 10−2 ÷ 2 b 1 × 3 + 5 c

6 × (9−6) d (12 + 6) ÷ 2 e

2 Decideifbothsidesofthesesimplestatementsareequal.

(2 + 3) −1 = 2 + 3−1 a

(3 + (−2)) (−1) = 3 + (−2) (−1) b

5 × (2 + (−3)) = 5 × 2 + (−3) c

−8 × 2− (−1) = −8 × (2− (−1)) d

−10 ÷ 2−4 = −10 ÷ (2−4) e

−2 × 3 + 8 ÷ (−2) = (−2 × 3) + (8 ÷ (−2)) f

3 Statethemissingnumberstocompletetheworkingforthesesubstitutions.

a + 2b (a = −3, b = 4)

a + 2b = −3 + 2 × 4 = + = a 3 × (a b)(a = 5, b = −1) 3 × (a b) = 3 × (5− (−1)) = 3 × = b

Fluency

Example21Usingorderofoperationswithpositiveintegers

Evaluatethefollowing.

10 + 5 × 3 a

15− (7−3) b

20 ÷ (2 × (5−3)) c

Solution

a 10 + 5 × 3 = 10 + 15 = 25

b 15− (7−3) = 15−4 = 11

c 20 ÷ (2 × (5−3))

= 20 ÷ (2 × 2)

= 20 ÷ 4 = 5

Nowyoutry

Evaluatethefollowing.

17−8 ÷ 4 a

10 × (15−9) b

27 ÷ (3 × (9−6)) c

Explanation

Multiplication (×) isdoneBEFOREaddition (+) 5 × 3 = 15

Bracketsneedtobedoneﬁrst (7−3) = 4.Thendothe subtraction 15−4

Startwiththeinnermostbrackets (5−3)

Finishworkingwiththebrackets-wefollowtheorderof operationswithinthebrackets (2 × 2)

Thenthedivision 20 ÷ 4.

4 Findtheanswerstoeachofthefollowing.

12 + 5 × 2 a

10 × 2 + 6 c

(9−2) × 4 e

28 ÷ (2 × 7) g

24−6 × 3 b

15 ÷ 3−2 d

18− (12−8) f

56−5 × 10 h

120 + 200 ÷ 5 i 88 × 2 ÷ 8 j

12 ÷ (18 ÷ 6) k

55 ÷ 11 × 5 m

240 ÷ 10 × 2 o

16−18 ÷ 9 l

55−25 ÷ 5 n

58 + 100 ÷ 20 p

100−25 ÷ 5 q (24−9) × 3 r

5 Evaluate.

56−4 × 6 a 96 ÷ 4 + 3 × 6 b

150−7 × (10−3 × 2) c 12 × (13−8) × (24−18) d

7 + 30 ÷ (10 ÷ (7−5)) e

13− (6− (5−3)) × 3−1 f

Example22Usingorderofoperationswithintegers

Evaluatethefollowing.

5−6 × (−2) a −21 ÷ (5− (−2)) b 2 × 102 ÷ 5 c

Solution Explanation a 5−6 × (−2) = 5− (−12) = 17

Dothemultiplicationbeforetheadditionand rememberthat 5− (−12) = 5 + 12

b −21 ÷ (5− (−2)) = −21 ÷ 7 = −3

c 2 × 102 ÷ 5 = 2 × 100 ÷ 5 = 200 ÷ 5 = 40

Nowyoutry

Dealwithbracketsﬁrstandrememberthat 5− (−2) = 5 + 2

Dealwithpowersbeforeotheroperations. (Note: 2 × 102 ¢ 202.)

Evaluatethefollowing. 12−15 ÷ (−3) a (−3−5) × (7 + (−4)) b 4 × 32 ÷ 2 c

6 Evaluatethefollowing.Remembertousethenormalorderofoperations.

−2 × 3 × 5 a −6−2 × 3 b

× (−1) c −3 ÷ (−1) + 7 × (−2) d 6 × (−2) −10 ÷ (−5) e

+ 8 × (−2) ÷ (−16) f 20−10 ÷ (−5) × 2 g 0 × (−3) + 2 × (−30) h 35−10 ÷ (−2) + 0 i

7 Useorderofoperationstoevaluatethefollowing.

3 × (2−4) a (7− (−1)) × 3 b (−8 + (−2)) ÷ (−5) c

40 ÷ (8− (−2)) + 3 d 0 × (38− (−4)) × (−6) e −6 × (−1 + 3) ÷ (−4) f ((−2) + 1) × (8− (−3)) g (−6−4) ÷ (50 ÷ (−10)) h −2 × (8−7 × (−2)) i

Useorderofoperationstoevaluatethefollowing.

5 × 22 ÷ 10 a

7 + 32 × 2 b

6−42 × (−2) c

8 + 13 ÷ (−3) d

22 −32 e

33 ÷ 9 + 1 f

Example23Substitutingintegers

Substitutethegivenintegerstoevaluatetheexpressions.

a −3b with a = −2 and b = −4 a (a + b) ÷ (−5) with a = −7 and b = 2 b

a2 b3 with a = −2 and b = −3 c

Solution

a a −3b = −2−3 × (−4) = −2− (−12) = −2 + 12 = 10

b (a + b) ÷ (−5) = (−7 + 2) ÷ (−5) = −5 ÷ (−5) = 1

c a2 b3 = (−2)2 (−3)3 = 4− (−27) = 4 + 27 = 31

Nowyoutry

HintforQ8:Multiplicationand divisioniscalculatedbeforeaddition andsubtraction.

Explanation

Substitute a = −2 and b = −4 andthenevaluate, notingthat −2− (−12) = −2 + 12

Substitute a = −7 and b = 2 andthendealwith thebracketsbeforethedivision.

Usebracketswhensubstitutingintoexpressions withpowers.

(−2)2 = −2 × (−2) = 4 (−3)3 = −3 × (−3) × (−3) = −27

Substitutethegivenintegerstoevaluatetheexpressions.

4a + b with a = −3 and b = −4 a

a + (b ÷ (−2)) with a = −10 and b = −6 b

a3 b2 with a = −2 and b = −3 c

9 Evaluatetheseexpressionsusing a = −2 and b = 1.

a + b a

a b b

2a b c

b a d

a −4b e

3b −2a f

b × (2 + a) g

a (2− b) h

(2b + a) (b −2a) i

10 Evaluatetheseexpressionsusing a = −3 and b = 5

a + b2 a

a2 b b

b2 a c

b3 + a d

Problem-solving and reasoning

11 True(T)orfalse(F)?

5 + 9 = 5 + 3 × 3 a

10 + 2 × 7 = 12 + 7 b

18−6 + 5 = 12 + 5 c

3 × 5 × 6 = 15 × 6 d

120 ÷ 6 × 2 = 20 × 2 e

(5 + 3) × 9 = 8 × 9 f

11–1312–15

12 Insertoperationsymbols (+, , ×, ÷) betweenthenumberstomakeeachofthefollowing statementstrue.

5 4 9 = 0 a

5 4 9 = 11 b

5 4 9 = 41 c

13 Insertbracketsinthesestatementstomakethemtrue.

−2 + 1 × 3 = −3 a

−10 ÷ 3− (−2) = −2 b

−8 ÷ (−1) + 5 = −2 c

−1−4 × 2 + (−3) = 5 d

−4 + (−2) ÷ 10 + (−7) = −2 e

20 + 2−8 × (−3) = 38 f

14 Evaluatethefollowing.

3 × (−2)2 a

−2 × (−2)3 b

−16 ÷ (−2)3 c

−4 + √25 d

7− √16 e

−26 + 3 √27 f

−4 + 2 × 3 √8 g

−8 ÷ 3 √−64 + 1 h

−3 × (−2)3 + 4 i

15 Writeeachofthefollowingsituationsintomathematicalsymbolsandnumbers,andthencalculate. MurrayreceivesfourdollarsfromhisMumandsevendollarsfromhisDadaspocketmoneyeach weekfor 12 weeks.Howmuchmoneydoeshehaveattheendofthe 12 weeks?

a Arafﬂeprizeconsistsof $5000 cashand 6 shoppingvoucherseachworth $500.Whatisthetotal valueoftherafﬂeprize?

b Sallyhasﬁftydollars.Shebuysfourpensattwodollarseachandeightexercisebooksatthreedollars each.HowmuchchangedoesSallyget?

Retailerofloungeroomfurniture

Afurnitureretailerneedsmanyskills includingbeingagoodcommunicatoranda successfulsalesperson.Theyneedtoapplythe mathematicsofmoneymanagementtostock orders,deliverycosts,insuranceandpayrates. Itisimportantthattheyknowtheirproducts andhaveoptionsforclients.Asuccessful retailerrelatestocustomersinaconﬁdent, friendly,cheerfulandhelpfulmanner.

1 Aloungeroomfurniturebusinessadvertisesthatallloungesarereducedby $250.Whatisthesale priceonthefollowingloungescurrentlyinstock?

2-seaterleatherloungemarkedat $2340 a

2.5-seaterleatherloungewithchaisemarkedat $2599 b

3-seater + 2-seatersofasetmarkedat $2099 c

2-seaterreclinerloungeinfabricmarkedat $2249 d

7-seatercornerloungeinfabricmarkedat $4130. e

2 Floorstockisatermdescribingfurniturethathasbeendisplayedintheshowroomforcustomers totryout.Itisoftendiscountedforaquicksaleandisusuallyavailableforimmediatedelivery. Completethetablebelowtoﬁndthesavingsoneachoftheseloungesandotheravailable productsfromthewholesalecentre.

Recliner 3-piece

Cornersuite

$4499

$2250

$3299 $1999

2.5-seaterleatherchaise $2295 $1999

Outdoorsofa

Occasionalchair

$1120 $895

$369 $149

3 Themodelsondisplayareavailableineitherleatherorfabric.

Findoutthedifferenceinthepricesofeachloungesuiteinleathercomparedwithfabric. a Onaverage,howmuchmoredoestheretailerchargefortheleathermodels? b

Model Leather Fabric Differenceinprice

ER 2 + chaise

$2099 $1499

DW 3450 R $2350 $1890

Ebony 3 + 2 $3495 $2599

Reclinerandconsole 3 $1149 $799

VictaEL + 1 $3297 $2599

4 JenandBraddecidetobuyagreyBostonChaiseloungesuite.Theyinvestigatetheiroptionsfromtwo differentoutletstoﬁndthebestoverallpriceincludingdelivery. Whichisthebestbuyandbyhowmuch?

Option 1:CustomSofas

Leatherlounge $1999

Upgradeoncolourofleatherto

grey $160

Delivery $100

Usingtechnology

Option 2:LeisureLounges

Lounge—greyleather $2199

Delivery $70

5 Aloungesuitebusiness,LuxuryLounges,usesaspreadsheetfororders. HintforQ5:

CopythefollowingExcelspreadsheet.Formatallthenumbercells toNumberwith 0 d.p.andallpriceandcostcellstoCurrency with 0 d.p.

Ifleatherfurniturecosts 25% morethanfabric,enterformulas intothe‘Priceinleather’columntocalculatetheseprices.Seethe hintboxforextraclues.

EnterformulasincolumnGtocalculate‘Costoffurniture’andthe ‘Totalcostoforder’.Costswillbe $0 untilthenumbersofitemsare enteredfrompart d

• Toincreaseapriceby 25%, multiplyitby 1.25.

• Toﬁllformulasdowna column,dragthe’ﬁllhandle’ down.

• CellG3 formula =

Useyourspreadsheettocalculatethetotalcostofeachofthefollowingorders.

Sellinggardengnomes

Wilburbuysgardengnomesfromalocalsupplierandsellsthemforaproﬁt.Therearethreesizes ofgnomes:

Wilbursetsupabalancesheettokeeptrackofhisexpenditureandrevenue.Thefollowingincomplete exampleshowsfourtransactionsstartingfromaninitialbalanceof $0.Negativenumbersareusedto indicatemoneyleavinghisaccount,andpositivenumbersareusedformoneyenteringhisaccount.

Presentareportforthefollowingtasksandensurethatyoushowclearmathematicalworkings, explanationsanddiagramswhereappropriate.

1Preliminarytask

a Explainwhythebalanceaftertheﬁrsttransactiononthebalancesheetis $100

b Explainwhythebalanceafterthesecondtransactiononthebalancesheetis $56

c Thetableabovehastwomissingnumbers.

i WhatistheeffectonWilbur’sbalancewhenhesells 6 smallgnomes?

ii Whatisthenewbalanceafterhesellsthesegnomes?

d IfWilburpurchasesafurther 12 mediumgardengnomesfromthesupplierdeterminethe balanceattheendofthistransaction.

2Modellingtask

a TheproblemistodeterminesalestargetssothatWilburwillbeinproﬁt(withapositivebalance) attheendofamonth.Writedownalltherelevantinformationthatwillhelpsolvethisproblem.

b Drawupanemptybalancesheetusingthesameheadingsastheexampleabove.Allow 7 rows fortransactionsbutleavealltherowsblanksothatafreshsetoftransactionscanbemade.You canassumehisinitialbalanceis $0

ForpartofoneparticularmonthWilburstartedwithnognomesofanysizeandmakesthefollowing gardengnomepurchasesandsales.

• Purchases 30 small

• Purchases 25 medium

• Sells 9 small

• Purchases 15 large

• Sells 15 medium

• Sells 6 small

• Sells 3 large

c Enterthesetransactionsintoyourbalancesheetandcalculatethebalanceaftereach transaction.

d Statetheﬁnalbalanceaftertheabovetransactionsarecompleted.

e DecidehowmanygnomesofeachtypeareremaininginWilbur’sstockattheendofthemonth.

f Byconsideringthebalancepositionfrompart d abovedetermineonecombinationofsalesusing anygnomesintheremainingstockthatmeansthatWilburwillmakeaproﬁtgreaterthan $200 inthemonth.Justifyyouranswerwithappropriatecalculations.

g Summariseyourresultsanddescribeanykeyﬁndings.

3Extensionquestions

a HowmuchproﬁtdoesWilburmakewhenbuyingandsellingeachsizeofgnome?

b Wilburwantstomakeamonthlyproﬁtascloseto $200 aspossible.Chooseacombinationof gnomesthathecanbuyandselltoachievethis.Justifyyourchoicewithworking.

c Isitpossibletoachieveabalanceequalto $200 exactly?Justifyyouranswerwithworking.

Formulate

Evaluate

Integertugofwar

Keytechnology:Programmingandspreadsheets

Inagameoftugofwar,aropeispulledinoppositedirectionsbytwoteams.Thecentreoftherope ismarkedbyaribbonandstartsatpositionzero.Thewinningteamwillmovetheribbonagiven distanceintheirdirection.Inthisactivity,wewilluseanumberlinetorecordthepositionofthe ribbonwhereoneteamispullinginthepositivedirectionandtheotherinthenegativedirection.The winneristheteamtomovetheribbonpastaparticularpoint.

1Gettingstarted

Let’splayagameoftugofwarwithtwoteamsofequalability. TeamPos triestopulltotheright onthisnumberlineand TeamNeg triestopulltotheleft.Wewillassumethateitherteamcan successfullymoveupto 3 unitsintheirdirectiononanygiven 3-secondinterval.

a Usearandomnumbergeneratortogenerateamoveupto 3 unitsleftorright:Suggestions:

• Spreadsheet:=RANDBETWEEN (−3, 3)

• CAS:RandInt (−3, 3)

b Continuetocreaterandommovesleftorrightwithyourrandomnumbergeneratorandaddyour resultsintothistable.(Yournumbersinsertedintorow 3 onwardswillbedifferentfromthesample shownhere.)

c Assumethatiftheribbonreaches −5 or 5 then TeamNeg or TeamPos wins.Continueaddingto yourtableuntilthepositionoftheribbonisateither −5 or 5.Whichteamhaswonyourtugof war?

2Applyinganalgorithm

a Studythisﬂowchartwhichdescribesanalgorithmforplayingthetugofwargameoutlinedin part 1 above.Describewhatyouthinkthevariables i, a and b represent.

b Completethistableshowingthevaluesof i, a and b througheachpassoftheloopusingyour randomnumbergeneratortocalculatethevalueof a eachtime.Keepgoinguntil b reaches −5 or 5 or i = 10 Pass

3Usingtechnology

Wewilluseaspreadsheettosimulatetheplayingoftheabovetugofwargamebutthistimetheribbon needstobepulledthroughto −10 or +10 foroneoftheteamstowin.

a Enterthefollowingintoaspreadsheet.

b FilldownatcellsA6,B6,C6,D6 andE6 uptoandincludingrow 25.Thiswillmeanthat 20 passes havetakenplace.

c EntertheformulaincellE1.Whatdoyouthinkthisformuladoes?

d PressFunctionF9 torunanothergameoftugofwar.Repeatforatotalof 10 gamesandrecord howmanytimesTeamPoswins,TeamNegwinsorthereisaDraw.

Puzzles and games

1 Hey,doyouknowwhatawisecrackeris?

Completethesumsabovetounlockthepuzzlecode.

2 Whatexplosiveeventwasintheyear 1000 CE?

Answerthefollowingmultiplicationsanddivisionstoworkoutthepuzzlecode.Writeyouranswer onanothersheetofpaper.

3 Usingthesymbols +, , ×, ÷,makeasmanysumsasyoucanthathave −5 astheiranswer.

Negative number operations

4 − (−3) = 4 + 3

4 + (−3) = 4 − 3

Multiplication and division

Substitution

a = −2, b = 5, c = −4

Whole numbers 0, 1, 2, 3, ....

Mental strategies

• 156 + 79 = 156 + 80 − 1 = 235

• 45 + 47 = 45 + 45 + 2 = 92

• 3 × 22 = 3 × 20 + 3 × 2 = 66

• 4 × 88 = 2 × 176 = 352

• 164 ÷ 4 = 82 ÷ 2 = 41

• 297 ÷ 3 = (300 ÷ 3) − (3 ÷ 3) = 99

2c − ab = 2 × (−4) − (−2) × 5 = −8 − (−10) = 2

use brackets with negatives

Primes (two factors)

2, 3, 5, 7, 11, 13 ...

factorisation

= 23 × 3

Properties

Order of operations

• Brackets, × and ÷ then + and − 10 × (−3) + 7 = −30 + 7 = −23

Multiples of 3: 3, 6, 9, 12 ...

Multiples of 4: 4, 8, 12 ∴ LCM = 12 HCF

Squares and cubes

= 16, √16 = 4 = 27, 3√27 = 3

than two factors 4, 6, 8, 12

Powers and indices

× 9 × 9 × 9 = 94

Factors of 8: 1, 2, 4, 8

Factors of 28: 1, 2, 4, 7, 14, 28

∴ HCF = 4

Divisibility

• 2 Last digit even.

• 3 Sum of digits divisible by 3.

• 4 Number from last 2 digits divisible by 4.

• 5 Last digit 0 or 5.

• 6 Divisible by 2 and 3.

• 8 Number from last 3 digits divisible by 8 or ends in 000.

• 9 Sum of digits divisible by 9.

• 10 Last digit 0.

Chapterchecklist

AversionofthischecklistthatyoucanprintoutandcompletecanbedownloadedfromyourInteractiveTextbook.

1A 1

1A 2

1A 3

Icanusementaladditionandsubtractiontechniqueseffectively e.g.Evaluatethefollowingmentally:

a 347−39 b 125 + 127

Icanusetheadditionalgorithmwithwholenumbers e.g.Useanalgorithmtoﬁndthissum. 938 + 217

Icanusethesubtractionalgorithmwithwholenumbers e.g.Useanalgorithmtoﬁndthisdifference. 141 −86

1B 4

1B 5

1B 6

1C 7

1C 8

1C 9

Icanusementalmultiplicationanddivisiontechniqueseffectively e.g.Findthefollowingmentally:

a 5 × 160 b 464 ÷ 4

Icanusethemultiplicationalgorithmwithwholenumbers e.g.Useanalgorithmtoevaluate 412 × 25

Icanusethedivisionalgorithmwithwholenumbers e.g.Useanalgorithmtoevaluate 938 ÷ 13.

Icanwriteproductsusingindexnotation e.g.Write 8 × 8 × 8 usingindexnotation.

Icanconvertfromindexnotationtoexpandednotation e.g.Write 54 inexpandedform.

Icanﬁndthesquareandcubeofwholenumbers e.g.Find:

a 62 b 23

1C 10

1D 11

1D 12

1D 13

Icanﬁndthesquarerootandcuberootofcertainsmallwholenumbers e.g.Find: a √81 b 3 √64

Icanclassifyanumberasprimeorcomposite(orneither) e.g.Decidewhethereachofthefollowingisprimeorcomposite:

a 29 b 63

Icanﬁndthelowestcommonmultiple(LCM)oftwowholenumbers e.g.FindtheLCMof: 6 and 8

Icanﬁndthehighestcommonfactor(HCF)oftwowholenumbers e.g.FindtheHCFof: 36 and 48

1E 14 Icanwriteanumberastheproductofprimefactorsusingafactortree e.g.Write 300 asaproductofprimefactors.

1E 15 Icanusedivisibilityteststodetermineifanumberisdivisible by 2, 3, 4, 5, 6, 8 or 9 e.g.Decidewhether 627 isdivisibleby 2, 3, 4, 5, 6, 8 or 9

1E 16 Icanﬁndthelowestcommonmultiple(LCM)andhighestcommonfactor (HCF)oftwowholenumbersusingprimefactorisation e.g.FindtheLCMandHCFofthefollowingusingprimefactorisation: 105 and 90

1F 17 Icanaddapositiveintegertoanegativeinteger e.g.Evaluate:

a −5 + 2 b −1 + 4

1F 18 Icansubtractapositiveintegerfromanotherinteger e.g.Evaluate:

a 3−7 b −2−3

1G 19 Icanaddnegativeintegerstoanotherinteger e.g.Evaluate:

a 10 + (−3) b −3 + (−5)

1G 20 Icansubtractnegativeintegersfromanotherinteger e.g.Evaluate:

a 4− (−2) b −11− (−6)

1H 21 Icanﬁndtheproductofintegers

e.g.Evaluate:

a 3 × (−7) b −4 × (−12)

1H 22 Icanﬁndthequotientofintegers

e.g.Evaluate: a −63 ÷ 7 b −121 ÷ (−11)

1H 23 Icanuseorderofoperationswithintegers

e.g.Evaluate −7 + 6 × (−5).

1I 24 Icanuseorderofoperationstoevaluatenumericalexpressions e.g.Evaluatethefollowing.

a 10 + 5 × 3 b −20 ÷ 22 −1

1I 25 Icanuseorderofoperationstoevaluatenumericalexpressionsinvolving groupingsymbols e.g.Evaluatethefollowing.

a (7 + 2) × 5−6 b 10 ÷ (7−2) (−1)

Short-answerquestions

1 1A Useamentalstrategytoevaluatethefollowing.

324 + 173 a 592−180 b

89 + 40 c 135−68 d 55 + 57 e

+ 998 g

2 1A Useamentalstrategytoﬁndthesesumsanddifferences.

3 1B Useamentalstrategyfortheseproductsandquotients.

4 1B Findtheseproductsandquotientsusingsettingout.

5 1B Findtheremainderwhen 673 isdividedbythesenumbers.

6 1C Writeusingpowers.

7 1D Evaluate.

8 1D Findallthefactorsof 60 a

Findallthemultiplesof 7 between 110 and 150 b

Findalltheprimenumbersbetween 30 and 60 c

FindtheLCMof 8 and 6 d

FindtheHCFof 24 and 30 e

9 1E Writethesenumbersinprimefactorform.Youmaywishtouseafactortree. 36 a 84 b 198 c

10 1E Usedivisibilityteststodecideifthesenumbersaredivisibleby 2, 3, 4, 5, 6, 8 or 9 84 a 155 b 124 c

11 1E Writethenumbers 20 and 38 inprimefactorformandthenusethistohelpﬁndthefollowing.

LCMof 20 and 38 a HCFof 20 and 38 b

12 1F Evaluate.

−6 + 9 a

5−13 c

−62−14 e

−111 + 110 g

13 1G Evaluate.

5 + (−3) a

(−6) e

14 1H Evaluate.

× 2 a

÷ (−5) e

15 1H Copyandcomplete.

+ (−6) b

−24 + 19 b

−7−24 d

−194−136 f

−328 + 426 h

÷ (−16) f

12 = a (−1)2 = b

22 = c (−2)2 = d

32 = e (−3)2 = f

16 1I Evaluateusingtheorderofoperations.

2 + 3 × (−2) a

−3 ÷ (11 + (−8)) b

−2 × 3 + 10 ÷ (−5) c

−20 ÷ 10−4 × (−7) d

17 1I Substitute a = −2, b = 3 and c = −5 andevaluatetheseexpressions. ab + c a a2 b b

ac b c a + b + c d

) d

+ (

÷ (−27) h

Multiple-choicequestions

1 1I 400 ÷ 5 × 2 isthesameas:

400 ÷ 10 A

× 2

2 1A Thesumanddifferenceof 97 and 49 are:

146 and 58 A

246 and 48 B

136 and 58 C

146 and 48 D

147 and 58 E

3 1E 561 isexactlydivisibleby:

4 1B 89 × 5 isthesameas:

90 × 4 A

90 × 5−1 × 5 B

89 × 10 × 2 C

178 × 10 D

450 E

5 1C 2 × 2 × 2 × 2 × 5 × 5 is:

24 × 52 A

2 × 4 + 5 × 2 B

24 + 52 C 107 D 1000 E

6 1D TheLCMof 22 × 3 × 5 and 2 × 7 is: 2 A

22 × 3 × 5 × 7 B

2 × 3 × 5 × 7 C

23 × 3 × 5 × 7 D 7 E

7 1F −15 + 12−3 equals:

8 1G −6 + (−4) isthesameas: −6−4 A

9 1C If 182 = 324,then √324 equals: 162 A

+ 6

10 1F If a = −2, b = 3 and c = −1 then a2 + bc equals:

Extended-responsequestions

1 Amonthlybankaccountshowsdepositsaspositivenumbersandpurchasesandwithdrawals (P + W) asnegativenumbers.

Details P + W Deposits

Openingbalance – – $250

Waterbill −$138 – a

Cashwithdrawal −$320 – b Deposit – c $115

Supermarket d – −$160

Deposit – $400 e

Findthevaluesof a, b, c, d and e

Ifthewaterbillamountwas $150,whatwouldbethenewvalueforletter e?

b Whatwouldtheﬁnaldepositneedtobeifthevaluefor e was $0?Assumetheoriginalwater billamountis $138 asinthetableabove.

2 Twoteamscompeteataclubgamesnight.TeamAhas 30 playerswhileteamBhas 42 players. Howmanyplayersarethereintotal? a

Writeboth 30 and 42 inprimefactorform. b FindtheLCMandHCFofthenumberofplayersrepresentingthetwoteams. c Teamsareaskedtodivideintogroupswithequalnumbersofplayers.Whatisthelargestgroup sizepossibleifteamAandteamBmusthavegroupsofthesamesize?

2 Anglerelationships andpropertiesof geometricalﬁgures

Essentialmathematics:Whyanglerelationshipsand propertiesofgeometricalﬁguresareimportant

Geometryknowledgeisappliedinpracticaloccupationsbyexpertssuchasarchitects,engineers, surveyors,jewellers,fashiondesigners,plumbers,builders,carpenters,sheetmetalworkersand urbanplanners.

Surveyorsuseparallellinegeometryfortheplacementofparallellinesonathleticstracks,sports courts,airportrunways,andtrainandtramtracks.

Fashiondesignersstartbysketchingafashionblock-ﬁgurewhichisahumanﬁguredrawnusing onlygeometricalshapes,mostlyquadrilateralsandtriangles.Geometryknowledgeisthe foundationforcreatingoriginalfashiondesigns.

Builderscheckifahousewallframeisrectangularbymeasuringitstwodiagonals.Anydifference indiagonallengthsmeansthewallframeisaparallelogram,andthehousewouldbe‘outof square’ifthiserrorwasnotﬁxed.

Inrobotics,engineersemploythree-dimensionalcoordinatesystemstoprogramrobotstofollow speciﬁcroutesandcompleteallocatedtasks.

Inthischapter

2AAnglesatapoint (Consolidating)

2BParallellines (Consolidating)

2CTriangles

2DQuadrilaterals

2EPolygons

2FSolids

2GThreedimensionalcoordinate systems

AustralianCurriculum9.0

SPACE

Establishpropertiesofquadrilaterals usingcongruenttrianglesandangle properties,andsolverelatedproblems explainingreasoning(AC9M8SP02)

Describethepositionandlocationof objectsin3dimensionsindifferent ways,includingusinga three-dimensionalcoordinatesystem withtheuseofdynamicgeometric softwareandotherdigitaltools (AC9M8SP03)

©ACARA

Onlineresources

Ahostofadditionalonlineresources areincludedaspartofyourInteractive Textbook,includingHOTmathscontent, videodemonstrationsofallworked examples,auto-markedquizzesand muchmore.

1 Nametheseobjects.Choosefrom: A line AB B segment AB C point A D angle ABC.

Warm-up

2 Chooseacorrectanglenamefrom A ÒDEF B ÒSTU C ÒABC foreachgivendiagram.

3 Nametheseanglesas A acute B right C obtuse D straight E reﬂex F revolution

4 Namethetrianglethatﬁtsthedescription.Choosefrom A scalene B isosceles C equilateral D acute E right F obtuse.Drawanexampleofeachtriangletohelp.

Oneobtuseangle a 2 equallengthsides b Allanglesacute c 3 differentsidelengths d 3 equal 60° angles e onerightangle f

5 Namethesixspecialquadrilateralswithfoursides.Choosefrom A circle B square C parallelogram D line E triangle F rectangle G rhombus H hexagon I kite J trapezium K tetrahedron

6 Findthevalueof a inthesediagrams.

7 Thisdiagramincludesapairofparallellinesandathirdline (transversal).

Whatisthevalueof a? a

Whichpronumerals (b, c, d, e, f or g) areequalto a?Listin alphabeticalorder. b

Whichpronumerals (b, c, d, e, f or g) areequalto 50?Listin alphabeticalorder. c

8 Findthevalueof x intheseshapes,usingthegivenanglesum. Anglesum = 180° x° 20°

° a Anglesum = 360°

2A 2A Anglesatapoint CONSOLIDATING

Learningintentions

• Tobeabletoclassifyanglesasacute,right,obtuse,straight,re exorarevolution.

• Tobeabletonameanglesinrelationtootherangles,forinstance,namingtheangleverticallyoppositeto agivenangle.

• Tobeabletodeterminetheanglesatapointusingangleproperties.

• Tobeabletorelatecompassbearingstoangles.

Keyvocabulary: acute,right,obtuse,straight,re ex,revolution,complementary,supplementary, verticallyopposite,perpendicular,compassbearing

Fromthreesimpleobjects–point,lineandplane–wecandevelopalltheelementsofgeometry, justastheGreekmathematicianEucliddidabout 2300yearsago.

Wecanstartbylookingattheanglesformed whenlinesmeetatapoint.

Lessonstarter:Howmanyangles?

Whentwolinescross,differentanglesareformed, likeinthisexample.

• Isthereanother 60° angle?Why?

• Whatisthesizeofoneoftheobtuseangles?How didyouworkthisout?

• Arethereanystraightanglesinthediagram?

• Arethereanyreﬂexanglesinthediagram?

• Whatisarevolutionangle?

Keyideas

Typesofangles

Euclidis considered bymanyas the’father ofgeometry’ havingwritten aboutitaround 300BCE.

Specialpairsofanglesatapointinclude:

Complementary angles (addupto 90°)

• Supplementary angles (addupto 180°)

• Verticallyopposite angles(equal) a° a° •

° b° a + b = 180

b° a° a + b = 90

Anglesina revolution addupto 360° .

Twolinesare perpendicular iftheyintersectatrightangles (90°)

Eightpointcompassbearing

• Bearingsareusuallymeasuredclockwisefromnorth. (360°) N (0°) (315°) NW (270°) W (225°) SW S (180°) SE (135°) E (90°) NE (45°)

Exercise2A

Und er stand ing

1 Writethemissingword.Choosefrom: equal,supplementary,complementary and perpendicular Anglesthataddto 90° arecalled angles. a Anglesthataddto 180° arecalled angles. b Iftwolinesmeetatrightangles (90°),thentheyaresaidtobe c Verticallyoppositeanglesare . d

2 Whattypeofanglearethefollowing?

3 Completethesesentencesforthisdiagram. b° and c° are angles. a a

HintforQ1:Choose from: acute,right,obtuse, straight,reﬂexorrevolution

and e

are angles. b

4 Estimatethesizeoftheseangles.

ÒAOB a ÒAOC b Reﬂex ÒAOE c

Fluency

Example1Namingangles

Nameananglewhichis:

verticallyoppositeto ÒDOE a complementaryto ÒCOB b supplementaryto ÒEOA c

Solution

5,6,7(½),85,7,8

Explanation

a ÒAOB ÒDOE and ÒAOB areequalandsitopposite eachother.

b ÒBOA ÒCOB and ÒBOA addto 90°

c ÒDOE (or ÒAOB)

Nowyoutry

Nameananglewhichis:

verticallyoppositeto ÒPOQ a complementaryto ÒMOR b supplementaryto ÒMOQ c

5 Nameananglewhichis: verticallyoppositeto ÒDOE a complementaryto ÒCOB

Pairsofanglesona straightlineare supplementary(addto 180°).

HintforQ5:Verticallyopposite anglesareoppositeandequal. Complementaryanglesaddto 90° Supplementaryanglesaddto 180° b supplementaryto ÒEOA c

Example2Findinganglesatapoint

Determinethevalueofthepronumeralsinthesediagrams.

Solution

a a + 30 = 90

a = 60

b + 90 = 360

b = 270

b a + 65 = 180

a = 115

b = 65

Nowyoutry

Explanation

a° and 30° makeacomplementarypairof anglesaddingto 90°.Anglesinarevolution addto 360°

a° and 65° makeasupplementarypairof anglesaddingto 180° b° isverticallyopposite the 65° angle.

Determinethevalueofthepronumeralsinthesediagrams.

6 Statethevalueofthepronumeral(letter)inthesediagrams.

7 Determinetheunknownanglesmarkedinthesediagrams.

HintforQ7:Anglesinarightangle addto 90°.Anglesonastraight lineaddto 180°.Anglesina revolutionaddto 360°

8 Givethecompassbearing,indegrees,forthesedirections.

West(W) a East(E) b North(N) c South(S) d NW e SE f SW g NE h

Problem-solving and reasoning

HintforQ8:Checkthe Keyideas forhelp.

9,109,10–11(½),12

9 Aroundbirthdaycakeiscutintosectorsforninefriends(includingJack)atJack’sbirthdayparty.After thecakeiscutthereisnocakeremaining.WhatwillbetheangleatthecentreofthecakeforJack’s pieceif:

everyonereceivesanequalshare? a

Jackreceivestwiceasmuchaseveryoneelse?(Inparts b, c and d assumehisfriendshaveequal sharesoftherest.) b

Jackreceivesfourtimesasmuchaseveryoneelse? c

Jackreceivestentimesasmuchaseveryoneelse? d

10 Inwhichdirection(e.g.north-eastorNE)wouldyoubewalking ifyouwereheadedonthesecompassbearings?

11 Findthevalueofthepronumeralsinthesediagrams.

12 Explain,withreasons,whatiswrongwiththesediagrams.

Clockgeometry

13 Hereisaclockfacewithanhourhand(shortarrow) andminutehand(longarrow).Thetimeshown is 8:20

a Howmanydegreesdoesthehourhandturnin: 6 hours? i 12 hours? ii 1 hour? iii 3 hours? iv

b Howmanydegreesdoestheminutehandturnin: 1 hour? i 30 minutes? ii 5 minutes? iii 20 minutes? iv

14 Whatistheanglebetweenthehourhandandminute handonaclockatthesetimes?

2:30 pm a 5:45 am b

1:40 am c 10:20 pm d

2:35 am e 12:05 pm f

4:48 pm g 10:27 am h

2B 2B Parallellines CONSOLIDATING

Learningintentions

• Tounderstandthatparallellinesneverintersectandthatarrowsareusedtoindicatethisonadiagram.

• Tobeabletousepropertiesofparallellinesto ndunknownangles.

Keyvocabulary: parallellines,transversal,corresponding,alternate,cointerior

Insimplelanguage,Euclid’s 5thaxiomsaysthat parallellinesdonotintersectormeetonaﬂat plane.

Allsortsofshapesandsolidsbothinthe theoreticalandpracticalworldscanbeconstructed usingparallellines.Iftwolinesareparallelandare cutbyathirdlinecalledatransversal,specialpairs ofanglesarecreated.

Lessonstarter:Arethey parallel?

Herearetwodiagramsthatshowapairoflines crossedbyathirdlinecalledatransversal.Two anglesaregiven.

• Doyouthinkthateachdiagramcontainsa pairofparallellines?

• Canyoudeterminealltheotheranglesin thediagrams?

• Howmanydifferentanglesaretherein eachdiagram?

Allsortsofshapesandsolidscanbeconstructedusing parallellinesandtransversals.

Keyideas

A transversal isalinecuttingatleasttwootherlines.

Pairsofanglesformedbytransversalscanbe: corresponding (in correspondingpositions) corresponding

alternate (onopposite sidesofthetransversaland insidetheothertwolines)

cointerior (onthesame sideofthetransversaland insidetheothertwolines). cointerior

Parallellinesneverintersect.

Linesare parallel iftheydonotintersect.

• Parallellinesaremarkedwiththesamenumberofarrows.

Iftwoparallellinesarecutbyatransversal:

• thecorrespondinganglesareequal(pairs)

• thealternateanglesareequal(2 pairs)

• thecointerioranglesaresupplementary(addupto 180°)(2 pairs)

Exercise2B

Und er stand ing

1 Twoparallellinesarecutbyatransversal.Writethemissingword. Correspondinganglesare a Cointerioranglesare b Alternateanglesare c

2 Nametheanglethatis: correspondingto ÒABF a correspondingto ÒBCG b alternate to ÒFBC c alternate to ÒCBE d cointeriorto ÒHCB e cointeriorto ÒEBC f vertically oppositeto ÒABE g verticallyopposite to ÒHCB h

HintforQ1:Choosefrom equal or supplementary.

HintforQ2:Nameangleslike this: ÒABC or ÒDEF

Example3Describingrelatedanglesinparallellines

Statewhetherthefollowingmarkedanglesarecorresponding,alternateorco-interior.Hence,state whethertheyareequalorsupplementary.

Solution

a Themarkedanglesarealternate. Thereforetheyareequal (a = b)

b Themarkedanglesareco-interior.Therefore theyaresupplementary (a + b = 180)

Explanation

Thetwoanglesareinsidetheparallellinesand onoppositesidesofthetransversal,sotheyare alternate.

Alternateanglesinparallellinesareequal.

Thetwoanglesareinsidetheparallellinesand onthesamesideofthetransversal,sotheyare co-interior.

Co-interioranglesinparallellinesare supplementary,addingto 180°

c Themarkedanglesarecorresponding. Thereforetheyareequal (a = b)

Nowyoutry

Thetwoanglesareincorrespondingpositions (bothtotheleftoftheintersectionpoints). Correspondinganglesinparallellinesareequal.

Statewhetherthefollowingmarkedanglesarecorresponding,alternateorco-interior.Hencestate whethertheyareequalorsupplementary.

3 Statewhetherthefollowingmarkedanglesarecorresponding,alternateorco-interior.Hence,state whethertheyareequalorsupplementary.

Example4Workingwithparallellines

Findthevalueofthepronumeralsinthesediagrams.Giveareasonforeachanswer.

Solution

Explanation

a a = 71,alternateanglesinparallellines.Alternateanglesinparallellinesareequal.

b b = 118,correspondinganglesin parallellines. Correspondinganglesinparallellinesareequal.

c c = 180−115 = 65,cointerioranglesin parallellines. Cointerioranglesinparallellinesaddto 180°

Nowyoutry

Findthevalueofthepronumeralsinthesediagrams.Giveareasonforeachanswer.

4 Findthevalueofthepronumeralsinthesediagrams.Giveareasonforeachanswer.

Example5Usingparallellinesinshapes

Findthevalueofthepronumeralsinthisdiagram,statingreasons.

HintforQ4:Correspondinganglesare equalinparallellines.Alternateangles areequalinparallellines.Cointerior anglesinparallellinesare supplementary(addto 180°).

Solution

a + 72 = 180

a = 108

b + 72 = 180

b = 108

Explanation

Thepairsofanglesarecointerior,whichare supplementaryifthelinesareparallel.

Thisshowsthatoppositeanglesina parallelogramareequal.

Cointerioranglesinparallellinesaresupplementary.

Nowyoutry

Findthevalueofthepronumeralsinthisdiagram,statingreasons.

5 Findthevalueofthepronumeralsinthesediagrams,statingreasons.

HintforQ5:Cointeriorangles addto 180

Problem-solving and reasoning

6 Findthevalueofthepronumeralsinthesediagrams,statingreasons.

7 Decideifthefollowingdiagramsincludeapairofparallellines.Giveareasonforeachanswer.

8 Findthevalueof a inthesediagrams.

HintforQ8:Extendinglinescan makeiteasiertoseethistype ofdiagram.

9 Sometimesparallellinescanbeaddedtoadiagramtohelpfindanunknownangle.Forexample, ÒAOB canbefoundinthis diagrambyfirstdrawingthedashedlineandfinding ÒAOC (40°) and ÒCOB (70°) So ÒAOB = 40°+ 70°= 110°

Applyasimilartechniquetoﬁnd ÒAOB inthesediagrams.

Pipenetworks

10 Aplanforanaturalgasplantincludesmanyintersectingpipelines,someofwhichareparallel. Helpthedesignersﬁnishtheplansbycalculatingthesizeoftheanglesmarked a, b,etc.

2C 2C Triangles

Learningintentions

• Tounderstandthattrianglescanbeclassi edbytheirsidelengthsasscalene,isoscelesorequilateral.

• Tounderstandthattrianglescanbeclassi edbytheirinterioranglesasacute,rightorobtuse.

• Toknowthattheanglesumofanytriangleis 180° .

• Tobeabletousetheanglesumofatriangleto ndunknownangles.

• Tobeabletousetheexteriorangletheoremto ndunknownangles.

Keyvocabulary: scalene,isosceles,apex,base,equilateral,acute,right,obtuse,exteriorangle,anglesum

Atriangleisashapewiththreestraightsides.Thetriangleisavery rigidshapeandthisleadstoitsuseintheconstructionofhousesand bridges.Itisoneofthemostcommonlyusedshapesindesignand construction.

Lessonstarter:Illustratingtheanglesum

Youcancompletethistaskusingapencilandrulerorusingdynamic geometrysoftware.

• Drawanytriangleandmeasureeachinteriorangle.

• Addallthreeanglestoﬁndtheanglesumofyourtriangle.

• Compareyouranglesumwiththeresultsofothers.Whatdo younotice?

Triangularshapesareoftenused tostrikingeffectinarchitecture,as shownbypartoftheNationalGallery ofCanada.

Ifdynamicgeometryisused,dragoneoftheverticestoaltertheinteriorangles.Nowchecktoseeif yourconclusionsremainthesame.

Keyideas

A triangle has:

• 3 sides

• 3 vertices(singular:vertex)

• 3 interiorangles.

Trianglesareclassiﬁedbysidelengths:

• Sideswiththesamenumberofdashesareofequallength.

Trianglesareclassiﬁedbyinteriorangles: acute (Allanglesacute) • right (1 rightangle) • obtuse (1 obtuseangle). •

The anglesum ofatriangleis 180°

The exteriorangle theorem: Theexteriorangleofatriangleisequal tothesumofthetwooppositeinterior angles.

exterior angle

Exercise2C

1 Givethecommonnameofatrianglewiththeseproperties.Refertothe Keyideas inthissection forhelp.

Onerightangle a 2 equalsidelengths b Allanglesacute c Allangles 60° d

Oneobtuseangle e 3 equalsidelengths f 2 equalangles g 3 differentsidelengths h

2 Statewhetherthesetrianglesarescalene,isoscelesorequilateral.

3 Statewhetherthesetrianglesareacute,rightorobtuse.

Example6Usingtheanglesumofatriangle

Findthevalueof a inthistriangle.

Solution

a + 38 + 92 = 180

a + 130 = 180 a = 50

Nowyoutry

Findthevalueof a inthistriangle.

Explanation

Theanglesumofthethreeinterioranglesofatriangleis 180° Also 38 + 92 = 130 and 180−130 = 50

4 Usetheanglesumofatriangletohelpﬁndtheunknownangleinthese triangles.

HintforQ4:Foreachone useamentalstrategyor startwithanequationlike a + 36 + 48 = 180.

Example7Workingwithisoscelestriangles

Findthevalueof a intheseisoscelestriangles.

Solution

a a + 35 + 35 = 180

a + 70 = 180 a = 110

b 2a + 26 = 180 2a = 154 a = 77

Nowyoutry

Findthevalueof a intheseisoscelestriangles.

Explanation

Thetwobaseanglesin anisoscelestriangleare equal.

Thetwobaseanglesinan isoscelestriangleareequal.

5 Thesetrianglesareisosceles.Findthevalueof a

HintforQ5:Thetwo baseanglesinan isoscelestriangle areequal.

Example8Usingtheexteriorangletheorem

Findthesizeoftheunknownangle.

Solution

a + 90 = 161

a = 161−90 = 71

Nowyoutry

Findthesizeoftheunknownangle.

6 Findthevalueof

Explanation

Usetheexteriorangletheoremforatriangle. Theexteriorangle (161°) isequaltothesumof thetwooppositeinteriorangles.

Problem-solving and reasoning

7 Decideifitispossibletodrawatrianglewiththegivendescription.Drawadiagramtosupport youranswer.

Rightandscalene a Obtuseandequilateral b Rightandisosceles c Acuteandisosceles d Acuteandequilateral e Obtuseandisosceles f

8 Atriangleisconstructedusingacircleandtwo radiuslengths.

Whattypeoftriangleis DAOB andwhy? a Nametwoanglesthatareequal. b

Find ÒABO if ÒBAO is 30° c

Find ÒAOB if ÒOAB is 36° . d

Find ÒABO if ÒAOB is 100

9 Findthevalueof a inthesediagrams.

HintforQ8a:Whatcanyousay aboutthelengths OA and OB?

Parallellinesandtriangles

10 Useyourknowledgeofparallellinesandtrianglestoﬁndtheunknownangle a

HintforQ9a:Whatisthesizeof eachangleinanequilateral triangle?

10–11

11 Toprovethattheanglesumofatriangleis 180°,workthroughthesestepswiththegivendiagram.

HintforQ11:Thinkbacktothe parallellinerulesfrom Section2B

a Usingthepronumerals a, b or c,givethevalueoftheseanglesandstateareason.

ÒABD i ÒCBE ii

b Whatistrueaboutthe threeangles ÒABD, ÒABC and ÒCBE andwhy?

c Whatdoparts a and b abovesay aboutthepronumerals a, b and c,andwhatdoesthissayabout theanglesumofthetriangle ABC?

2D 2D Quadrilaterals

Learningintentions

• Tobeabletoclassifyquadrilateralsasparallelograms,rectangles,rhombuses,squares,kitesand/ortrapezia.

• Toknowthattheanglesumofanyquadrilateralis 360° .

• Tobeabletousetheanglesumofaquadrilateralto ndunknownangles.

• Tounderstandthatpropertiesofanglesinparallellinescanbeusedto ndunknownanglesintrapezia andparallelograms.

Keyvocabulary: quadrilateral,parallelogram,square,rectangle,rhombus,kite,trapezium,parallel

Shapeswithfoursidesarecalledquadrilaterals.Allquadrilateralshavethesameanglesum.Theirother propertiesdependonsuchthingsaspairsofsidesofequallength,parallelsidesandlengthsofdiagonals. Allquadrilateralscanbedividedintotwotriangles.Sincethesixanglesinsidethetwotrianglesmake upthefouranglesofthequadrilateral,theanglesumis 2 × 180°= 360° .

Lessonstarter:Whichquadrilateralssuit?

Quadrilaterals ofmanykinds areusedby architects.

Nameallthedifferentquadrilateralsyoucanthinkofthathavethepropertieslistedbelow.Theremay bemorethanonequadrilateralforeachpropertylisted.Draweachquadrilateraltoillustratetheshape anditsfeatures.

4 equallengthsides

2 pairsofparallelsides

pairofparallelsides

Keyideas

Quadrilaterals arefour-sidedshapes.

Parallelograms arequadrilateralswithtwopairsof parallel sides.

Parallelogram

Specialparallelograms Square Rectangle Rhombus

Otherspecialquadrilaterals Kite Trapezium

Theanglesumofanyquadrilateralis 360° .

Quadrilateralswithparallelsidesincludetwopairsof cointeriorangles.

Und er stand ing

1 Whatarethesixspecialtypesofquadrilaterals?

2 Writethemissingnumberorword.Choosefrom: 90° , equal, 360° or two Theanglesumofaquadrilateralis . a Thesidelengthsofarhombusare inlength. b Akitehas pairsofequalsides. c Thediagonalsofsquares,rhombusesandkitesintersectat d

3 Answertrue(T)orfalse(F)tothesestatements.Refertothediagramsinthe Keyideas oraccurately drawyourownshapes,includingthediagonals.

a Square

Allsidesareofequallength. i Diagonalsarenotequalinlength. ii Allsidesareparalleltoeachother. iii Diagonalsintersectatrightangles. iv

b Rectangle

Diagonalsmustintersectatrightangles. i Allinterioranglesare 90° ii Allsidesmustbeofequallength. iii Therearetwopairsofparallelsides. iv

c Rhombus

Allinterioranglesmustbeequal. i Allsidesareofequallength. ii Diagonalsintersectatrightangles. iii

d Parallelogram

Therearetwopairsofparallelsidesofequallength. i Diagonalsmustbeequalinlength. ii Diagonalsmustintersectatrightangles. iii

e Kite

Therearetwopairsofsidesofequallength. i Therearetwopairsofparallelsides. ii Diagonalsintersectatrightangles. iii

f Trapezium

Diagonalsareequalinlength. i Therearetwopairsofparallelsides. ii

HintforQ3:Thereddashed linesarethediagonals.

Example9Usingtheanglesumofaquadrilateral

Findthevalueofthepronumeralsinthesequadrilaterals.

Solution

a a + 100 + 90 + 115 = 360 a + 305 = 360 a = 55

b x + 265 + 30 + 25 = 360 x + 320 = 360 x = 40

Nowyoutry

Explanation

Thesumofanglesinaquadrilateralis 360° Useamentalstrategyorsolvetheequation.

Usetheanglesumofaquadrilateral.

Findthevalueofthepronumeralsinthesequadrilaterals.

4 Usethequadrilateralanglesumtoﬁndthevalueof a inthesequadrilaterals.

HintforQ4: isa 90° angle.Theangle sumofaquadrilateralis 360

Example10Workingwithparallelograms

Findthevalueof a and b inthisparallelogram.

Solution

a + 77 = 180

a = 103

b = 180−103 = 77

Nowyoutry

Findthevalueof a and b inthisparallelogram.

Explanation

Twoanglesinsideparallellinesarecointerior andthereforeaddupto 180° Notethatoppositeanglesinaparallelogram areequal.

5 Findthevalueofthepronumeralsinthesequadrilaterals.

HintforQ5:Oppositeangles inaparallelogramareequal. Otherpairsofanglesare cointerior(addto 180°).

6 Namethespecialquadrilateralswhichhave: allsidesofequallength a onepairofparallellines b twopairsofequalsides c diagonalsmeetingatrightangles d diagonalsofequallength. e

7 Useyourknowledgeofgeometryfromtheprevioussectionstoﬁndthe valuesof a

HintforQ7:Anglesinarevolution addto 360°.Anglesonastraight lineaddto 180°.Vertically oppositeanglesareequal.

8 Considerthepropertiesofspecialquadrilaterals.Decideifthefollowingaretrue(T)orfalse(F). Asquareisatypeofrectangle. a Arectangleisatypeofsquare. b Asquareisatypeofrhombus. c Arectangleisatypeofparallelogram. d Aparallelogramisatypeofsquare. e Arhombusisatypeofparallelogram. f

9 Considerthepropertiesofthegivenquadrilaterals.Givethevaluesofthe pronumerals.

HintforQ9: Akite a Arhombus b Aparallelogram c

10 Youcanconﬁrmthattheanglesumofaquadrilateralis 360° bytearingoffthecornersofany cut-outquadrilateral. Usearulertodrawanyquadrilateralandthencutitout. a Tearoffthefourcorners. b Arrangethefourpiecessothecorners(verticesofquadrilateral)allmeetatonepointasshown. Whatdoyounotice? c Whatdoesthistellyouabouttheanglesumofthequadrilaterals? d

1 2A Determinethevalueofthepronumeralsinthesediagrams.

2 2A Findthevalueofthepronumeralsinthesediagrams.

3 2B Whichofthefollowingdiagramshaveapairofcorrespondinganglesmarked?

4 2B Findthevalueofthepronumeralsinthesediagrams.Giveareasonforeachanswer.

5 2B Findthevalueofthepronumeralsinthesediagrams,statingreasons.

6 2C Usetheanglesumofatriangletohelpﬁndtheunknownangleinthesetriangles.

7 2C Findthesizeoftheunknownangles.

8 2D Findthevalueofthepronumeralsinthesequadrilaterals.

9 2D Findthevalueofthepronumeralsinthesequadrilaterals.

10 2D Decideifthefollowingaretrue(T)orfalse(F). Allsquaresarerectangles. a Allquadrilateralsarerectangles. b Allrectanglesarerhombuses. c

2E 2E Polygons

Learningintentions

• Toknowthenamesofdifferenttypesofpolygonswithupto 12 sides.

• Tounderstandwhataregularpolygonis.

• Tobeableto ndtheanglesumofapolygon.

• Tobeabletousetheanglesumofapolygonto ndunknownangles.

Keyvocabulary: polygon,regularpolygon,pentagon,hexagon,heptagon,octagon,nonagon,decagon, hendecagon,dodecagon

Trianglesandquadrilateralsarebothexamplesofpolygons. Theword‘polygon’comesfromtheGreekwords poly, meaning‘many’,and gonia,meaning‘angles’.Thenumber ofinterioranglesequalsthenumberofsides.Theanglesum ofeachtypeofpolygondependsonthisnumber.Inthis sectionwewillexploretherulefortheanglesumofa polygonwith n sides.

Lessonstarter:Developingtherule

Thefollowingprocedureusesthefactthattheanglesumof atriangleis 180°.Completethetableandtrytowritethe generalruleintheﬁnalrow.

ThePentagonisafamousgovernmentofﬁce buildinginWashingtonDC,USA.

Keyideas

Polygons areshapeswithstraightsides.

Pentagon

Octagon

Decagon

Polygonsarenamedaccordingtotheirnumberofsides.

Theanglesum S ofapolygonwith n sidesisgivenbytherule: S = (n −2) × 180°

A regularpolygon hassidesofequallengthandequalinteriorangles.

1 Namethepolygonwiththefollowingnumberofsides.

2 Statethenumberofsidesonthesepolygons.

Hexagon a Quadrilateral b Decagon c Heptagon d Pentagon e Dodecagon f

3 Evaluate (n −2) × 180° if: n = 6 a n = 10 b n = 22 c

4 Whatisthecommonnamegiventothesepolygons?

Regularquadrilateral a Regulartriangle b

Fluency

Example11Findingtheanglesum

Findtheanglesumofaheptagon.

Solution

S = (n −2) × 180°

= (7−2) × 180°

= 5 × 180° = 900°

Nowyoutry

Findtheanglesumofanoctagon.

5 Findtheanglesumofthesepolygons.

Explanation

Aheptagonhas 7 sidesso n = 7.Simplify (7−2) beforemultiplyingby 180°

Pentagon (n = 5) a Octagon (n = 8) b Decagon (n = 10) c Hexagon d Nonagon e Heptagon f

Example12Findinganglesinpolygons

Findthevalueof a inthispentagonbyusingthegivenanglesum.

Anglesum = 540°

Solution

a + 170 + 80 + 90 + 95 = 540

a + 435 = 540 a = 105

Nowyoutry

Explanation

Addalltheanglesandsetthisequaltothe anglesumof 540°.Thensimplifyandsolvefor a oruseamentalstrategy.

Findthevalueof a inthishexagonbyusingthegivenanglesum.

Anglesum = 720°

6 Findthevalueof a inthesepolygons,byusingthegivenanglesum.

HintforQ6:Writean equationusingthe givenanglesum,then ﬁndthevalueof a

7 Regularpolygonshaveequalinteriorangles.Findthesizeofaninterioranglefortheseregularpolygons withthegivenanglesum.

Decagon (1440

HintforQ7:First ﬁndtheanglesum thendividebythe numberofsides. Pentagon (540

Example13Findinginterioranglesofregularpolygons

Findthesizeofaninteriorangleinaregularoctagonbyﬁrstﬁndingtheanglesum.

Solution

S = (n −2) × 180° = (8−2) × 180° = 6 × 180°= 1080°

Interioranglesize = 1080 ÷ 8 = 135°

Nowyoutry

Explanation

Firstcalculatetheanglesumofanoctagon using n = 8 and S = (n −2) × 180°

All 8 anglesareequalinsizesodividethe anglesumby 8

Findthesizeofaninteriorangleinaregularnonagonbyﬁrstﬁndingtheanglesum.

8 Findthesizeofaninteriorangleintheseregularpolygonsbyﬁrstlyﬁndingtheanglesum.Roundthe answertoonedecimalplacewherenecessary. Regularpentagon a Regularheptagon b Regularhexagon c Regulardecagon d Regularoctagon e Regularhendecagon f

Problem-solving and reasoning

9 Findthevalueof a intheseshapesbyﬁrstlyﬁndingtheanglesum.

9,109(½),10–12

10 Findthenumberofsidesofapolygonwiththegivenanglesums. HintforQ10:Theanglesum

11 Findthenumberofsidesofaregularpolygonifeachinteriorangleis: 120° a 162° b 147.272727…° c

12 Consideraregularpolygonwithaverylargenumberofsides (n). Whatshapedoesthispolygonlooklike? a Istherealimittothesizeofapolygonanglesumordoesitincreasetoinﬁnityas n increases? b Whatsizedoeseachinteriorangleapproachas n increases? c

13 Findthevalueof x inthesediagrams.

2F 2F Solids

Learningintentions

• Tobeabletonamesolidsusingappropriateterminology(e.g.hexagonalprism,squarepyramid).

• Tobeabletocountthenumberoffaces,verticesandedgesinasolid.

Keyvocabulary: polyhedron,prism,pyramid,cross-section,face,vertex(plural:vertices),edge,apex,cylinder, sphere,cone,cube,hexahedron,cuboid,tetrahedron

Asolidisanobjectthatoccupiesthree-dimensionalspace.Theoutsidesurfacescouldbeﬂatorcurved. Asolidwithallﬂatsurfacesiscalledapolyhedron(plural polyhedra or polyhedrons).Theword‘polyhedron’ comesfromtheGreekwords poly,meaning‘many’,and hedron,meaning‘faces’.

ThetopofthisCanaryWharfbuildinginLondon(left)isalarge,complexpolyhedron.Polyhedraalsooccurin nature,particularlyinrockormineralcrystalssuchasquartz(right).

Lessonstarter:Namingchallenge

Solidscanbenamedbytheirnumberoffaces.Herearetwoexamples.

Hexahedron(6 faces)Octahedron(8 faces)

Nameanddrawasolidwiththefollowingnumberoffaces.

• 7 faces

• 5 faces

• 10 faces

Keyideas

A polyhedron (plural:polyhedra)isaclosedsolidwithﬂatsurfaces(faces), vertices and edges

• Polyhedracanbenamedbytheirnumberoffaces.

Hexahedron (orrectangular prismorcuboid) – 6 faces – 12 edges – 8 vertices

Tetrahedron (ortriangularpyramid) face vertex edge – 4 vertices – 4 faces – 6 edges

Prisms arepolyhedrawithtwoidentical(congruent)ends.The congruentendsdeﬁnethe cross-section oftheprismandalso itsname.Theotherfacesareparallelograms.Ifthesefacesare rectangles,asshown,thenthesolidisa rightprism

Pyramids arepolyhedrawithonefacethatisthebaseandall otherfacesmeetingatthesamevertexpointcalledthe apex Theyarenamedbytheshapeofthebase.

Somesolidshavecurvedsurfaces.Commonexamplesinclude:

Cylinder Sphere Cone

A cube isahexahedronwithsixsquarefaces.

Hexagonal prism

hexagon cross-section

apex

Square-based pyramid

square base

Anothernameforarectangularprismis cuboid.Itisalsoahexahedron.

Exercise2F

1 Writethemissingwordornumberinthesesentences.Choosefrom: congruent, cube, six, seven, vertices, octagonal, circle, seven. Ahexahedronhas faces. a Theﬂatfaceatthebaseofacylinderisa b Ahexahedronwithsixsquarefacesisalsocalleda c Apolyhedronhasfaces, andedges. d Aheptahedronhas faces. e Aprismhastwo ends. f Apentagonalprismhas faces. g Thebaseofapyramidhas 8 sides.Thepyramidiscalledan pyramid. h

2 Namethreesolidsthathavecurvedsurfaces.

3 Whichofthesesolidsarepolyhedra(i.e.haveonlyﬂatsurfaces)?

Cube A Pyramid B Cone C Sphere D

Cylinder E Rectangularprism F Tetrahedron G Hexahedron H

Example14Countingfaces,verticesandedges

Statethenumberoffaces,verticesandedgesforthesesolids.

Octahedron(orHexagonalprism) a

Solution

a 8 faces

12 vertices

Hexahedron(orPentagonalpyramid) b

Explanation

18 edges Facesaretheﬂatsurfaces. Verticesarethecorners. Edgesarethelinesonthediagram.

b 6 faces

6 vertices

10 edges

Thereisonepentagonalfaceandﬁvetriangular faces.

Fiveverticesareonthebaseplusoneapex. Fiveedgesareonthebaseandﬁvemeetthe apex.

Nowyoutry

Statethenumberoffaces,verticesandedgesforthesesolids.

Decahedron(Octagonalprism) a

Pentahedron(Square-basedpyramid) b

4 Countthenumberoffaces,verticesandedges(inthatorder)on thesepolyhedra.

HintforQ4:Facesareﬂat surfaces,verticesarecorners, andedgesarethelinesdrawn onthediagram. a b c

Example15Classifyingsolidsusingfaces

Classifythesesolidsbyconsideringthenumberoffaces. a b c

Solution

a Hexahedron

b Heptahedron

c Pentahedron

Nowyoutry

Namethepolyhedronthathas 8 faces.

Explanation

Thesolidhas 6 faces.

Thesolidhas 7 faces.

Thesolidhas 5 faces.

5 Namethepolyhedronthathasthegivennumberoffaces.

HintforQ5:Usenamessuchas pentahedron,hexahedron.

6 Howmanyfacesdothesepolyhedrahave?

Octahedron a Hexahedron b Tetrahedron c Pentahedron d Heptahedron e Nonahedron f Decahedron g Hendecahedron h

Example16Namingprisms

Namethesesolidsasatypeofprism. a b

Solution

a Rectangularprism

b Hexagonalprism

Nowyoutry

Namethissolidasatypeofprism.

Explanation

Thecross-sectionisarectangle.

Thecross-sectionisahexagon.

7 Nametheseprisms.

Example17Namingpyramids

Namethissolidasatypeofpyramid.

Solution Explanation Pentagonalpyramid Thebaseisapentagon.

Nowyoutry

Namethissolidasatypeofpyramid.

8 Namethesepyramids.

Problem-solving and reasoning

9 Nameeachofthesesolidsintwodifferentways.

HintforQ7:Name usingtheshapeof thecross-section.

HintforQ8:Nameusingthe shapeofthebase.

HintforQ9:Asanexample:a pentagonalpyramidisalsoa hexahedron(6 faces).

10 Decideifthefollowingstatementsaretrue(T)orfalse(F).Makedrawingstohelp. Atetrahedronisapyramid. a Allsolidswithcurvedsurfacesarecylinders. b Acubeandarectangularprismarebothhexahedrons. c Ahexahedroncanbeapyramid. d Therearenosolidswith 0 vertices. e Therearenopolyhedrawith 3 surfaces. f Allpyramidswillhaveanoddnumberoffaces. g

11 Nameeachofthesesolidsinthreedifferentways. a b

12 Investigateifthefollowingstatementistrue(T)orfalse(F). Forallpyramids,thenumberoffacesisequaltothenumberofvertices.

Euler’srule

13a Copyandcompletethistable.

Numberof Numberof Numberof Solid faces (F ) vertices (V ) edges (E ) F + V Cube Squarepyramid Tetrahedron

b Comparethenumberofedges (E ) withthevalue F + V foreachpolyhedron.Whatdoyounotice?

14a Apolyhedronhas 16 facesand 12 vertices.Howmanyedgesdoesithave?

b Apolyhedronhas 18 edgesand 9 vertices.Howmanyfacesdoesithave?

c Apolyhedronhas 34 facesand 60 edges.Howmanyverticesdoesithave?

TheNationalLibraryofBelarusisarhombicuboctahedronwith 8 triangularfacesand 18 squarefaces.

2G 2G Three-dimensionalcoordinatesystems

Learningintentions

• Toknowthatthepositionofobjectsinthree-dimensional (3D) spacecanbedescribedusinga 3Dcoordinatesystem

• Tobeabletodescribethepositionofanobjectorapointusinga 3Dcoordinatesystem

• Tobeabletosolvesimplegeometryproblemsin 3Dspaceusingcoordinates

Keyvocabulary: Three-dimensional(3D)space,coordinatesystem

Whenwefindourwayarounda multi-storeycarparkorworkwith a 3D-printer,weneedtoconsider three-dimensionalspace.Two dimensionsarerequiredtodefine aspaceatonelevel,likezonesat groundlevelinacarpark,buta thirddimensionisrequiredtodefine spacesabovethatlevel.

Lessonstarter:Wheremight 3Dcoordinatesbeuseful?

Three-dimensional coordinatesystems areusedinmany differenttechnologies suchas 3Dprinting.

Weknowthatwhentryingtodescribethepositionofazoneinacarparkyoumighttrytoremember threecharacters.TwocharactersforthezoneonalevellikeD4 andanothercharacterforthelevelinthe building.

• Listtwoorthreeotherexampleswherea 3Dcoordinatesystemmightbeusedtodescribeposition.

• Chooseoneofyourchosen 3Dsystemsandresearchhowthe 3Dcoordinatesystemisusedtodescribe position.Writedownthreeorfourdotpointstosummariseyourﬁndings.

Keyideas

Thepositionofanobjectinthree-dimensional (3D) spacecanbedescribedusingacoordinatesystem usingthreecomponents.

Threeaxes: x, y,and z areusedtodeﬁne 3Dspace.

• Theorigin O hascoordinates (0, 0, 0).

• Thethreeaxesareallatrightanglestoeachother.

Thepositionofacubicunitofspacecanbedescribed usinggridspacesasshown.

• Theredcubeisthisdiagramcouldbedescribedas (B, c, 3)

Apointinthree-dimensionalspaceisdescribedusing (x, y, z) coordinates.

• Thepoint P isthisdiagramhascoordinates (1, 2, 4)

Exercise2G

Und er stand ing 1,2 1,2

1 Considerthissetoftwoblockssittingin 3Dspacebelow.

a Whichofthefollowingdescribesthepositionofthesquareatthebaseoftheblocks?

(A, a) A (A, c) B (B, c) C

(b, A) D (C, c) E

b Whichof thefollowingdescribesthepositionoftheblockwhichisred?

(A, a, 2) A (B, c, 3) B (B, c, 2) C

(A, c, 2) D (A, c, 1) E

c Howmanyblockswould berequiredtobuildastackwheretheblockatthetopispositionedat (B, c, 6)?

2 Considerthis 3Dcoordinatesystemwithpoint P asshown.

a Onthe x-y plane,point A hascoordinates (2, 0).Whichofthefollowingisthecoordinates ofpoint A inthe 3Dsystem?

(2, 0, 1) A (2, 0, 3) B

(2, 0, 0) C (0, 2, 0) D

(0, 0, 2) E

b Whatarethe 3Dcoordinatesofthefollowingpoints? B i C ii P iii

c Pencilthefollowingpointsontothegivensetofaxes.

Fluency

Example18Usinggridspacestodescribepositionin 3Dspace

Eachcubeinthis 3Dspacecanbedescribedusingan x(A, B, C or D) , y(a, b, c or d) and z(1, 2, 3 or 4) coordinatein thatorder.Givethecoordinatesofthecubeswiththegiven colour. red a yellow b green c

Solution

a (A, b, 4)

b (D, b, 1)

c (B, d, 2)

Explanation

Usingtheorder x,then y then z,wehavethe baseofthecolumninthesquare (A, b) andthe redcubeis atlevel 4 onthe z-axis.

Thebaseoftheyellowcubeisthesquare (D, b) anditisatlevel 1 onthe z-axis.

Thegreencubehasabasesquareat (B, d) and isatlevel 2 onthe z-axis.

Nowyoutry

Eachcubeinthis 3Dspacecanbedescribedusingan

x(A, B, C or D) , y(a, b, c or d) and z(1, 2, 3 or 4) coordinate inthatorder.Givethecoordinatesofthecubeswiththe givencolour. red a yellow b green c

3 Eachcubeinthis 3Dspacecanbedescribedusingan

x(A, B, C or D) , y(a, b, c or d) and z(1, 2, 3 or 4) coordinatein thatorder.Givethecoordinatesofthecubeswiththegiven colour. red a yellow b green c

4 Eachcubeinthis 3Dspacecanbedescribedusingan

x(A, B, C or D) , y(a, b, c or d) and z(1, 2, 3 or 4) coordinateinthat order.Givethecoordinatesofthecubeswiththegivencolour. red a

Example19Using 3Dcoordinatestodescribeapoint

Writedownthecoordinatesofthe followingpointsshownonthis 3Dgraph.

Solution

a (3, 0, 0)

b (3, 2, 0)

c (0, 2, 4)

d (3, 2, 4)

Nowyoutry

Writedownthecoordinatesofthefollowing pointsshownonthis 3Dgraph.

Explanation

Point A ispositionedat 3 onthe x-axis, 0 onthe y-axisand 0 onthe z-axis.

Point B isonthe x-y planeatpoint (3, 2) and hasa z-coordinateof 0

Point C isonthe y-z planeandsothe x-coordinateis 0

Ontherectangularprismshown,point P is alignedwiththenumber 3 onthe x-axis, 2 on the y-axisand 4 onthe z-axis.

5 Writedownthecoordinatesofthefollowingpointsshownonthis 3Dgraph.

A a B b C c P d

6 Writedownthecoordinatesofthefollowingpointsshown onthis 3Dgraph.

7 Plotthegivenpointsontothis 3Dcoordinatesystem.

A (1, 2, 4) a

B (3, 1, 2) b

C (0, 2, 3) c

D (4, 0, 1) d

E (3, 3, 0) e

F (4, 4, 3) f

HintforQ5:Coordinatesare listedintheorder(x,y,z).

Problem-solving and reasoning

8 Choosethecorrectsquareonthe 2Ddiagram (i, ii, iiioriv) thatmatchesthebaseoftheblockfrom the 3Ddiagram.

9 Imaginethecubeat (A, a, 1) thatcanonlymoveinthree directions:East (x), North (y) orUp (z).Howmanyspaces wouldthecubemoveintotalifitwasshiftedtothe positionswiththefollowingcoordinates?

(C, c, 2) a

(B, a, 3) b

(A, c, 4) c

(D, b, 2) d

(C, d, 4) e

(B, c, 1) f

10 Amulti-storeycarparkhas 5 levelsandeachlevelisdividedintodifferentzonesinthesameway.The zonesforeachlevelareshowninthisdiagram.Azonewhichisinthe 4thcolumn(East)and 2ndrow (North)andonthe 3rdlevelisgiventhecoordinates (D,b, 3). a Givethecoordinatesofazonewhichis:

i inthe 3rdcolumnEast, 2ndrowNorthandonthe 4thlevel. ii inthe 2ndcolumnEast, 3rdrowNorthandonthe 3rdlevel.

b Acarisparkedinzone (C,a, 2) andistomovetwospacestotheWest,threespacestotheNorth andupthreelevels.Whatarethecoordinatesofthezonewherethecaristobemoved?

c Acarisparkedinzone (B,d, 5) andistomoveonespacetotheEast,twospacestotheSouthand downthreelevels.Whatarethecoordinatesofthezonewherethecaristobemoved?

11 Explainwhythepositionofadrone,forexample,needsthreecoordinatesratherthanjusttwo coordinates.

12 Apoint P canbecombinedwiththeorigintoforma rectangularprismwithacertainvolume.Thisdiagram showsthepoint P producingarectangularprismwith volume 4 × 3 × 2 = 24 cubicunits.Findthevolumeofthe rectangularprismsformedbythefollowingpoints.

P (2, 3, 1) a

P (4, 2, 2) b P (5, 3, 2)

7, 2, 4)

3DDrones 13

13 AdroneisprogrammedtoonlymoveNorth/South,East/Westandup/downinasystemusingmetres asitsunits.

a Findtheminimumdistancethedronewilltravelifitmovesfromtheorigin (0, 0, 0) tothefollowing points.

(10, 5, 7) i (20, 35, 40) ii (12, 29, 18) iii

b Findtheminimumdistancethedronewilltravelifitmovesfromthepoint (8, 20, 6) tothefollowing points.

(10, 28, 12) i (2, 15, 26) ii (0, 20, 30) iii

c Findtheminimumdistancethedronewilltravelifitmovesfromthepoint (−2, 16, 21) tothe followingpoints.

(3, 7, 10) i (10, 0, −4) ii (−40, 4, −20) iii

Jewellerydesigner

Adesignerofjewelleryisacreativeperson whoneedsexcellenteyesight,asteadyhand andgoodﬁne-motorskillsfortheirday-to-day work.Theyworkwithmaterialssuchasgold, silverandgemstones,tocreatesomethingnew andinteresting.Designersneedtounderstand thechemistryoftheirmaterialsandthe geometryofdesign.Theanglesatwhich gemstonessuchasdiamondsarecutcan accountforpricevariationandcanevenbethe differencebetweenabeautiful,sparkling diamondoronedamagedbyachiporcrack.

1 Thewaythatgemstonesarecutcreatesmanydifferentdesigns.Thecutsformshapesandangles whichcontributetothebrightness,sparkleandvalueofthegemordiamond.Lookatthefollowing topviewsandlistallthegeometricalshapesthatyoucanseeineachdesign.

2 Gemstonesanddiamondsaremadefrommineralsandcrystalswhichnaturallyformmanydifferent geometricalshapes.Choosethecorrectnameforeachsolidbelowfromthefollowinglist:

Tetrahedron(4 faces),Icosahedron(20 triangularfaces),Heptahedron(7 faces),Octahedron(8 faces), Pentahedron(5 faces),Dodecahedron(12 pentagonalfaces).

Usingtechnology

3 Usegeometrysoftwaretodigitallydrawthetopviewandsideviewofadiamond-cutdesign. Youcouldresearchdiamond-cutdesignsorchooseonefromtheimagesbelow.

Half dutch rose Pentagon

English round cut Double

Ninjawarriorlogo

Athisworkshop,ShaneiscuttingoutsoftplasticninjastarstogivetopeopletoplayonhisNinjaWarrior course.

Shanemakesstarswhicharequadrilateralswithoneinterioranglegreaterthan 180° liketheoneshown. Henoticesthatsomeofthestarsaremorepopularthanothersandthisseemstodependontheangles formedateachvertex.

Shaneworksonlyinmultiplesof 10 degreesbecauseofthelimitationoftheequipmentthatheuses.

Presentareportforthefollowingtasksandensurethatyoushowclearmathematicalworkingsand explanationswhereappropriate.

1Preliminarytask

a Shane’s 40–140 Ninjastarhasacute ÒADC = 40° andobtuse ÒABC = 140°

i Witha protractorandastraight-edge/ruler,drawanaccuratediagramofastarmatching thisdescription,andmarkinthe 40° and 140° angles.

ii Findreﬂex ÒABC

iii If ÒBAD = 60° ﬁnd ÒBCD

iv If ÒBCD = 50° ﬁnd ÒBAD.

b Shane alsohasa 50–150 Ninjastarwhichincludesacute ÒADC = 50° andobtuse

ÒABC = 150°.Drawan accuratediagramofthestarandmarkinalltheinternalanglesif:

i ÒBAD = 60°

ii ÒBCD = 50°

c Whatdoyou noticeaboutthe 40–140 and 50–150 stars?Explainwhytheyhavesome matchingangles.

A B C D Ninja star

2Modellingtask

Formulate a TheproblemistodeterminealltheanglesinarangeofpopularstarssothatShaneknows howtomanufacturethem.Writeanddrawalltherelevantinformationthatwillhelpsolvethis problem,includingtherulefortheanglesumofaquadrilateral.

Solve

Evaluate and verify

b Thepopular‘2x’starhasthepropertywhereobtuse ÒABC = 2ÒADC,asshown inthe diagrambelow.

i Findalltheinternalanglesif ÒADC = 70° and ÒBAD = 60°.Drawan accuratediagramto illustrateyourstar,showingtheangleat ÒBCD

ii Forthistypeofstar Shaneknowsnottochoose ÒADC = 40°.Explainwhy thiswouldnot produceasuitablestar.(Hint:Trytodrawone,markinginalltheangles.)

c Anotherpopulartypeisthe‘2.5x’star,whichhasthepropertythatobtuse ÒABC = 2.5ÒADC

i Findallthe internalanglesif ÒADC = 60° and ÒBAD = 50°.Drawan accuratediagramto illustrateyourstar.

ii IfShaneonlyworkswithmultiplesof 10° anduses ÒADC = 60°,determinethe number ofpossiblestarsthathecouldmake.

d Intheend,Shanedecidesthatthebeststaristhe‘isoscelesstar’,whichhastheproperty that ÒBAD =ÒBCD.Drawsomepossibleisoscelesstarsthathecouldproduce,giventhatit shouldcontainoneinterior anglegreaterthan 180° andallanglesaremultiplesof 10°

Communicate e Summariseyourresultsanddescribeanykeyﬁndings.

3Extensionquestion

a IfShanehastheabilitytousemultiplesof 5°,describesomenew 2x, 2.5x andisoscelesstars thatarenowpossibletocreatethatwerenotpossiblebefore.

b Onetypeofisoscelesstariscalledthe‘perpendicularstar’.Ithasthesegment BC perpendicular to AD.Iftheanglesat A and C are x°,determinetheanglesat B and D intermsof x fora perpendicularstar.

3Dcitymodel

Keytechnology:Dynamicgeometry

Frompreviousstudies,weknowthatthetwo-dimensionalCartesianplaneallowsustoconstructand analysetwo-dimensionalshapesandgraphrelationshipsbetweentwovariables.Whenathirddimension isadded,wecanconstructandanalysethree-dimensional (3D) solids.Thisisachievedbyusinga three-dimensionalcoordinatesystem (x, y, z) whereallthreeaxesareperpendiculartoeachotherand intersectattheorigin (0, 0, 0).Such 3Dcoordinatessystemsareusedbyengineersandarchitectsto explorebuildingdesignsandbuildcitylandscapes.

1Gettingstarted

The 3Dcoordinatesystemusescoordinateswiththreecomponents (x, y, z).Apoint P in 3Dspacewillthereforehavean x-coordinate,a y-coordinateanda z-coordinate.

a Forthepoint P (3, 4, 3) showninthissetof 3Daxes,answerthefollowing. Statethevalueofthe x-coordinate, y-coordinateand z-coordinate. i Whatistheheightofthepoint P abovethe x-y plane? ii

Howfaristhepoint P fromthe x-z plane? iii

Howfaristhepoint P fromthe y-z plane? iv

b Usea 3Dcoordinatesystemtoplotthesepoints. (0, 2, 4) i (2, 5, 3) ii (4, 1, 0) iii

Technology and computational thinking

2Usingtechnology

a Finda 3Ddynamicgeometrypackage,likeGeogebra 3D,andspendafewmomentsbecoming familiarwithitsfunctions.

b Tryconstructingarectangularprismasshownbelow.Thisexampleisshowntobeacubeifyou considerthecoordinatesoftheverticesandsidelengths.

c Labelthecoordinatesoftheeightverticesandconﬁrmthattheyarecorrectbycheckinghowthey alignwiththescaleonthethreeaxes.

3Applyinganalgorithm

a Wewillconstructasimplecitylandscapebyaddingtwoorthreeotherrectangularprismstoyour 3Dgraph.Followthisalgorithmforeachprism.

Step 1:UsethePolygontooltoconstructafour-sidedpolygononthe x-y plane.Thisisusually achievedbyselectingthetoolthenclickingatfourpointsonthe x-y planeandalsoatthestarting pointagaintoﬁnish.

Step 2:UsethePrismorExtrudetoprismtooltoconstructarectangularprismofacertainaltitude ofyourchoice.

Step 3:Adjustthepositionofyourprism(representingbuildings)bymovingoneoftheverticesof thepolygonbase.Alsoaddgapsbetweenprismstorepresentcitystreets.

b Listthecoordinatesoftheverticesofeachaddedbuilding(prism).

1 Thisshapeincludes 12 matchsticks.(Tosolvethesepuzzlesallmatchesremaining mustconnecttoothermatchesatbothends.)

Remove 2 matchstickstoform 2 squares. a Move 3 matchstickstoform 3 squares. b

2a Use 9 matchstickstoform 5 equilateraltriangles. b Use 6 matchstickstoform 4 equilateraltriangles.

3 WhoamI?

Iwasafemalemathematicianfamousformyworkandpublicationsongeometry. Useyouranswerstothefollowingtounlockthepuzzlecode.

4 Findtheanglebetweenthehourandminutehandsonaclockat thesetimes.

Complementary angles

Vertically opposite angles

= 40 40 ° Triangles

(Scalene, Isosceles, Equilateral, Acute, Right, Obtuse) a + 135 = 180

Angle sum = 180 ° Scalene

Supplementary angles

Revolution

Angles at a point

Lines, shapes and solids

Parallel lines

• Corresponding angles are equal (a = b )

• Alternate angles are equal (a = c )

• Cointerior angles are supplementary (a + d = 180) c° a° b

Polygons

Regular pentagon

Quadrilaterals

(Square, Rectangle, Rhombus, Parallelogram, Kite, Trapezium)

Angle sum = 360° Quadrilateral

Parallelogram

Solids

Triangular prism

Rectangular prism (or cuboid or hexahedron) (or pentahedron) (or pentahedron)

Square pyramid

Chapterchecklist

AversionofthischecklistthatyoucanprintoutandcompletecanbedownloadedfromyourInteractiveTextbook.

2A 1 Icannameanglesinrelationtootherangles

e.g.Nameananglewhichis(a)verticallyoppositeto ÒDOE,(b)complementaryto ÒCOB, and(c)supplementaryto ÒEOA

2A 2 Icanﬁndanglesatapointusingcomplementary,supplementaryorvertically oppositeangles

e.g.Determinethevalueofthepronumeralsinthisdiagram.

2B 3 Icanﬁndanglesusingparallellinesandexplainmyanswersusingcorrect terminology

e.g.Findthevalueof c inthisdiagram,givingareasonforyouranswer.

2B 4 Icanuseparallellinestoﬁndmissinganglesinshapes

e.g.Findthevalueofthepronumeralsinthisdiagram,statingreasons.

2C 5 Icanﬁndunknownanglesinatriangleusingtheanglesum e.g.Findthevalueof a inthistriangle.

Chapter checklist

2C 6 Icanﬁndunknownanglesinisoscelestriangles e.g.Findthevalueof a inthistriangle. a° 26°

2C 7 Icanusetheexteriorangletheorem e.g.Findthevalueof a inthisdiagram.

2D 8 Icanusetheanglesumofaquadrilateraltoﬁndunknownangles e.g.Findthevalueofthepronumeralsinthesequadrilaterals.

2D 9 Icanﬁndunknownanglesinquadrilateralswithparallellines e.g.Findthevalueof a and b inthisparallelogram.

2E 10 Icanﬁndtheanglesumofapolygon e.g.Findtheanglesumofaheptagon.

2E 11 Icanusetheanglesumofapolygontoﬁndunknownangles e.g.Giventhatpentagonshaveananglesumof 540°,ﬁndthevalueof a inthisdiagram.

2E 12 Icanfindthesizeofeachinteriorangleinaregularpolygon e.g.Findthesizeofaninteriorangleinaregularoctagonbyfirstfindingtheanglesum.

2F 13 Icancountthefaces,verticesandedgesofasolid e.g.Statethenumberoffaces,verticesandedgesforthesolidshown.

2F 14 Icanclassifysolidsbytheirnumberoffaces e.g.Classifythissolidbyconsideringhowmanyfacesithas.

2F 15 Icannameprismsbyconsideringtheircross-section e.g.Namethissolidasatypeofprism.

2F 16 Icannamepyramidsbyconsideringtheirbase e.g.Namethissolidasatypeofpyramid.

2G 17 Icanusegridspacestodescribeapositionin3Dspace. e.g.Usingthegivencoordinatesystem,givethecoordinatesoftheredcube.

Chapter checklist

2G 18 Icandescribeapointinthree-dimensionalspaceusingan (x, y, z) coordinate system.

e.g.Whatarethecoordinatesofthepoints A, B, C and P?

Short-answerquestions

1 2A Findthevalueof a inthesesimplediagrams.

2 2B Thesediagramsincludeparallellines.Findthevalueof a

3 2B Decideifthesediagramsincludeapairofparallellines.Givereasons.

4 2C Giveanameforeachtriangleandﬁndthevalueof a.

5 2C Thesetrianglesincludeexteriorangles.Findthevalueof a

6 2D Namethequadrilateral(s)whichhave: allsidesequalinlength a onepairofparallellines b twopairsofequallengthsides c diagonalsintersectingatrightangles d equallengthdiagonals. e

7 2D Findthevalueof a and b inthesequadrilaterals.

8 2E Findtheanglesumofthesepolygonsusing S = (n −2) × 180° Heptagon a Nonagon b Dodecagon c 9 2E Findthesizeofaninteriorangleoftheseregularpolygonsbyﬁrstlyﬁndingtheanglesum. Regularpentagon a Regulardodecagon b

2F Namethepolyhedronthathas:

6 faces a 10 faces b 11 faces c 11 2F Whattypeofprismorpyramidarethesesolids?

12 2G Usingthegivencoordinatesystem,givethecoordinateoftheredcube.

13 2G Givethecoordinatesofthefollowingpoints.

Multiple-choicequestions

1 2A Whatisthenamegiventotwoanglesthataddupto 90°? Right A Supplementary B Revolutionary C Complementary D Verticallyopposite E

2 2A Twoanglesonastraightlineaddto:

3 2B Thevalueof a inthisdiagramisequalto:

4 2B Apairofalternateanglesinparallellines: arenotequal A areverticallyopposite B areequal C arecomplementary D aresupplementary E

5 2E Therulefortheanglesum S ofapolygonwith n sidesis:

6 2A Thecompassbearingfornorth-eastis:

7 2E Thenamegiventoaneleven-sidedpolygonis: heptagon A elevenagon B decagon C dodecagon D hendecagon E

8 2E Theanglesum(using S = (n −2) × 180°)ofahexagonis:

9 2E Thesizeofoneinteriorangleofaregularhexagonis:

10 2F Howmanyedgesdoesarectangularprismhave?

Extended-responsequestions

1 Thisregularpolygonhas 9 sides.

a Findtheanglesumusing S = (n −2) × 180°

b Findthesizeofitsinterioranglescorrecttothenearest degree.

c Findthesizeofitsexterioranglescorrecttothenearest degree.

d Thepolygonisusedtoformtheendsofaprism. Forthisprismﬁndthenumberof: faces i vertices ii edges. iii

interior angle

exterior angle

2 Amodernhouseplanisshownhere.

Listthenamesofatleastthreedifferentpolygonsthatyousee. a Findthevaluesofthepronumerals a f b

Hints:

• a isanangleinsidearectangle

• For b,ﬁrstﬁndtheanglenextto b insidethesmalltriangle

• c isalternateto 29° insideparallellines

• d isinsideahexagon,soﬁrstuse S = (n −2) × 180°

• e isoutsideaquadrilateral(anglesumis 360°)andanglesinarevolutionaddto 360°

• Try f withoutahint!

3 Fractions,decimals andpercentages

Essentialmathematics:whyskillswithfractions,decimals andpercentagesareimportant

Machinistsbuildintricatemetalpartsincludingforlocks,vehicleenginepartsandsurgical instruments.Digitalcalipersthatmeasureinmmto 2 or 3 decimalplacesareused.

Supposeacarpentryapprenticeisaskedforthedrill-bitsizeinthemiddleof 1 8 and 5 32.The equivalentfractionsare 8 64 and 10 64,givingtherequireddrill-bitsizeof 9 64

Plumbersbendcoppertubingintoanglesthatareoftenspokenofasfractionsof 360°.For example, 1 6 is 60° and 3 4 is 270° .

Computersrequireclocktimescodedasadecimal.E.g. 11h 25m 18s = 11 + 25 60 + 18 3600

= 11 + 0.416 + 0.005 = 11.4216 = 11.4217 hours.

Inthischapter

3AEquivalentfractions (Consolidating)

3BOperationswithfractions

3COperationswithnegativefractions

3DUnderstandingdecimals (Consolidating)

3EOperationswithdecimals

3FTerminating,recurringandrounding decimals

3GConvertingfractions,decimalsand percentages

3HFindingapercentageandexpressingas apercentage

3IDecreasingandincreasingbya percentage

3JCalculatingpercentagechange

3KSolvingpercentageproblemsusingthe unitarymethod

AustralianCurriculum9.0

NUMBER

Recogniseirrationalnumbersinapplied contexts,includingsquarerootsand p (AC9M8N01)

Recogniseterminatingandrecurringdecimals, usingdigitaltoolsasappropriate(AC9M8N03)

Usethe4operationswithintegersandwith rationalnumbers,choosingandusingefﬁcient strategiesanddigitaltoolswhereappropriate (AC9M8N04)

Usemathematicalmodellingtosolvepractical problemsinvolvingrationalnumbersand percentages,includingﬁnancialcontexts; formulateproblems,choosingefﬁcient calculationstrategiesandusingdigitaltools whereappropriate;interpretandcommunicate solutionsintermsofthesituation,reviewingthe appropriatenessofthemodel(AC9M8N05)

©ACARA

Onlineresources

Ahostofadditionalonlineresourcesare includedaspartofyourInteractiveTextbook, includingHOTmathscontent,video demonstrationsofallworkedexamples, auto-markedquizzesandmuchmore.

Warm-up quiz

1 Matchthefollowingwordstothetypesoffractions: A wholenumber B improperfraction C properfraction D mixednumeral 1 2 5 a

2 Howmanyquartersarein:

3 Completethefollowing.

4 Fillintheblanks.

5 Whatfractionisshaded? a b

6 Matchthefractionsontheleft-handsidetotheirdecimalformontheright.

7 Find:

8 Writeas i simplefractions ii decimals.

9 Find 10% of:

10 Find:

11 Copyandcompletethefollowingtable.

3A 3A Equivalentfractions

Learningintentions

• Tounderstandwhatequivalentfractionsare.

CONSOLIDATING

• Tounderstandthatfractionshaveanumericalvalue,andtwodistinctfractionscanhavethesamenumericalvalue, like 3 5 and 6 10

• Tobeabletosimplifyfractions.

Keyvocabulary: fraction,numerator,denominator,equivalent,simplify,simplestform

Fractionsaremadewhenwholenumbersarebrokenintoparts. Thisdiagramshowsthepartsofafraction.

4 7 numerator: parts taken from the whole denominator: number of equal parts the whole is broken into Think‘u’for‘upthetop’and‘d’for‘downthebottom.

4 parts selected The whole is divided into 7 parts. 4 7 Denominator Numerator

Thereare 7 equalpartsinthewholeand 4 ofthemareshaded.

Equivalentfractionsarefractionsthatrepresentequalportionsofawhole amountandsoareequalinvalue.Theskillofgeneratingequivalentfractionsis neededwheneveryouaddorsubtractfractionswithdifferentdenominators.

Ruralpropertiesare brokenupintopaddocks. Thepaddocksarea fractionalproportionof thepropertyasawhole. Farmerscanusethis designtodiversifytheir crops.

Fractionsareveryimportantwheneverwemeasureorcompare.Chefsusethemwhenbaking. Buildersusethemwhenmixingconcrete.Amusiciancanevenusefractionswhencomposingmusic.

Lessonstarter:Knowyourterminology

Itisimportanttoknowandunderstandkeytermsassociatedwiththe studyoffractions.

Asaclassgiveadeﬁnitionorexampleofeachofthefollowingkeyterms.

Numerator

• Improperfraction

• Lowestcommonmultiple(LCM)

• Denominator

• Equivalentfraction

• Multiple

• Lowestcommondenominator

• Mixednumeral • Factor

• Properfraction

Hintforlessonstarter:

•

• Highestcommonfactor(HCF)

Hintforlessonstarter: 7 3 numerator denominator Thisisanimproperfraction. Doyouknowwhy?

Keyideas

Equivalentfractions areequalinvalue.Theymarkthesameplaceonanumberline.

e.g., 3 5 and 6 10 areequivalentfractions.

Equivalentfractionsaremadebymultiplyingordividingthe numerator and denominator bythe samenumber.

e.g.

Afractioncanbe cancelleddown (simpliﬁed)ifthetop(numerator) and bottom(denominator) haveacommonfactor,otherthanone.

e.g. 12 18 = 2 × 6 3 × 6 = 2 3

HintforKeyideas: 6 isthe HCFof 12 and 18. 6 6 ‘cancels’ to 1 because 6 ÷ 6 = 1

The simplestform ofafractioniswhenthenumeratoranddenominatorhavenocommonfactors otherthanone.

• Twofractionsareequivalentiftheyhavethesamesimplestform.

Exercise3A

Und er stand ing

1 Fillinthemissingnumberstocompletethefollowingstringsofequivalentfractions.

HintforQ1:Rememberalways × or ÷ thetopandthebottom bythesamenumber!

2 Whichofthesetwofractionsisequivalentto 2 3: 10 15 or 3 4?

3 Whichofthefollowingfractionscanbecancelleddown(simpliﬁed)?

4 Arethefollowingstatementstrue(T)orfalse(F)?

1 2 and 1 4 areequivalentfractions. a 3 6 and 1 2 areequivalentfractions. b Thefraction 8 9 iswritteninitssimplestform. c

14 21 canbesimpliﬁedto 2 3 d 11 99 and 1 9 and 2 18 areallequivalentfractions. e 4 5 canbesimpliﬁedto 2 5 f

Fluency

Example1Generatingequivalentfractions

Rewritethefollowingfractionswithadenominatorof 40

Dividebothnumeratoranddenominatorby 3.

Nowyoutry

Rewritethefollowingfractionswithadenominatorof 32

5 Writethemissingnumberinthesefractionswithadenominatorof 24

HintforQ5:Multiply ordividetopand bottombythesame number.

6 Writethemissingnumberinthesefractionswithadenominatorof 30.

7 Findthemissingvaluetomakeequivalentfractions.

Example2Convertingfractionstosimplestform

Writethefollowingfractionsinsimplestform.

Nowyoutry

Writethefollowingfractionsinsimplestform.

8 Writethefollowingfractionsinsimplestform.

Explanation

TheHCFof 8 and 20 is 4

Boththenumeratorandthedenominatorare dividedbytheHCFof 4

TheHCFof 25 and 15 is 5

Boththenumeratorandthedenominatorare dividedbytheHCFof 5

HintforQ8:Divide bytheHCFofthe numeratorand denominator.

9 Manycalculatorsgivethesimpliﬁedfractionafteryoupress‘=’.Useacalculatortosimplify thesefractions.

Problem-solving and reasoning

10 Eachdiagrambelowshowsafractionofthewhole.Writeeachshadedfractioninmorethanoneway.

Thomasate 1 4 ofa 250 gramblockofchocolate.Maryate 3 6 ofher 250 gramblock.Whoatethe mostchocolate?

Apizzaiscutintoeightequalpieces.Sianate 2 slicesofpizza,Callumhad 4 slices.Whatfraction ofthepizzaisleft?

12 Writedownfourfractionsthatwhensimpliﬁedequal 1 5

13 BozzotheClown,shownbelow,isholdingﬁveballoonswithfractionsonthem.Whichballoonisthe oddoneout?

Trickyshading 14

14 Inthesediagramsassumethatthesidesaredividedevenly.Findthefractionthatisshaded. a b

3B 3B Operationswithfractions

Learningintentions

• Tobeabletoaddandsubtractfractionsby rst ndingthelowestcommonmultiple(LCM)ofthedenominators.

• Tobeabletomultiplyanddividefractions.

• Tobeabletoperformthefouroperationsonmixednumerals,convertingtoimproperfractionsasrequired.

Keyvocabulary: lowestcommonmultiple(LCM),lowestcommondenominator(LCD),reciprocal,numerator, denominator,mixednumeral,improperfraction

Thissectionreviewsthedifferenttechniquesinvolvedinadding,subtracting,multiplyingand dividingfractions.

Properfractions,improperfractionsandmixednumeralswillbeconsideredforeachofthefour mathematicaloperations.

Lessonstarter:Shadingfractions

Inpairs,drawandshadethisgridtohelpworkoutthefollowingfractionquestions.

Whatrulesdoyouknowforadding,subtracting,multiplyinganddividingfractions?

Keyideas

Addingandsubtractingfractions

• Toaddorsubtractfractions,converttothesame denominator. Whenthedenominatorsarethesame,justaddorsubtractthe numerators

e.g.

• The lowestcommonmultiple(LCM) ofthedenominatorsisusedifthedenominatorsare different.Thisiscalledthe lowestcommondenominator(LCD).

Multiplyingfractions

• Converttoimproperfractions(ifneeded)

• Multiplythenumeratorstogether

• Multiplythedenominatorstogether

• Simplifyyouranswer

Dividingfractions

• Todividebyafraction,remembertoﬂipthefractionafter thedivisionsignupsidedown,thenmultiply.

e.g. 1 2 ÷ 1 4 = 1 2 × 4 1 = 2

Toﬂipafraction a b upsidedowntobecome b a iscalledtakingits reciprocal.

Exercise3B

Und er stand ing

Thereare 2 quarters inonehalf.

1a Whichtwooperationsrequirethedenominatorstobethesamebeforeproceeding?

b Whichtwooperationsdonotrequirethedenominatorstobethesamebeforeproceeding?

2 Statethelowestcommondenominatorforthefollowingpairsoffractions. 1 5 + 3 4 a

3 Rewritethefollowingequationsandﬁllintheemptyboxes.

4 Statethereciprocalofthefollowingfractions.

HintforQ4:Reciprocalmeans toturnupsidedown.Convert mixednumeralstoimproper fractionsﬁrst!

Example3Addingandsubtractingfractions

Evaluate.

Explanation a 3 5 + 4 5 = 7 5 = 1 2 5

Thedenominatorsarethesamesosimplyadd thenumerators.

Three fifths plusfour fifths equalsseven fifths Thefinalanswercanbewrittenasamixednumeral.

Continuedonnextpage

Nowyoutry

Evaluate.

5 Evaluate.

Lowestcommonmultipleof 3 and 4 is 12 WriteequivalentfractionswithaLCDof 12.

Thedenominatorsarenowthesame,sosubtractthe numerators.

HintforQ5:Converttoacommon denominatorifrequired.

Example4Addingandsubtractingmixednumerals

Evaluate. 3

Explanation

Convertmixednumeralstoimproperfractions. Thelowestcommonmultipleof 8 and 4 is 8 WriteequivalentfractionswithLCD. Addnumeratorstogether,denominator remainsthesame.Converttheanswerbackto amixednumeralifrequired.

Convertmixednumeralstoimproperfractions. Thelowestcommonmultipleof 2 and 6 is 6 WriteequivalentfractionswithLCD. Subtractnumeratorsandsimplifytheanswer.

Nowyoutry

Evaluate.

6 Evaluate.Writeﬁnalanswersasmixednumerals. 3 1 7 + 1 3 7

HintforQ6:Youcanaddthe wholesﬁrstifyouprefer.

Example5Multiplyingfractions

Evaluate.

Solution Explanation

Multiplythenumeratorstogether. Multiplythedenominatorstogether. Theanswerisinsimplestform.

Useimproperfractions, 1 3 4 = 7

Cancelanynumeratorwithanydenominator. Multiplythenumerators 2 × 7 = 14 Multiplythedenominators 5 × 1 = 5

Nowyoutry Evaluate.

7 Evaluate.

HintforQ7:Looktocancelﬁrst ifpossible.

8 Evaluate.

Example6Dividingfractions

Evaluate.

Solution

a 2 5 ÷ 3 7 = 2 5 × 7 3 = 14 15

Nowyoutry

Evaluate.

9 Evaluatethefollowing.

Explanation

Change ÷ signtoa × signandﬂipthedivisor. Multiplybythereciprocal. Proceedasformultiplication. Multiplynumeratorstogetherandmultiply denominatorstogether.

Convertmixednumeralstoimproperfractions. Change ÷ signtoa × signandﬂipthedivisor. Multiplybythereciprocal.Thereciprocalof 4 3 is 3 4 Multiplyandsimplify.

10 Evaluatethefollowing.

HintforQ10:Convertto improperfractionsﬁrst.

Problem-solving and reasoning 11 11,12

11 Thereare 30 studentsinMissMac’smathsclass.Findouthowmanystudentsthereareineachofthe followinggroups.

1 3 oftheclasshadbrownhair. a 1 2 oftheclasscametoschoolbybus. b

5 6 oftheclassspokeEnglishathome. c 1 10 oftheclasslikedmaths. d

12 MaxandTanyaarepaintingalargewalltogether,whichissplitequallyintotwohalves.Maxpaintsthe lefthalfofthewallatthesametimeasTanyapaintstherighthalf.Maxhaspainted 3 7 ofhishalfand Tanyahaspainted 2 5 ofherhalf.

Whatfractionoftheoverallwallhasmaxpainted?Calculate 3 7 × 1 2 toﬁndthis.

WhatfractionoftheoverallwallhasTanyapainted?

13 Useacalculatortoﬁndtheanswerstothefollowing.

3C 3C Operationswithnegativefractions

Learningintentions

• Tounderstandthatthetechniquesforadding,subtracting,multiplyinganddividingpositivefractionsalsoapplyto negativefractions

• Tounderstandthattherulesforpositiveandnegativeintegersalsoapplytofractions

• Tobeabletoadd,subtract,multiplyanddividenegativefractionsandmixednumerals

Keyvocabulary: negative,LCM,LCD,reciprocal,numerator,denominator,mixednumeral,improperfraction

TheEnglishmathematiciannamedJohnWallis (1616–1703) inventedanumberlinethatdisplayed numbersextendinginboththepositiveandnegativedirections.

So,justaswecanhavenegativeintegers,wecanalsohavenegativefractions.Infact,eachpositive fractionhasanopposite(negative)fraction.Twoexamplesarehighlightedonthenumberlinebelow:

Youaregivenastartingpointandasetofinstructionstofollow.Youmustdeterminewheretheﬁnishing pointis.Theﬁrstsetofinstructionsreviewstheadditionandsubtractionofintegers.Theothertwosets involvetheadditionandsubtractionofpositiveandnegativefractions.

• Startingpointis 1.Add 3,subtract 5,add −2,subtract −4,subtract 3

Finishingpoint =

• Startingpointis 0.Subtract 3 5,add 1 5,add 4 5,subtract 2 5,subtract 3 5.

Finishingpoint =

• Startingpointis

Finishingpoint =

Keyideas

Thetechniquesfor +, , ×, ÷ positivefractionsalsoapplytonegativefractions.

Thearithmeticrulesweobservedforintegers(Chapter1)alsoapplytofractions. Subtractingalargerpositivefractionfromasmallerpositivefractionwillresultinanegativefraction.

Forexample:

Addinganegativefractionisequivalenttosubtractingitsopposite.

Forexample:

Subtractinganegativefractionisequivalenttoaddingitsopposite.

Forexample: 1 2 1 3 = 1 2 + + 1 3 = 1 2 + 1 3

Theproductorquotientoftwofractionsofthesamesign(positiveornegative)isapositivefraction.

•

Theproductorquotientoftwofractionsoftheoppositesign(positiveandnegative)isanegative fraction.

Exercise3C

Und er stand ing

1 Usinganumberlinefrom −4 to 4,indicatethepositionsofthefollowingnegativeandpositivefractions.

2 Statethemissingfractionstocompletethesesentences.

Adding 1 4 isequivalenttosubtracting a

Adding 1 3 isequivalenttosubtracting . b

Subtracting 3 5 isequivalenttoadding c

Subtracting 2 7 isequivalenttoadding d

3 Statewhethertheanswerforthefollowingexpressionswillbepositiveornegative.Donotevaluate theexpressions. 3 5 × 1 3 a

Chasbeenrecorded.Scientistscalculatethedifferenceas:

Fluency

Example7Addingandsubtractingnegativefractions

Simplify:

Explanation

Adding 5 7 isequivalenttosubtracting 5 7

Subtracting 4 3 isequivalenttoadding 4 3

Adding 1 4 isequivalenttosubtracting 1 4

TheLCMof 5 and 4 is 20

WriteequivalentfractionswithLCDof 20

Subtractthenumerators.

Subtracting −3 2 3 isequivalenttoadding 3 2 3

Convertmixednumeraltoimproperfraction. Denominatorsarethesame,thereforeadd numerators −7 + 11 = 4

Nowyoutry

Simplify:

5 Simplify:

6 Simplify:

HintforQ6:First ﬁndacommon denominator.

Example8Multiplyingwithnegativefractions

Simplify:

Explanation

Thetwofractionsareofoppositesignsothe answerisanegative.

Thetwofractionsareofthesamesign,sothe answerisapositive.

Cancelwherepossible,thenmultiply numeratorsandmultiplydenominators.

Nowyoutry

Simplify:

7 Simplify:

Example9Dividingwithnegativefractions

Explanation

Thereciprocalof 3 4 is 4 3

Thetwofractionsareofthesamesignsothe answerisapositive.

Theanswershouldbeinsimplestform.

Thereciprocalof 3 is 1 3

Thetwonumbersareofoppositesign,sothe answerisanegative.

Nowyoutry

8 Simplify:

HintforQ8:Flipthe secondfractionto ﬁnditsreciprocal.

Problem-solving and reasoning

9 Toolapoolhasanaveragemaximumtemperature of 13 1 2 °Candanaverageminimumtemperature of −3 1 4 °C.Averagetemperaturerangeis calculatedbysubtractingtheaverageminimum temperaturefromtheaveragemaximum temperature.Whatistheaveragetemperature rangeforToolapool?

10 Arrangethesefractionsinincreasingorder. 3 4 , 1 2 , 5 3 , 3 4 , −1 1 2 , 1 16 , 1 5 , 3 1 10

11 Xaioaimstoget,onaverage, 8 hoursofsleepperweeknight.OnMondaynighthesleptfor 6 1 3 hours, onTuesdaynight 7 1 2 hours,onWednesdaynight 5 3 4 hoursandonThursdaynight 8 1 4 hours.

StatethedifferencebetweentheamountofsleepXaioachievedeachnightandhisgoalof 8 hours. Giveanegativeansweriftheamountofsleepislessthan 8 hours. a Afterfournights,howmuchisXaioaheadorbehindintermsofhissleepgoal? b IfXaioistoexactlymeethisweeklygoal,howmuchsleepmusthegetonFridaynight? c

12 Maria’smotherwantstomake 8 curtainsthateachrequire 2 1 5 metresofmaterialinastandardwidth, butsheonlyhas 16 1 4 metres.SheasksMariatobuymorematerial.Howmuchmorematerialmust Mariabuy?

13 Placeaninequalitysign(< or >)betweenthefollowingfractionpairstomakeatruestatement.(Note: Youcanuseanumberlinetodecidewhichvalueisgreater.)

Interestingfractionproducts

14 Itispossibletomultiplytwonegativefractionsandobtainapositiveinteger.Forexample, 4 5 × 5 2 = 2.Foreachofthefollowing,giveanexampleorexplainwhyitisimpossible.

Apositiveandanegativefractionmultiplyingtoanegativeinteger a

Apositiveandanegativefractionaddingtoapositiveinteger b

Apositiveandanegativefractionaddingtoanegativeinteger c Threenegativefractionsmultiplyingtoapositiveinteger d

14,15

15 Donotevaluatethefollowingexpressions,juststatewhethertheanswerwillbepositiveornegative. 2 7 × 1 7 × 3 11

3D 3D Understandingdecimals CONSOLIDATING

Learningintentions

• Tounderstandplacevalueinadecimal.

• Tobeabletocomparetwoormoredecimalstodecidewhichislargest.

• Tobeabletoconvertdecimalstofractions.

• Tobeabletoconvertfractionstodecimals,incaseswherethedenominator’sonlyprimefactorsare 2 and/or 5

Keyvocabulary: decimal,decimalpoint,placevalue,fraction

Decimalsalsorepresent‘partsofawhole’.Theyrepresentfractionswithdenominatorsof 10, 100, 1000 Thedecimalpointisusedtoseparatethewholenumberandthefractionpart.

3 17 100 = 3.17

Inthissection,wereviewsomedecimalconceptsfromYear 7

Lessonstarter:Decimalsaroundus

Inpairs,listatleastﬁveplaceswheredecimalsareusedandgiveaspeciﬁcexampleforeachone.

Keyideas

Value increasing by 10 times for each place movement

decreasing by 10 times for each place movement

Comparingandorderingdecimals.

Tocomparetwodecimalnumberswithdigitsinthesameplace-valuecolumns,comparethe left-mostdigitsﬁrst.Continuecomparingdigitsasyoumovefromlefttorightuntilyouﬁnd twodigitsthataredifferent.

e.g.Compare 362.581 and 362.549

362.581

362.549 1 8 4 thestdigit thatisdifferent: > So 362.581362.549 >

Convertingdecimalsto fractions

• Countthenumberofdecimalplacesused.

• Thisisthenumberofzeroesthatyoumustplaceinthedenominator.

• Simplifythefractionifrequired.

e.g. 0.64 = 64 100 = 16 25

Convertingfractionstodecimals

• Ifthedenominatorisapowerof 10,simplychangethefractiondirectlytoadecimalfromyour knowledgeofitsplacevalue.

e.g. 239 1000 = 0.239

• Ifthedenominatorisnotapowerof 10,trytoﬁndanequivalentfractionforwhichthe denominatorisapowerof 10 andthenconverttoadecimal.

e.g.

• Iftheabovetwomethodsarenotsuitable,dividethebottom(denominator)intothetop (numerator).

e.g. 8

Exercise3D

1 Whichofthefollowingisthemixednumeralequivalentof 8.17?

2 Whichofthefollowingisthemixednumeralequivalentof 5.75?

3 Whichdecimalnumberisequalto 4 + 1 10 + 5 100?

4 Whichdecimalnumberisequalto 20 + 3 10 + 7 1000?

Fluency

Example10Comparingdecimals

Whichislarger?

57.89342 or 57.89631

Solution

Explanation

57.89631 islarger. Writeunderneatheachother. 57.89 342 57.89 631 ↑ 1stdigitdifferentfromlefttoright 6 > 3 3 1000 < 6 1000

Nowyoutry

Whichislarger? 2.14073 or 2.14039

5 Writedownthelargerdecimalineachpair.

36.48537.123 a

21.95321.864 b

0.03720.0375 c

4.217534.21809 d

65.411264.8774 e 9.52813529.5281347 f

6 Decideifeachofthefollowingstatementsistrue(T)orfalse(F). 3

HintforQ5:Comparedigitsfrom lefttoright.

Example11Convertingdecimalstofractions

Convertthefollowingdecimalstofractionsintheirsimplestform.

Solution

a 725 1000 = 29 40

b 5 12 100 = 5 3 25

Nowyoutry

Explanation

Threedecimalplaces,thereforethreezeroesindenominator. 0.725 = 725 thousandths.

Twodecimalplaces,thereforetwozeroesindenominator. 0.12 = 12 hundredths.

Convertthefollowingdecimalstofractionsintheirsimplestform. 0.58 a 12.65 b

7 Convertthefollowingdecimalstofractionsintheirsimplestform. 0.31

HintforQ7:Don’tforgetto simplify.

Example12Convertingfractionstodecimals

Convertthefollowingfractionstodecimals.

Explanation

a 239 100 = 2 39 100 = 2.39

Convertimproperfractiontoamixednumeral. Denominatorisapowerof 10 b 9 25 = 36 100 = 0.36

Nowyoutry

Convertthefollowingfractionstodecimals.

8 Convertthefollowingfractionstodecimals.

9 Convertthefollowingfractionstodecimals. 3

HintforQ9:Converttoa

10 Convertthefollowingmixednumeralstodecimalsandthenplacethemin descendingorder.

HintforQ10: 3 descending is going down from largest to smallest 2 1 2 2 5 , 2 1 4 , 2 9 50 , 2 3 10

Problem-solving and reasoning

11,12 12–14

11 ThedistancesfromNam’slockertohissixdifferentclassroomsarelistedbelow:

•

LockertoroomB5 (0.186 km)

LockertoroomA1 (0.119 km) •

LockertoroomP9 (0.254 km) •

Lockertogym (0.316 km)

LockertoroomC07 (0.198 km)

LockertoBWTheatre (0.257 km)

ListNam’ssixclassroomsinorder,fromtheclosestclassroomtotheonefurthestawayfromhislocker.

12 ThePrimeMinister’sapprovalratingis 0.35,whiletheOppositionLeader’sapprovalratingis 3 5 Whichleaderisaheadinthepopularitypollsandbyhowmuch?

13 Jeromewishestodigahole 1.5 metres deep.

Michaelhasdugahole 1 3 4 metresdeep. Whoseholewillbethedeepest?

a Howmanycentimetresdifferenceis thereinthedepthofeachhole?

14 Writedowna decimal thatliesbetweenthepairsoffractions. 1 2 and 9 10 a 3 4 and 1 b 1 4 and 3 4

HintforQ14:Anumberlinemayhelp

15 Completethefollowingmagicsquaresusingamixtureoffractionsanddecimals.

HintforQ15:Eachrow, columnanddiagonaladd tothesamenumberin eachmagicsquare.It’s theMAGICnumber!

HintforQ13:

3E 3E Operationswithdecimals

Learningintentions

• Tobeabletoaddandsubtractdecimals.

• Tobeabletomultiplydecimals.

• Tounderstandthatmultiplyinganddividingbypowersof 10 involvesmovingthedigitsleftorrightofthe decimalpoint.

• Tobeabletodividedecimals.

Keyvocabulary: decimalpoint,powerof 10,divisor,dividend,quotient

Thissectionreviewsthedifferenttechniquesinvolvedin adding,subtracting,multiplyinganddividingdecimals.

Lessonstarter:Matchthephrases

Therearesevendifferentsentencebeginningsand sevendifferentsentenceendingsbelow.Yourtaskisto matcheachsentencebeginningwithitscorrectending. Whenyouhavedonethis,writedownthesevencorrect sentences.

Precisionelectronicmeasuringinstrumentsgivea decimalread-out.

Sentencebeginnings

Sentenceendings

Whenaddingorsubtractingdecimals thedecimalpointmovestwoplacestotheright.

Whenmultiplyingdecimals thedecimalpointinthequotientgoesdirectlyabove thedecimalpointinthedividend.

Whenmultiplyingdecimalsby 100 makesureyoulineupthedecimalpoints.

Whendividingdecimalsbydecimals thenumberofdecimalplacesinthequestionmust equalthenumberofdecimalplacesintheanswer.

Whenmultiplyingdecimals thedecimalpointmovestwoplacestotheleft.

Whendividingby 100 startbyignoringthedecimalpoints.

Whendividingdecimalsbyawholenumber westartbychangingthequestionsothatthedivisor isawholenumber.

Keyideas

Addingandsubtractingdecimals

• Ensuredigitsarecorrectlyalignedinsimilarplace-valuecolumns.

• Ensurethedecimalpointsarelinedupdirectlyunderoneanother. 37.56 + 5.23137.560 + 5.231 ✔ 37.56

Multiplyinganddividingdecimalsby powersof10

• Whenmultiplying,thedecimalpointmovestothe right thesamenumberofplacesasthere arezeroesinthemultiplier.

e.g. 13.753 × 100 = 1375.3 13.753

Multiply by 10 twice.

• Whendividing,thedecimalpointmovestothe left thesamenumberofplacesasthereare zeroesinthe divisor

e.g. 586.92 ÷ 10 = 58.692 586.92 Divide by 10 once.

Multiplyingdecimals

• Initiallyignorethedecimalpointsandmultiplythewholes.

• Placethedecimalpointintotheanswerusingtherule: ‘Thenumberofdecimalplacesintheanswermustequalthetotalnumberofdecimalplacesin thequestion.’

e.g. 5.738.6 × 573 86 49278 × 5.73 8.6 49.278 ×= (3 decimal places in question, 3 decimal places in answer)

Dividingdecimals

Thedecimalpointinthe quotient goesdirectlyabovethedecimalpointinthe dividend

e.g. 56.34 ÷ 3 )

Quotient(answer)

Weavoiddividingdecimalsbyotherdecimals.Insteadwechangethedivisorintoawholenumber. Whateverchangewemaketothedivisorwemustalsomaketothedividend,soitisequivalentto multiplyingby 1 andthevalueofthequestionisnotchanged.

Exercise3E

1 Whichofthefollowingisthecorrectset-upforthefollowingadditionproblem?

2 Thecorrectanswertotheproblem 2.731 ÷ 1000 is:

3 If 56 × 37 = 2072,thecorrectanswertotheproblem 5.6 × 3.7 is:

Hintfor3EUnderstanding:The Keyideas sectionanswersall thesequestions.

4 Whichofthefollowingdivisionswouldprovidethesameanswerasthedivisionquestion

Fluency

Example13Addingandsubtractingdecimals

Calculate. 23.07 + 9.8 a 9.7−2.86 b

Solution

a 23.07 + 9.80 32.87

b 89.16710 −2.86 6.84

Nowyoutry

Explanation

Lineupthedecimalpoints.Fillinanyzeroes thenaddvertically.

Aligndecimalpointsdirectlyunderoneanother andﬁllinmissingdecimalplaceswithzeroes. Carryoutsubtractionfollowingthesame procedureasforsubtractionofwhole numbers.

Calculate. 1.05 + 12.96 a 3.2−1.74 b

5 Calculatethefollowing.

6 Calculatethefollowing.

HintforQ5:Aligndigits inmatchingplace-value columns. 5.6 +

Example14Multiplyinganddividingbypowersof 10

Calculate.

a 9.753 ÷ 100 = 0.09753

b 27.58 × 10000 = 275800

Explanation

Dividingby 100 (2 zeroes),thereforethe decimalpointmustmovetwoplacestothe left.Additionalzeroesareinsertedasnecessary. 09.753

Multiplyingby 10000 (4 zeroes),therefore thedecimalpointmustmovefourplacesto theright.Additionalzeroesareinsertedas necessary. 27.5800

Nowyoutry

Calculate.

7 Calculate.

Example15Multiplyingdecimals

Calculate 25.7 × 0.3.

12257 × 3 771 25.7 × 0.3 = 7.71

Nowyoutry

Calculate 4.5 × 1.6

8 Calculate.

HintforQ7:Move leftorrightbythe numberofzeroes.

Performmultiplicationignoringthedecimal point. (257 × 3 = 771) Therearetwodecimalplacesinthequestion,so twodecimalplacesintheanswer.

HintforQ8:Firstignorethe decimalpoint.

Example16Dividingdecimals

Calculate.

Carryoutdivision,rememberingthatthe decimalpointintheanswerisplaceddirectly abovethedecimalpointinthedividend.

Insteadofdividingby 0.03,multiplyboththe divisorandthedividendby 100,togetawhole number, 3

(Move each decimalpointtwoplacesto theright.)Carryoutthedivisionquestion 6413.7 ÷ 3

Nowyoutry Calculate.

9 Calculate.

10 Completethesedivisionsbyﬁllinginthemissingnumbers.

11 TheheightsofMrsBuchanan’sﬁvegrandchildrenare 1.34 m, 1.92 m, 0.7 m, 1.5 m,and 1.66 m. Ifthegrandchildrenlaiddowninarow,headtotoe,howlongwouldtherowbe?

12 Lookcloselyatthetableofcanteenpricesbelow.

Canteenprices

Findthecostofeachperson’slunch. a

Vaughn Charlotte Reece 1 pie 1 sandwich 1 pie 1 sauce 1 chocolate 2 colas 1 apple 1 juice 1 sandwich 2 milks 1 pktchips

Whohadthemostchangefrom $20? b

13 Analternativemethodformultiplyingdecimalsistoconvertthemtofractionsandthenmultiplythem. Forexample, 0.3 × 1.2 =

= 36

,whichcanbeconvertedinto 0.36.Usethismethod tomultiplythefollowingdecimals.

0.3 × 0.7 a

0.1 × 3.07 b

0.2 × 0.05 c

14 Answereachofthe 12 questionsbelowtounlockthecodeandﬁndouthowSamalanswers Sally’squestion.

Hey Samal, do you know why they teach us decimals?

3F 3F Terminating,recurringand roundingdecimals

Learningintentions

• Tounderstandthedifferentnotationsforrecurringdecimals(involvingdotsanddashes)

• Tobeabletoconvertafractiontoaterminatingdecimalusingdivision.

• Tobeabletoconvertafractiontoarecurringdecimalusingdivision.

• Tobeabletorounddecimalstoagivennumberofdecimalplacesby rst ndingthecriticaldigit.

Keyvocabulary: terminatingdecimal,recurringdecimal(orrepeatingdecimal),rounding Notallfractionsconverttothesametypeofdecimal.

Forexample:

1 2 = 1 ÷ 2 = 0.5 (onlyhasonedecimalplace) 1 3 = 1 ÷ 3 = 0.33333… (keepsgoingandgoing) 1 7 = 1 ÷ 7 = 0.142857142857… (thepatternrepeats)

Decimalsthatstop(orterminate)areknownasterminatingdecimals,whereasdecimalsthatcontinue onforeverwithsomeformofpatternareknownasrepeatingorrecurringdecimals.

Lessonstarter:Decimalpatterns

Useacalculatortoperformthesedivisions.Canyouseeapattern?

Withoutyourcalculator,writedown 5 9 and 6 9 asdecimals.Whatdowecallthesetypesofdecimals?

Keyideas

A terminatingdecimal hasaﬁxednumberofdecimalplaces(i.e.itterminates).

e.g. ) 5 8 5 8 0.62585.000 0.625 524 =÷= Terminating decimal (3 decimal places only)

A recurringdecimal (orrepeatingdecimal)keepsgoingandthedecimalplacesrepeat.

e.g. ) 1 3 1 30.33331.0000 0. 333... 111 1 =÷= Recurring decimal

Aconventionistousedotsplacedabovethedigitstoshowthestartandﬁnishofarepeatingcycle ofdigits.

e.g. 0.55555…= 0.5 and 0.3412412412…= 0.3412

Anotherconventionistouseahorizontallineplacedabovethedigitstoshowtherepeatingcycle ofdigits.

e.g. 0.55555…= 0.5 and 0.3412412412…= 0.3412

Rounding decimals

Decimalscanbewrittenwithfewerdecimalplacesbyrounding. Toroundwemustlookatthedigitimmediatelyafterthenumberofplaceswewant.(It’sthe critical digit!)

These values are closer to 7.4 and round to 7.4.

These values are closer to 7.5 and round to 7.5.

Exercise3F

1 Statewhetherthefollowingareterminatingdecimals(T)orrecurringdecimals(R).

2 Foreachlinegiven,whichcircleddecimalisthedecimalinthetriangleclosestto?

3 Expressthefollowingrecurringdecimalsusingtheconventionofdotsorabartoindicatethestartand ﬁnishoftherepeatingcycle.

HintforQ3:Write 0.7555 as 0.75 0.33333 a 6.21212121… b 8.5764444… c 2.135635635 d 11.2857328573… e 0.003523523 f

4 Writedownthe‘critical’digit(thedigitimmediatelyaftertheroundingdigit)foreachofthefollowing. 3.5724 (roundingto 3 decimalplaces) a 15.89154 (roundingto 1 decimalplace) b 0.004571 (roundingto 4 decimalplaces) c 5432.726 (roundingto 2 decimalplaces) d

Example17Writingterminatingdecimals

Convertthefollowingfractionstodecimals.

Write

Dividethebottom(denominator)intothetop(numerator).

Nowyoutry

Convertthefollowingfractionstodecimals.

5 Convertthefollowingfractionstodecimals.

HintforQ5:Theseareall terminatingdecimals.

Example18Writingrecurringdecimals

Expressthefollowingfractionsasrecurringdecimals.

Thispatterncontinues,itisarepeatingdecimal.

Thispatterncontinues.

Nowyoutry

Expressthefollowingfractionsasrecurringdecimals.

6 Expressthefollowingfractionsasrecurringdecimals.

1 3 a 5 9 b 5 6 c 7 9 d 3 7 e 1

Example19Roundingdecimals

Round 14.258 to 1 decimalplace. a Round 0.671 to 2 decimalplaces. b

HintforQ6:Rememberto usetherepeatingnotation. 0.444…= 0.4.

a 14.3 14.2 58 roundedto 1 decimalplace—lookatnextdigit (5).Criticaldigitis 5.Roundup 14.258 ¥ 14.3

b 0.67 0.67 1 roundedto 2 decimalplaces—lookatthenext digit (1).Criticaldigitis 1.Rounddown 0.671 ¥ 0.67

Nowyoutry

Round 24.9349 to 2 decimalplaces. a Round 0.048561 to 3 decimalplaces. b

7 Roundeachofthefollowingdecimalsto 1 decimalplace.

8 Writeeachofthefollowingdecimalscorrectto 2 decimalplaces (thenearesthundredth).

HintforQ7:Theﬁrstdecimal placeisalsocalledthetenths column.

9a Choosethecorrectanswertoeachofthefollowing. Is 7.9 closerto 7 or 8? i Is 7.99 closerto 7.9 or 8.0? ii Is 4.96 closerto 4.9 or 5.0? iii

b Roundthefollowingto 1 decimalplace.

10 Findouthowyourcalculatorrounds.Useittoroundeachofthefollowingdecimalstothenumberof decimalplacesgiveninthebrackets.

11 SimoneandGreeraretwoverygoodjuniorsprinters.Simoneran 100 min 12.83 seconds,whileGreer ranitin 12.77 seconds. Whocameﬁrst,andbyhowmuch? a Roundeachtimetoonedecimalplace.Canyoustilldecidewhocameﬁrst? b

12 Petrolissoldat 147.9 centsperlitre.Find,correcttothenearestcent,thecostof:

1 litre a

5 litres b

12 litres c 39.7 litres d

Neitherterminatingorrecurring 13

13 Therearesomenumberswhenwrittenasadecimalneitherterminatenorrecur.Onesuchsetofnumbers arecalledsurds,whichincludea √ sign,e.g. √2 Write √2 correctto 7 decimalplacesusingacalculator. a Findsomeothersurdsandwritethemcorrectto 3 decimalplaces. b Othernon-surdnumberswhichdonotterminateorrecurincludepiandphi.Researchthesenumbers andwriteabriefreportaboutwhytheyareimportant.

1 3A Writethefollowingfractionsinsimplestform.

2 3B Evaluate.

3 3B Evaluate.

4 3C Evaluate.

5 3C Evaluate.

6 3D Convertthefollowingdecimalstofractionsintheirsimplestform.

7 3D Convertthefollowingfractionstodecimals.

8 3E Calculate.

9 3E Calculate.

10 3F Convertthefollowingfractionstodecimals,expressingyouranswersasrecurring decimalsifnecessary.

11 3F Roundeachofthefollowingdecimalstotwodecimalplaces.

3G 3G Convertingfractions,decimals andpercentages

Learningintentions

• Tounderstandthatapercentage(%)isanumberoutof 100.

• Tobeabletoconvertpercentagestofractionsanddecimals.

• Tobeabletoconvertfractionsanddecimalstopercentages.

Keyvocabulary: percentage,percent,fraction,decimal

A percentage isaparticularfractionwherethedenominatorisalways 100

Percent isLatinfor‘outof 100’.

7%= 7 percent = 7 outof 100 = 7 100 = 0.07

Peopleusepercentageseverydayinbanking,salesandschooltests.

Lessonstarter:Estimatingpercentages

• Listthedrinksinorderfromthemosttotheleastamountleftintheglass.

• Estimatethepercentageofdrinkremainingineachoftheglassesshownabove.

• Discussyourestimateswithapartner.

Keyideas

A percent sign (%) means‘outofonehundred’.

23%= 23 100

Percentages canbeconvertedto fractions and decimals

35%½ 35 100 = 7 20 (fraction)

35%½ 35 ÷ 100 = 0.35 (decimal)

Fractionsanddecimalscanbeconvertedtopercentages.

1 4 or 0.25 asapercentage ½ 1 4 × 100 = 25 and 0.25 × 100 = 25

so 1 4 = 0.25 = 25%

Commonpercentagesandtheirequivalentfractionsareshowninthetablebelow. Itishelpfultoknowthese.

Creamy soda Cola Raspberry Chocolate LemonMilkshake Orange Lime

Und er stand ing

1 Thefractionequivalentof 27% is:

2 Thedecimalequivalentof 37% is:

3 Thepercentageequivalentof 47 100 is:

4 Copyandcompletethetableordiscussasagroup.

Fluency

Example20Convertingpercentagestofractionsanddecimals

Write 72% as: asimplefraction a adecimal b

Solution

a 72%= 72 100 = 18 × 4 25 × 4 = 18 25

b 72%= 72 ÷ 100 = 0.72

Nowyoutry

Explanation

Change % signtoadenominatorof 100

Cancelthehighestcommonfactorof 4 tosimplify.

Tochangea % toadecimal,divideby 100

Write 65% as: asimplefraction a adecimal b

5 Writethesepercentagesassimplefractions.

39%

HintforQ5:Writeeachnumber outof 100 a 11% b 17% c 99

6 Writethesepercentagesasdecimals.

HintforQ6:Movethedecimal pointtwoplacestotheleft.

Example21Convertingfractionsanddecimalstopercentages

Writethefollowingaspercentages.

Explanation a 4 5 × 100 = 80 So 4 5 = 80%

Multiplythefractionby 100 Simplify 300 ÷ 8 = 37 remainder 4

Multiplyby 100 toexpressasapercentage b 3 8 × 100 = 300 8 = 37 1 2 So 3 8 = 37 1 2 %

c 0.81 × 100 = 81

So 0.81 = 81%

Nowyoutry Writethefollowingaspercentages.

7 Writethesefractionsaspercentages.

Multiplythedecimalby 100 0.8181% =

HintforQ7:Multiplyby 100 andsimplify.

8 Writethesedecimalsaspercentages.

Problem-solving and reasoning

9a If 1 5 = 20%,whatdoes 3 5 equalasapercentage?

b If 1 8 = 12.5%,whatdoes 7 8 equalasapercentage?

c If 1 3 = 33 1 3 %,whatdoes 2 3 equalasapercentage?

HintforQ9:Whatcanwe multiplyeachfractionby?

10 Completethefollowingconversiontablesinvolvingcommonfractions,decimalsandpercentages.

11 TheSharkshockeyteamhaswon 13 outof 17 gamesfortheseasontodate.Theteamstillhasthree gamestoplay.WhatisthesmallestandthelargestpercentageofgamestheSharkscouldwinfor theseason? Moneyandpercentages

12 Copyandcompletethistable.Canyouseeaconnection?

3H 3H Findingapercentageandexpressingas apercentage

Learningintentions

• Tobeabletoexpressonequantityasapercentageofanother.

• Tobeabletoconvertunitsinordertoexpressonequantityasapercentageofanother.

• Tobeableto ndacertainpercentageofaquantity.

Keyvocabulary: percentage,fraction,units

Percentagesareusefulwhencomparingdiscounts,interestratesandevenmarksinatest.

Forexample,Huen’sreportcardcouldbewrittenasmarksoutofeachtotalorinpercentages.

Inthissectionwelookatexpressinganumberasapercentageofanothernumberaswellasﬁndinga percentageofanamount.

Shopsoftendisplaydiscountedpricesasapercentageoftheoriginalprice. Calculatingthenewpricehelpsdetermineifthediscountoffersasigniﬁcant saving.

Lessonstarter:Whatpercentagehaspassed?

Answerthefollowingquestions.

• Whatpercentageofyourdayhaspassed?

• Whatpercentageofthecurrentmonthhaspassed?

• Whatpercentageofthecurrentseasonhaspassed?

• Whatpercentageofyourschoolyearhaspassed?

• Whatpercentageofyourschooleducationhaspassed?

• Ifyoulivetoanaverageage,whatpercentageofyourlifehaspassed?

• Whenyouturned 5,whatpercentageofyourlifewasoneyear?

• Whenyouare 40,whatpercentageofyourlifewilloneyearbe?

Keyideas

Toexpressonequantityasa percentage ofanother:

1 Writethequantitiesasa fraction.(The‘whole’amountisalwaysthedenominator.)

2 Multiplythisfractionby 100 e.g.Expressatestscoreof 14 outof 20 asapercentage. 14

Toﬁndacertainpercentageofaquantity:

1 Expresstherequiredpercentageasafraction.(Youcanalsousedecimals.)

2 Changethe‘of’toamultiplicationsign.

3 Expressthenumberasafraction.

4 Followtherulesformultiplicationoffractions.

e.g.Find 20% of 80

Exercise3H

1 Thecorrectworkinglinetoexpress 42 asapercentageof 65 is:

2 Thecorrectworkinglinetoﬁnd 42% of 65 is:

HintforQ2:Thenumber withthepercentsignis writtenwiththe 100 inthe denominator.

3 Whatisthepercentagefor: ascoreof 20 outof 40? a ascoreof 0 outof 10? b ascoreof 50 outof 50? c

4 Copyandcompletethefollowingsentences.

Finding 1% ofaquantityisthesameasdividingthequantityby a

Finding 10% ofaquantityisthesameasdividingthequantityby b

Finding 20% ofaquantityisthesameasdividingthequantityby c

Finding 50% ofaquantityisthesameasdividingthequantityby d

Finding 25% ofaquantityisthesameasdividingthequantityby . e

Example22Expressingonequantityasapercentageofanother

Express 34 outof 40 asapercentage.

Solution

34 40 × 100 1 = 17 ✚✚ 20 1 × ✟✟ 100 5 1 = 17 1 × 5 1 = 85 So 34 40 = 85%

Nowyoutry

Express 13 outof 20 asapercentage.

5 Expresseachofthefollowingasapercentage.

outof

Explanation

Writeasafraction,withtheﬁrstquantityas thenumeratorandsecondquantityasthe denominator.Multiplyby 100 Cancelandsimplify.

HintforQ5:Multiplyby 100 20 outof 25 a 13 outof 20 b

Example23Convertingunitsbeforeexpressingasapercentage

Express 60 centsasapercentageof $5

× 100 1 = 60 5 = 12

60 500 = 12%

So 60 centsis 12% of $5

Nowyoutry

Express 700 gasapercentageof 4 kg.

6 Express:

40casapercentageof $8 a

50casapercentageof $2

3 mmasapercentageof 6 cm. c

400 masapercentageof 1.6 km. d

200 gasapercentageof 5 kg. e

200 masapercentageof 8 km. f

Explanation

Unitsneedtobethesame.

Convert $5 to 500 cents. Writequantitiesasafractionandmultiply by 100.Cancelandsimplify.

7 Expresseachquantityasapercentageofthetotal. 28 lapsofa 50 lapracecompleted. a Saved $450 towardsa $600 guitar. b 172 fansinatraincarriageof 200 people. c Level 7 completedofa 28 levelvideogame. d 36 studentsabsentoutof 90 total. e 21 kmmarkofa 42 kmmarathon. f

Example24Findingacertainpercentageofaquantity

Find 25% of 48

Solution

25% of 48 = 25 100 × 48 1 = 1 4 × 48 1 = 12

Nowyoutry

Find 7% of 50

8 Find:

Explanation

Writethepercentageasafractionover 100 ‘Of’meansmultiply. Cancelandsimplify.

Problem-solving and reasoning

9 Find:

10% of $750

5% of 2 km b

30% of 150 kg c

20% of 90 minutes d

10% of 5 litres e

25% ofonehour f

HintforQ9:Youmayliketochange theunitsinthequestiontomake iteasiertoworkwith. 3% of 1 km = 3% of 1000 metres.

HintforQ9:Rememberto puttheunitsinyour answer. 10% of $50 = 10 100 × $50 1 = $5 a

50% of $6.50 g 2% of $8 h

7% of 1 2 kg i

10 Copyandcompletethetableofsportingchoices.

11 Calculatorsmakeworkingwithpercentageseasier.Usea calculatortoanswerthesequestions.

Find 8% of $8.40 a

Find 13% of 2 km. b

Find 7 1 4 % of $500 c

Find 24% of 1 hour. d

Find 31.5% of $45960 e

4% ofaclassof 25 studentsareawaywiththeﬂu. Howmanystudentsareatschool?

49.5% ofbabiesbornatthelocalhospitalaregirls.Of the 200 borninthemonth,howmanywereboys?

Seanpays 42% ofhis $86400 incomeintax.How muchisleftafterhepayshistax?

12 Find:

33 1 3 % of 15 litresoforangejuice.

a 66 2 3 % of 3000 marbles.

b 12 1 2 % ofa $64 pairofjeans.

37.5% of 120 donuts.

HintforQ12: 33 1 3 %= 1 3

13 Mostbanksrequirea 10% depositbeforelendingyouanymoney.AshleeandMatthave 7% of the $450000 theirhomecosts.

HowmuchdoAshleeandMatthaveastheirdeposit? a Howmuchdothebanksneedthemtohave? b Howmuchmoredotheyneedtosave? c Iftheygetagovernmentgrantof $14000,willtheyhavethe 10% needed? d

Learningintentions

• Tobeableto ndthenewvalueifanamountisincreasedordecreasedbyapercentage.

• Tounderstandthatpercentagemark-upsanddiscountscorrespondtoincreasinganddecreasingaprice byapercentage.

• TounderstandthatGSTrepresentsa 10% mark-up.

Keyvocabulary: reduction,discount,mark-up,pro t,loss,sellingprice,costprice

Percentagesareusedeveryday,oftenwhendealingwithmoney.Intheworldofﬁnance,calculationsand percentageincreasesanddecreasesarecommonplace.

Theoriginalamountofsomethingcanbethoughtofas 100%

Lessonstarter:Whatdoesitmean?

Inpairs,answerthefollowing:

• Whatdoesitmeantobuyapairofshoes‘onsale’?

• Whatdoesitmeanifthesaleis‘20% off’?

• Whatdoesitmeanto‘paythemarkedprice’?

• Whatdoesitmeantobuyanitemonsale?

• Is $10 offbetterthan 10% off?Discuss.

• Whenwouldyoupaymorethantheoriginalprice?

• Whatcurrentsalesarebeingadvertisedintoday’spaper?

Keyideas

Toincreasebyagivenpercentage:

• ﬁndthepercentageoftheamount

• addthisamounttotheoriginal.

Todecreasebyagivenpercentage:

• ﬁndthepercentageoftheamount

• subtractthisamountfromtheoriginal.

Keywords:

• Decrease:reduction,discount,sale,percentageoff,loss

• Increase:mark-up,proﬁt

• Sellingprice = costprice + proﬁt or costprice loss

Piechartsusepercentagesof 360° to divideupacircle.

• GST:GoodsandServicesTax(InAustraliathisisa 10% mark-up.)

Exercise3I

1 Decideifeachoftheseshowsanincreaseoradecrease. Mark’s $1650 returnairfaretoLosAngeleswasreducedby 10% a Sonyamade 15% proﬁtwhenshesoldherhouse. b Theshopdiscountedallofitscomputersby 10% c Thomasreceivedapayriseof 5% onhiswageof $570 perweek. d Ataxof 15% isaddedtothecostofeverythingintheUnitedKingdom. e

2 Addorsubtractthesepercentages.

3 Calculatethenewpricewhen:

anitemmarkedat $15 isdiscountedby $3 a anitemmarkedat $25.99 ismarkedupby $8 b anitemmarkedat $17 isreducedby $2.50 c anitemmarkedat $180 isincreasedby $45 d

Fluency

Example25Increasingavaluebyapercentage

Findthenewvaluewhen $160 isincreasedby 40% Solution

40% of 160 = 40 100 × 160 1 = $64

Newprice = $160 + $64 = $224

Nowyoutry

Findthenewvaluewhen $85 isincreasedby 30%

4 Findthenewvaluewhen:

Calculate 40% of $160 Cancelandsimplify.

Newprice = originalprice + increase

$400 isincreasedby 10% a $240 isincreasedby 10% b

$250 isincreasedby 10% c $700 isincreasedby 20% d

$500 isincreasedby 1% e $800 isincreasedby 25% f

$84 isincreasedby 25%. g $90 isincreasedby 50%. h

Example26Decreasingavaluebyapercentage

Findthenewvaluewhen $63 isdecreasedby 20%

20% of $63 = 20 100 × 63 1 = $12.60

Newprice = $63−$12.60 = $50.40

Nowyoutry

HintforQ4:Addtheincreaseto theoriginalamount.

Calculate 20% of $63 Cancelandsimplify.

Newprice = originalprice decrease

Findthenewvaluewhen $120 isdecreasedby 15%

5 Findthenewvaluewhen:

$400 isdecreasedby 10% a

$250 isdecreasedby 10% c

$200 isdecreasedby 15% e

$1000 isdecreasedby 50% g

6a Find 8% of $2500

b Increase $2500 by 8%

c Decrease $2500 by 8%

$240 isdecreasedby 10% b

$90 isdecreasedby 20% d

$840 isdecreasedby 25% f

$60 isdecreasedby 15% h

Example27Calculatingdiscounts

Findthecostofan $860 televisionthathasbeendiscountedby 25%

Solution

Discount = 25% of $860 = 25 100 × 860 1 = $215

Sellingprice = $860−$215 = $645

Nowyoutry

Explanation

Calculate 25% discount. Cancelandsimplify.

Sellingprice = costprice discount

Findthecostofa $450 tablethathasbeendiscountedby 40%.

7 Findthecostofthefollowing.

A $600 televisionthathasbeendiscountedby 20% a

A $150 lipstickthathasbeenreducedby 15% b

A $52 jumperthathasdepreciatedby 25%. c

8 Calculatethesellingpricesofthefollowingitemsiftheyaretobereducedby 25%

$16 thongs a $32 sunhat b

$50 sunglasses c $85 bathers d $130 boogieboard e

$6.60 surfboardwax f

Example28Calculatingmark-ups

Findthecostofamicrowaveoventhatwasoriginally $250 thenmarkedupby 12%

Solution

Mark-up = 12% of $250 = 12 100 × 250 1 = $30

Sellingprice = $250 + $30 = $280

Nowyoutry

Explanation

Calculate 12% of $250 Cancelandsimplify.

Sellingprice = costprice + mark-up

Findthecostofatoasterthatwasoriginally $64 thenmarkedupby 15%.

9 Findthecostofthefollowing.

An $80 framedPinkposterthathasbeenmarkedupby 30% a

A $14 mealthathasbeenincreasedby 10% b

A $420 stereothathasbeenmarkedupby 50% c

10 Calculatethesellingpricesofthefollowingitemsiftheyneedtohave 10% GSTaddedtothem.

$35 T-shirt a $75 backpack b

$42 massage c $83 ﬁshingrod d

$52.50 toaster e $149.99 cricketbat f

HintforQ10:Remember 10% GSTadds/increasestheprice ofanitem.

Problem-solving and reasoning

11 Answerthefollowingproblemsinvolvingpercentages.

Anne’sannualsalarywas $86000.Hernewsalaryis 5% more.WhatisAnne’snewsalary?

a Thestategovernmentincreasesthecostofa $9.60 traintripby 5%.Whatisthenewfare?

Acarworth $47000 droppedinvalueby 20% duringtheyear.Whatisthecarnowworth?

c The 10% GSTneedstobeaddedtothecostofameal.Whatdoesa $74 mealcostoncethe GSTisaddedin?

Taxof 40% reducesSaul’swageof $1600.WhatamountdoesSaulreceive?

e Sallymakesa 24% proﬁtonherhouse.Shepaid $500000.Whatdidshesellitfor? f

12 Twoshopsadvertisethesamebike.Bothhavearecommendedretail priceof $1800.Shoponeoffersa 10% discount.Shoptwooffers $200 offallbikes.

a Howmuchdiscountdoesshoponeofferonthisbike?

b Howmuchdoyoupayforthebikeateachshop?

c Whichshopwouldyourecommendandwhy?

d Ifthesamedealappliestoeachofthefollowing bikes,wouldyoustillbuyitfromthesameshop? $2000 bike i

HintforQ12:Findthe priceofeachbikeatshop oneandtwobefore answeringpart d. Areyousurprised byyouranswers?

13 Theword depreciation isusedwhenthevalueofanitem,suchasacar,boatorasetofgolfclubs, reducesinvalueeachyear.

a Rick’ssetofgolfclubsworth $2000 depreciatesatarateof $250 ayear. Copyandcompletethetableshowinghowthevalueoftheclubschangesovertime.

i Drawupasetofaxes(likethoseshownbelow)andgraphthevaluesshowninthetable. 12 Year Value ($) 500 0 2000 3 45678 1000 1500 ii

2000 1750

Whatshapeisyourgraph? iii

Afterhowmanyyearsisthevalueoftheclubszero? iv

b Rick’swifehasasetofgolfclubs,alsovaluedat $2000,whichdepreciateat 12 1 2 % eachyear.

i Willherclubseverbeworthless? ii

Endofyear 0 1 2 3 4

Value ($) 2000 1750 1531.25

HintforQ13:Useacalculator tohelpyouﬁndthevaluesin thistable! Completeasimilartableshowinghowthevalueofherclubschanges.

3J 3J Calculatingpercentagechange

Learningintentions

• Tounderstandthatpro tandlossrepresentthedifferencebetweenthesellingpriceandcostpriceofanitem.

• Tobeabletocalculatethepercentagechange(increaseordecrease)whenpricesareincreasedordecreased.

Keyvocabulary: percentagechange,percentagepro t,percentageloss,pro t,loss,sellingprice,costprice

Whensellingsomething,everyonelikestomakeaproﬁt.Thisiswhenyousellitformorethanyou paidforit.

Unfortunately,peopleoftendotheoppositeandmakealoss.

Thepercentagechangedependsonwhattheitemisoriginallyworth.Forexample:

Carboughtfor $1000

Carsoldfor $200

Lessonstarter:Hangon!

Carboughtfor $16000

Carsoldfor $15200

Discusshowyoucouldcheckifthecorrectpriceforthejeanshadbeenpaid.

Keyideas

Proﬁt = sellingprice costprice

Loss = costprice sellingprice

Calculatingapercentagechangeinvolvesthetechniqueofexpressingonequantityasapercentage ofanother(see Section3G).

• Percentagechange = change originalvalue × 100

• Percentageproﬁt = proﬁt originalvalue × 100

• Percentageloss = loss originalvalue × 100

Loss $800,percentageloss 80%

Loss $800,percentageloss 5%

1 Decidewhethereachofthefollowingrepresentsaproﬁtoraloss.

2 Calculatetheproﬁtmadeineachofthefollowingsituations.

= $14,Saleprice = $21

= $75,Saleprice = $103

= $25.50,Saleprice = $28.95

Costprice = $499,Saleprice = $935

3 Calculatethelossmadeineachofthefollowingsituations.

Costprice = $22,Saleprice = $9 a

Costprice = $92,Saleprice = $47 b

Costprice = $71.10,Saleprice = $45.20 c Costprice = $1121,Saleprice = $874

4 Whichofthefollowingisthecorrectformulaforworkingout percentagechange? %change = change originalvalue A %change = originalvalue change × 100

% change = change × 100%

%change = change originalvalue × 100

Fluency

Example29Calculatingpercentagechange:proﬁt

Calculatethepercentageproﬁtwhen $25 becomes $32

Solution

Proﬁt = $7

%Proﬁt = 7 25 × 100 1 % = 28%

Nowyoutry

Explanation

Thisispercentage proﬁt becauseitwassoldfor morethantheoriginal $25

Proﬁt = $32−$25

Percentageproﬁt = proﬁt originalvalue × 100%

Calculatethepercentageproﬁtwhenasecond-handtoypurchasedfor $36 issoldfor $45

5 Findthepercentageproﬁtwhen:

$20 becomes $36

$40 becomes $50 c

$12 becomes $20 e

$10 becomes $13 b

$25 becomes $30 d

$8 becomes $11 f

original × 100 1 a

$10 becomes $15 g

$6 becomes $12 h

Example30Calculatingpercentagechange:loss

Calculatethepercentagelosswhen $60 becomes $48

Solution

Loss = $12

%Loss = 12 60 × 100 1 = 20%

Nowyoutry

Explanation

Thisispercentage loss becauseitwassoldfor lessthantheoriginal $60

Loss = $60−$48

Percentageloss = loss originalvalue × 100

Calculatethepercentagelosswhena $140 chairissoldfor $84

6 Findthepercentagelosswhen:

$40 becomes $30 a

$6 becomes $3 c

$12 becomes $8 e

$25 becomes $20 g

$25 becomes $21 b

$8 becomes $2 d

$10 becomes $9 f

$20 becomes $18 h

Example31Solvingwordedproblems

Rossbuysatickettoaconcertfor $125,butislaterunabletogo.Hesellsittohisfriendfor $75 CalculatethepercentagelossRossmade.

Solution

Explanation

Loss = $125−$75 = $50 Loss = Costprice Sellingprice

% Loss = 50 125 × 100 1 = 40%

Rossmadea 40% lossontheconcertticket.

Nowyoutry

Percentageloss = loss costprice × 100

After 5 yearsanoilpainting’svalueincreasesfrom $2000 to $4500.Calculatethepercentageincrease initsvalueduringthistime.

7 Copyandcompletethetablesbelow.

(

8 Findthepercentagechange(increaseordecrease)when: 15 kgbecomes 18 kg. a 18 kgbecomes 15 kg. b 4 kgbecomes 24 kg. c 12 kgbecomes 30 kg. d

9 Findthepercentagechangeinpopulationwhen: atownof 4000 becomesatownof 5000 a acityof 750000 becomesacityof 900000 b acountryof 5000000 becomesacountryof 12000000 c

Problem-solving and reasoning

10 Garibuysatickettoaconcertfor $90,butisunabletogo.Hesellsittohisfriendfor $72.Calculatethe percentagelossGarimade.

11 Xavierpurchasedmaterialsfor $48 andmadeadogkennel. Helatersoldthedogkennelfor $84

CalculatetheproﬁtXaviermade. a CalculatethepercentageproﬁtXaviermade. b

12 Gemmapurchaseda $400 foal,whichshelatersold for $720

CalculatetheproﬁtGemmamade. a

CalculatethepercentageproﬁtGemmamade. b

13 Lee-Senpurchaseda $5000 car,whichshelatersold for $2800

CalculatethelossLee-Senmade. a

CalculatethepercentagelossLee-Senmade. b WhatshouldLee-Sensellthecarfortomakea 10% proﬁt?

14 TheAustralianBureauofStatisticstracksthepopulationgrowthofthecountryandofeachindividual stateandterritory.

Copyandcompletethetablebelow,roundingthe % changetoonedecimalplace.

HintforQ14:Usea calculatortohelpyou withthisquestion. Place March2016

ResearchthecurrentgrowthrateofAustraliaandoneothercountryofyourchoice.

3K 3K Solvingpercentageproblemsusing theunitarymethod

Learningintentions

• Tounderstandthattheunitarymethodinvolves ndingthevalueof‘oneunit’asanintermediatestep.

• Tobeabletousetheunitarymethodto ndaquantitywhenonlyapercentageisknown.

• Tobeabletousetheunitarymethodto ndanewpercentagewhenadifferentpercentageisknown.

• Tobeabletoapplytheunitarymethodto ndtheoriginalpricewhenapricehasbeenincreasedordecreasedby apercentage.

Keyvocabulary: unitarymethod,percentage,discount,saleprice,originalprice

Youprobablydidproblemslikethisinprimaryschool:‘Ifﬁveapplescost $6,whatisthecostof 7 apples?’

Withquestionslikethis,weﬁndthecostofoneappleﬁrst.

Thisishelpfulinpercentagesaswell.Onceweknowwhat 1% is worth,wecanﬁndanypercentageamount.Thisiscalledthe unitary method ? If 5 applescost $6 thenoneapple costs $1.20

Lessonstarter:Usingtheunitarymethod

• Fourticketstoaconcertcost $100.Whatdoesoneticketcost?Howmuchwillthreeticketscost?

• Tenworkerscandig 40 holesinanhour.Howmanycanoneworkerdiginanhour?Howmanyholes cansevenworkersdiginanhour?

• Sixpizzascost $54.Whatdoesonepizzacost?Howmuchwouldtenpizzascost?

• Ifeightpairsofsockscost $64,howmuchwould 11 pairsofsockscost?

• Fivepassionfruitcost $2.00.Howmuchwillninepassionfruitcost?

• Ifaworkertravels 55 kmin 5 tripsfromhometotheworksite,howfarwilltheytravelin 7 trips?

Keyideas

The unitarymethod involvesﬁndingthevalueof‘oneunit’andthenusingthisinformation toanswerthequestion.

Whendealingwith percentages,ﬁnding‘oneunit’meansﬁndingonepercent (1%)

Oncethevalueof 1% ofanamountisknown,itcanbemultipliedtoﬁndthevalueofany desiredpercentage.

Exercise3K

1a Whatdoyoudividebytogofrom 8% to 1%?

b Whatdoyoudividebytogofrom 25% to 1%?

c Whatdoyoumultiplybytogofrom 1% to 100%?

d Whatdoyoumultiplybytogofrom 1% to 50%?

2 If 1% ofanamountis $3,whatis: 2% oftheamount? a 10%

3 If 1% ofanamountis $8,whatis:

10% oftheamount? a 100% oftheamount? b

4 Copyandcomplete:

If 4% ofanamount = $16 then 1% ofanamount = and 100% ofanamount =

Fluency

Example32Usingtheunitarymethodtoﬁndthefullamount

If 8% ofanamountofmoneyis $48,whatisthefullamountofmoney?

Solution

8% of amount is $48 1% of amount is $6 100% of amount is $600 ÷ 8 × 100 ÷ 8 × 100

Fullamountofmoneyis $600

Nowyoutry

Explanation

Remembertoﬁnd 1% ﬁrst.

Divideby 8 toﬁndthevalueof 1% (48 ÷ 8 = 6)

Multiplyby 100 toﬁndthevalueof 100% (6 × 100 = 600)

If 15% ofanamountis $45,whatisthefullamount?

5 Calculatethefullamountofmoneyforeachofthefollowing.

3% ofanamountofmoneyis $27

5% ofanamountofmoneyis $40 b

HintforQ5:Firstﬁndthevalue of 1%. a

12% ofanamountofmoneyis $132 c

60% ofanamountofmoneyis $300 d

8% ofanamountofmoneyis $44 e

6% ofanamountofmoneyis $15 f

Example33Usingtheunitarymethodtoﬁndanewpercentage

If 11% ofthefoodbillis $77,howmuchis 25% ofthefoodbill?

Solution

11% of food bill is $77

1% of food bill is $7

25% of food bill is $175 ÷ 11 × 25 ÷ 11 × 25

Explanation

Find 1% first. Divideby 11 tofindthevalueof 1% (77 ÷ 11 = 7). Multiplyby 25 tofindthevalueof 25% (7 × 25 = 175)

Nowyoutry If 20% ofasalarybonusis $6000,howmuchis 75% worth?

6 If 4% ofthetotalbillis $12,howmuchis 30% ofthebill?

7 Calculate:

20% ofthebill,if 6% ofthetotalbillis $36 a

80% ofthebill,if 15% ofthetotalbillis $45. b

3% ofthebill,if 40% ofthetotalbillis $200 c

7% ofthebill,if 25% ofthetotalbillis $75 d

HintforQ6andQ7:Firstﬁndthe valueof 1% ofthebill.

Problem-solving and reasoning

Example34Usingtheunitarymethodtoﬁndtheoriginalprice

Apairofshoeshasbeendiscountedby 20%.Ifthesalepricewas $160,whatwastheoriginalprice oftheshoes?

Solution

Onlypaying 80% oforiginalprice:

Â80% oforiginalpriceis $160

Â1% oforiginalpriceis $2

Â100% oforiginalpriceis $200

Explanation

Theoriginalpriceoftheshoeswas $200. 20% discount,sopaying (100−20)%

Nowyoutry

Wepay 80% afterthe 20% discount.

Divideby 80 toﬁndthevalueof 1% (160 ÷ 80 = 2)

Multiplyby 100 toﬁndthevalueof 100% (2 × 100 = 200).

Anewbedwasdiscountedby 30%.Ifthesalepricewas $294,whatwastheoriginalprice ofthebed?

8 Anecklaceinajewellerystorehasbeendiscountedby 20% Ifthesalepriceis $240,whatwastheoriginalpriceofthenecklace?

9 Findtheoriginalpriceofthefollowingitems.

Apairofjeansdiscountedby 40% hasasalepriceof $30 (youpay 60%) a

Ahockeystickdiscountedby 30% hasasalepriceof $105 (youpay 70%)

Asecond-handcomputerdiscountedby 85% hasasale priceof $90 (youpay 15%)

Asecond-handtextbookdiscountedby 80% hasasale priceof $6

Astandardrosebushdiscountedby 15% hasasaleprice of $8.50

Amotorbikediscountedby 25% hasasalepriceof $1500 f

10 OncetheGSTof 10% isaddedtoabill,thepriceis 110%.Ifthepriceof amealatacafe,includingtheGST,is $55,howmuchGSTispaid?

11 Apairofjeans,including 10% GST,comesto $88.WhatisthecostwithouttheGST?

12 If 22% ofanamountis $8540,whichofthefollowingwouldgivethevalueof 1% oftheamount?

× 100 A

÷ 100 B

13 Herearethreereal-lifereceipts(withthenamesofshopschanged).GSTrateis 10% Answerthequestionsbelowbasedoneachone.

Superbarn

GYMEAFRUITMARKET

HowmuchwasspentatSuperbarn?

a Howmanykilogramsoftomatoeswerebought?

b WhichitemincludedtheGST,andhowdoyoutellbylookingatthereceipt?

c WhatisthecostoftheitemiftheGSTisnotincluded? d

GymeaFruitMarket

Whatwasthecostofbananasperkilogram? a Onwhatdatewasthepurchasemade? b WhatdoesROUNDmean? c Whatwasthetotalpaidfortheitems? d Howmuchtaxwasincludedinthebill? e Whatpercentageofthebillwasthetax? f

Xmart

Howmanytoyswerepurchased? a Whatwasthecostofthemostexpensiveitem? b WhichofthetoysattractedGST? c HowmuchGSTwaspaidintotal? d WhatpercentageofthetotalbillwastheGST? e

Ownerandmanagerofa fruitandvegetableshop

Owningandrunningyourownbusiness requireshardworkandlonghours.Theskills youneedarevaried,frombeingwellorganised andhavinggoodcommunicationskillstobeing abletomanagestaffandstock.Perseveranceis alsoarequirement,asmanybusinessesfail withintheirﬁrstthreeyears.

Skillwithnumbersisimportantforthe day-to-dayrunningofafruitandvegetable shoptoensureaprofit.Stockneedstobe orderedandkeptfresh.Enoughstockneedsto besoldtocoverwages,storerental,delivery costsand,ifpossible,alsosomeprofit.Prices mustbeadjustedforseasonalfluctuationsas wellasunexpectedcosts.

1 Imaginethatyouarestartingupafruitandvegetableshopinthetownorsuburbwhereyoulive. Listsomevegetablesandfruitsthatcouldbesuppliedfromyourlocalmarketsorfarms. a Whatcostswouldyourbusinesshave,otherthanbuyingproduce?

b Whatdoyouthinkwouldbethemainchallengestosuccessfullyrunningyourshop?

2 Whenorderingorbuyingproducefrommarkets,pricesperkgorperitemareusuallystated. Calculatetheunitpriceorcostperkilogramforeachofthefollowing.

15 kgbagofpotatoescosts $14.55 a 25 kgboxofapplescosts $34.50 b Boxoflettucecontaining 20 headsoflettucecosts $18

c 8 kgbagofcarrotscosts $6.40

20 kgboxofnavelorangescosts $25.40

3 Freshproduceisgenerallyboughtatawholesalepricefromsuppliersandsoldatahigherretail pricetocovercostsandprovideaproﬁttothestoreowner.Findtheretailprice/kgoftheitems below,giventheretailpriceis 250% ofthewholesaleprice.Roundtothenearestcent.

Freshasparaguswholesalepriceis $1.42 perkg a Beanswholesalepriceis $2.18 perkg b Broccoliwholesalingat $2.39 perkg c Apples 2.5 kgbagwholesalingat $1.83 d Celerywholesalingat 85 cperkg e

4 Findthecostofanindividualpieceoffruitusingtheinformationinthistable.Roundtothe nearestcent.

5 Findthecostofthefollowingorder foracustomer.

6 Comparethecostof 100 gofapplesforeachoption A and B andstatewhichisthebetterbuy. AtrayofPinkLadyappleswithaweightof 600 gramsfor $2.94,or A 0.75 kgofloosePinkLadyapplesfor $3.30 B

7 ArestaurantinDaintree,northQueensland,orderedthefollowingexoticfruitsfromanearbytropical fruitshop.Calculatethetotalcostofthisorderincludinga $45 packinganddeliveryfee.

3.8 kg PurpleMangosteens $10.99/kg Dragonfruitareverynutritious.

Usingtechnology

8 Manypeopleworkoverseasandwishtocomparelivingcosts. SetupthefollowingExcelspreadsheettoconvertAmerican producepricesperlb(pound)toAustraliandollars(AUD)perkg. WeusethelettersUSDforAmericancurrency.

a Usetheseconversionfactors: HintforQ8:

• IncellB2 enter 2.2 asthereare 2.2 lbperkg.

• IncellC2 enter 0.75 (i.e. 1 AUD = 0.75 USD)orusethe currentexchangerate.

• IncellD2 entertheformula = 1/C2.Thisgivesthenumber ofAUDforUSD.

• Toﬁllformulasdownacolumn, dragdownthe‘ﬁllhandle’.

b IncolumnC,multiplythepricesinUSD/lbby 2.2 togiveUSD/kg.Use $ signstoanchorthe B2 cellvalueasintheHint.

c IncolumnD,convertUSDtoAUD.MultiplybycellD2 (i.e.thenumberofAUDfor 1 USD).

d SelectanyﬁveoftheaboveitemsandcomparetheconvertedAustralianprices/kgtocurrent Australianonlinesupermarketprices/kg.Whatobservationcanyoumake?Givesomepossible reasonsforthedifferenceinprices.

Upsizedphonescreen

TheSamsumphonecompanyisconsideringmakinganewupsizedphonescreen comparedtooneofitssmallermodels.Thesmallermodelhasdimensions 6.3 cmby 11.3 cm.

Marketresearchhasindicatedthatatotalincreaseinscreenareaof 30% shouldbeenough tomeetthedemandinthemarket.Toincreasethescreensize,however,thelengthand thewidthneedtoincreasebythesamepercentage,sothelengthandwidthareinthe sameproportion.

Presentareportforthefollowingtasksandensurethatyoushowclearmathematical workings,explanationsanddiagramswhereappropriate.

1Preliminarytask

a DeterminetheareaoftheoriginalSamsumphonescreenwithalengthof 11.3 cmandawidth of 6.3 cm.

b Findthelengthandthewidthofanupsizedphoneifthedimensions(lengthandwidth)are increasedby 10%

c Findtheareaofanupsizedphonescreenifthedimensionsareincreasedby 10%

d Whatisthechangeinareabetweenthenewandoldscreenswhenthedimensionsareincreased by 10%?

e Findthepercentagechangeintheareaoftheup-sizedphoneifthedimensionsareincreased by 10%

Rememberthat: Percentagechange = change originalvalue × 100%

2Modellingtask

Formulate a Theproblemistodeterminethepercentageincreasewhichshouldbeappliedtothelength andwidthtoachievea 30% increaseinareaofthephonescreen. Writedownalltherelevantinformationthatwillhelpsolvethisproblem.

b MakeanaccuratedrawingoftheoriginalSamsumphonescreenwithdimensions 6.3 cmby 11.3 cm.Onyourdiagramincludeanillustrationofhowthedimensionsmightbeincreased.

c Makethefollowingcalculationstofindtheplanned,up-sizedscreenarea: findtheoriginalscreenarea i find 30% ofthisarea ii increasetheoriginalscreenareabythischangetogivethenew,up-sizedscreenarea. iii

d CalculatethedimensionsandscreenareaofanupsizedphoneiftheoriginalSamsumscreen’s dimensionsareincreasedbythefollowingpercentages.Roundyouranswerstothreedecimal places.

5% i 15% ii 25% iii

e Determinewhichoftheabovepercentagedimensionincreasesleadstoanincreaseof more than 30% intotalarea.Justifyyouranswerbycalculatingpercentageincreasesinscreenarea foreachsetofnewdimensionsthatyoucalculatedinpart d above.

f Examineyourresultsfromparts d and e andusetrialanderrortodeterminetherequired percentageincreaseinphonedimensionstoachievea 30% increaseinarea.Answercorrect toonedecimalplace.Rememberthatthelengthandthewidthneedtoincreasebythesame percentage.

g Summariseyourresultsinatableliketheonebelowanddescribeanykeyﬁndings.

3Extensionquestion

OnesalesexecutiveatSamsumsaysthattoincreasetheareaby 30% youshouldincreasethe dimensionsby 30%.Demonstratethatthesalesexecutiveiswrong.

Evaluate and verify

Communicate

PayingforAustralia

Keytechnology:Spreadsheets

Asat 2023,Australia’stotalpopulationisabout 25.5 million,itstotalgovernmentexpenditureisabout $600 billionandthetotaltaxationrevenueisabout $550 billion.AbouthalfofAustralia’sgovernment revenuecomesfrompersonalincometaxwithmostpeoplepayingtaxbasedonthefollowingtax table.

1Gettingstarted

Forthisactivity,assumethefollowingfactsaboutAustralia.

• Thecurrentpopulationis 25.5 million,including 20 millionadults.

• Thetotalgovernmentexpenditureis $600 billion.

• Theaveragepersonalincometaxis $12500 peradult.

• Thetotalrevenuefromothertaxes(notincometax)is $300 billion.

a WhatisthetotalrevenuetotheAustraliangovernmentfrompersonalincometaxpaidbyadult Australians?

b Bycombiningthetotalpersonalincometaxandothertaxes,doesAustraliaearnenoughrevenue topayforitstotalexpenditure?Givereasons.

c Usethetaxtableabovetocalculatethetotaltaxpayableonthefollowingtaxableincomes. (WewillignoretheMedicarelevyfornow.)

$30000 i

$100000 ii

$150000 iii

2Usingtechnology

ThefollowingspreadsheetisasimplebalancesheetfortheAustralianeconomyusingtheinformation frompart 1

a Enteralltheinformationintoaspreadsheetandanswerthefollowing. ThetotalrevenuefrompersonalincometaxiscalculatedinC9.Fromwhichcelldoesthe formulareceivetheaveragetaxpaidbyeachadult?

i Thepopulationincreaseyearbyyeariscurrentlysetat 0.5%.Explainhowtheformulaincell B10 usesthisinformationtocalculatethecorrecttotalﬁgureaftertheincrease.

ii ExplainhowtheformulasincellsD10 andE10 work. iii

b FilldownatcellsA10,B10,C9,D10,E10 andF9 toyear 2033.Whatdoyounoticeabouttheoverall ﬁnancialbalanceoverthistime?

c ChangetheaverageamountofincometaxpaidbyAustralianadultsincellD1 to $13000.Does thisimproveAustralia’sbalancesheet?

d Let’snowassumethattheaverageAustraliantaxableincomeis $75000 Usethetaxtableonthepreviouspagetocalculatetheaveragetaxpaidusingthistaxable income. i

ii 3Applyinganalgorithm

Useyourresultfrompart di inyourspreadsheettoupdatecellD1.Doesthisimprove Australia’sbalancesheet?

a Wewillsystematicallyapplythisalgorithmtoexplorewhataveragetaxableincomeisneededto provideAustraliawithenoughtaxrevenue.Startwith I = 50000.

• Step 1:CalculatethetaxpaidbyanAustralianearning$I.

• Step 2:EnterthistaxamountintocellD1 ofyourspreadsheet.

• Step 3:RecordAustralia’sbalanceatyear2033.

• Step 4:Increase I by 5000

• Step 5:ReturntoStep 1.

b Tothenearest $5000,whataveragetaxableincomeprovidesAustraliawithapositivebalanceat year2033?

c ChangetherateofAustralia’spopulationgrowthincellB4.Explorehowthischangeaffectsthe ﬁnancialbalanceatyear 2033 comparedtotheoriginal 0.5% rate.

2 Jillhasﬁvecoinsinherpocket:a $2 coin, $1 coin, 50 centpiece, 20 centpiece,andone 10 cent coin.IfJillchoosesjusttwocoinsfromherpocketwithoutlookingatthem,ornoticingtheirsizeor shape,howmanydifferentamountscouldshearriveat?

3 Writeonehalfintendifferentways.

4 Completethesemagicsquares.Allrows,columnsandthetwodiagonalsaddupto thesametotal.

5 Atangramconsistsofsevengeometricshapes(tans)asshownon theright.Thetangrampuzzleispreciselyconstructedusingvertices, midpointsandstraightedges.

a Writeeachoftheseparatetanpiecesasapercentage, afractionandadecimalamountoftheentirepuzzle.

b Checkyourseventansadduptoatotalof 100%

c Startingwithasquare,makeanewversionofa‘modern’ tangrampuzzle.Youmusthaveatleastsixpiecesinyourpuzzle. Anexampleofamodernpuzzleisshownbelow.

d Writeeachoftheseparatepiecesofyournewpuzzleas apercentage,afractionandadecimalamountofthewholepuzzle.

e Separatepiecesoftangramscanbearrangedtomakemorethan 300 creativeshapesanddesigns,someofwhichareshown. Youmayliketoresearchtangramsandattempttomake someoftheimages.

A fraction is part of a whole. numerator (parts from the whole) denominator (parts in the whole)

Equivalent fractions What is a fraction?

• represent the same amount, they are the same decimal.

Simplify fractions by dividing the numerator and denominator by their highest common factor..

Simplifying fractions =

Fractions

Adding or subtracting fractions

Mixed and improper fractions × 1 4 3 2

To add or subtract fractions you should use the lowest common denominator.

To take the reciprocal of a proper or improper fraction is to turn it upside Reciprocal

• cre ate equivalent fractions by multiplying or dividing the top (numerator) and bottom (denominator) by the same number.

Multiplying fractions

• use proper or improper fractions 3 × 1 2 × 4

• multiply the numerators • multiply the denominators • cancel a ny numerator with any denominator

8 • simplify

Dividing fractions • change the ÷ to × (its reciprocal) and multiply.

Fractions to decimals

(1) Change denominator to 10, 100 or 1000. or (2) Divide the denominator into the numerator.

Types of decimals

Decimals can terminate (stop).

Decimals can repeat/recur.

0.5, 0.75, 0.125 0.3333... = 0.3

= 0.61

The decimal point moves the same number of places as the

6.413 ×

= 641.3 Multiplying by tens

Dividing by tens 71.3 ÷

Dividing decimals

Multiply both numbers by 10, 100, 1000 etc. This is dividing by a whole number.

Decimals

Multiplying decimals

The number of decimal places in the question is the same as in the answer. 0.04 × 0.3 = 0.012

Percentages and decimals 0.45 = 45%

Adding and subtracting

When adding and subtracting with decimals make sure the decimal points line up.

Rounding decimals

If the digit after the place you want is 0, 1, 2, 3 or 4, round down; if the digit is 5, 6, 7, 8 or 9, round up.

Percentage change

Percentage change =×

Percentages out of 100

Find

Percentages and fractions

Chapterchecklist

AversionofthischecklistthatyoucanprintoutandcompletecanbedownloadedfromyourInteractiveTextbook.

3A 1 Icangenerateequivalentfractions

e.g.Rewrite 3 6 asanequivalentfractionwithadenominatorof 40

3A 2 Icanconvertafractiontosimplestform

e.g.Writethefraction 8 20 initssimplestform.

3B 3 Icanaddandsubtractfractions,includingmixednumerals

e.g.Simplify:

3B 4 Icanmultiplyfractions,includingmixednumerals

e.g.Simplify: a 2 5 × 3 7 b 8 5 × 1 3 4

3B 5 Icandividefractions,includingmixednumerals

e.g.Simplify:

3C 6 Icanaddandsubtractnegativefractions.

e.g.Simplify:

3C 7 Icanmultiplyanddividenegativefractions.

e.g.Simplify: 6 5 × 3 4 and −1 1 3 ÷ 3

3D 8 Icancomparedecimals

e.g.Comparethefollowingdecimalsandplacethecorrectinequalitysignbetweenthem: 57.8934257.89631

3D 9 Icanconvertdecimalstofractions

e.g.Convert 5.12 toafractioninitssimplestform.

3D 10 Icanconvertsimplefractionstodecimals

e.g.Convert 9 25 toadecimal.

3D 11 Icanaddandsubtractdecimals

e.g.Calculate: a 9.7−2.86 b 2.4 + 4.24

3E 12 Icanmultiplyanddividedecimalsbypowersof10

e.g.Calculate: a 9.753 ÷ 100 b 27.58 × 10000

3E 13 Icanmultiplydecimals

e.g.Calculate 25.7 × 0.3

3E 14 Icandividedecimals

e.g.Calculate 64.137 ÷ 0.03

3F 15 Icanconvertfractionstoterminatingdecimals

e.g.Write 7 8 asaterminatingdecimal.

3F 16 Icanconvertfractionstorecurringdecimals

e.g.Write 3 5 7 asarecurringdecimal.

3F 17 Icanroundterminatingdecimals

e.g.Round 14.258 to 1 decimalplace.

3F 18 Icanconvertpercentagestofractionsandtodecimals

e.g.Write 72% asasimplefractionandasadecimal.

3G 19 Icanconvertfractionsanddecimalstopercentages

e.g.Convertthefollowingtopercentages:

a 4 5 b 0.81

3H 20 Icanexpressonequantityasapercentageofanother,convertingunits ifrequired

e.g.Express:

a 34 outof 40 asapercentage b 60 centsasapercentageof $5

3H 21 Icanﬁndacertainpercentageofaquantity

e.g.Find 25% of 48.

3I 22 Icanﬁndtheresultwhenavalueisincreasedbyapercentage e.g.Findthenewvaluewhen $160 isincreasedby 40%

3I 23 Icanﬁndtheresultwhenavalueisdecreasedbyapercentage

e.g.Findthenewvaluewhen $63 isdecreasedby 20%.

3I 24 Icancalculatethecostofanitemafteradiscount

e.g.Findthecostofan $860 televisionthathasbeendiscountedby 25%

3I 25 Icancalculatethecostofanitemafteramark-up

e.g.Findthecostofa $250 microwaveoventhathasbeenmarkedupby 12%

3J 26 Icancalculatethepercentageproﬁtwhenpricesareincreased e.g.Calculatethepercentageproﬁtwhen $25 becomes $32

3J 27 Icancalculatethepercentagelosswhenpricesaredecreased e.g.Calculatethepercentagelosswhen $60 becomes $48

3K 28 Icanusetheunitarymethodtoﬁndthefullamount e.g.If 8% ofanamountis $48,whatisthefullamountofmoney?

3K 29 Icanusetheunitarymethodtoﬁndanewpercentage e.g.If 11% ofthefoodbillwas $77,howmuchis 25% ofthefoodbill?

3K 30 Icanusetheunitarymethodtoﬁndtheoriginalprice e.g.Apairofshoeshasbeendiscountedby 20%.Ifthesalepricewas $120,whatwastheoriginal priceoftheshoes?

Short-answerquestions

1 3A Copyandcomplete.

2 3A Simplify.

3 3B Evaluate.

4 3B Find:

5 3B Find:

6 3B Find:

7 3B Calculatethesedivisions.

8 3C Evaluateeachofthefollowing.

9 3D Convertthesefractionstodecimals.

10 3D Writethesedecimalsassimplefractions.

11 3E Evaluate.

3H Expresseachofthefollowingasapercentage. $35 outof $40 a 6 outof 24 b

$1.50 outof $2 c 16 cmoutof 4 m d

Multiple-choicequestions

1 3D 0.36 expressedasafractionis:

2 3B 1 8 + 5 8 isequalto:

3 3E When 21.63 ismultipliedby 13.006,thenumberofdecimalplacesintheansweris:

4 3A 2 1 3 isthesameas:

5 3B Thereciprocalof 3 4 is:

6 3D Whichdecimalhasthelargestvalue?

7 3E 9.46 × 1000 is:

8 3H 75% of 84 isthesameas:

4 × 3 A

9 3K If 1% ofthetotalequals 8,then 5% ofthetotalequals:

10 3I $790 increasedby 10% gives:

Extended-responsequestions

1a A $320 statuehastheGST (10%) addedtotheprice.Whatistheﬁnalprice?

b Thepriceofa $670 stoveincludesGST.Whatisthepricenon-inclusiveofGST?Roundtothe nearestcent.

c InanothercountrytheGSTis 13.5%.Ifacoatispricedat $145.28 andthispriceincludesGST, whatisthepriceexcludingGST?

2 ThefollowingtableshowsthevalueofA$1 (oneAustraliandollar)inforeigncurrency. GenevieveisplanninganextendedholidaytoAsia.SheplansonvisitingIndia,Singapore,Phuket andHongKong.

a ShehasdecidedtochangesomeAustraliandollarstoeachoftheabovecurrenciesbeforeshe ﬂiesout.HowmuchofeachcurrencywillshereceiveifshechangesA$500 toeachcurrency?

b Ifshespent 70% ofherThaibahtonhotels,howmuchThaibahtdoesshehavelefttospend?

c AftervisitingHongKong,Genevievehas $42 HKDleft.Whatdoesthisconvertbacktoin Australiandollars?