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Integrated Curriculum for Secondary SchoolsCurriculum Specifications

MATHEMATICSFORM4

CurriculumDevelopmentCentreMinistryofEducationMalaysia2006


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Copyright 2006CurriculumDevelopmentCentreMinistryof Education Malaysia

Aras 4- 8,BlokE9KompleksKerajaanParcel EPusatPentadbiranKerajaanPersekutuan62604

Putrajaya

First published2006

Copyright reserved. Except for usein areview, thereproductionorutilisation ofthiswork

in anyformorbyanyelectronic, mechanical,orothermeans,now knownor hereafter invented, includingphotocopying,and recording is forbiddenwithoutthe prior written permission fromtheDirector ofthe CurriculumDevelopment Centre, Ministryof EducationMalaysia.


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CONTENTSPageRUKUNEGARAivNationalPhilosophy

ofEducationvPrefaceviIntroductionviiStandard Form1QuadraticExpressions and Equations2

Sets4MathematicalReasoning8The StraightLine16Statistics20ProbabilityI24

Circles III26TrigonometryII29Anglesof Elevation andDepression33Lines and Planes in3-Dimensions34


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RUKUNEGARA DECLARATIONOUR NATION, MALAYSIA, beingdedicatedto achievinga greater unity

ofall her peoples;to maintainingademocratic way of life;to creatinga just societyinwhich thewealth of the nation

shall beequitably shared;to ensuring a liberal approach to her richanddiverseculturaltraditions;to building a progressivesocietywhich shall beoriented

tomodernscienceandtechnology;WE, her peoples,pledge our united efforts to attain theseends guided by these principles:BELIEFIN GODLOYALTY TOKING ANDCOUNTRYUPHOLDING THECONSTITUTIONRULEOFLAWGOODBEHAVIOUR AND MORALITY RUKUNEGARA DECLARATION

OUR NATION, MALAYSIA, beingdedicated


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to achievinga greater unityofall her peoples;to maintaininga

democratic way of life;to creatinga just societyinwhich thewealth of the nationshall beequitably shared;to ensuring a liberal approach to her richand

diverseculturaltraditions;to building a progressivesocietywhich shall beorientedtomodernscienceandtechnology;

WE, her peoples,pledge our united efforts to attain theseends guided by these principles:BELIEFIN GODLOYALTY TOKING ANDCOUNTRYUPHOLDING THECONSTITUTIONRULEOFLAWGOODBEHAVIOUR AND MORALITY


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Educationin Malaysiais an ongoingefforttowardsfurther

developingthepotentialofindividuals in aholisticandintegratedmannerso as toproduceindividuals who

areintellectually, spiritually, emotionally andphysically balancedandharmonious, basedon afirmbeliefinGod.Suchaneffort is

designedtoproduceMalaysiancitizenswhoare knowledgeable and competent,whopossess highmoralstandards, andwhoareresponsibleandcapableofachievingahighlevelofpersonalwell-being as well

asbeingable


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to contribute to the bettermentofthefamily, the society andthe nationatlarge.

Educationin Malaysiais an ongoingefforttowardsfurtherdevelopingthepotentialofindividuals in aholistic

andintegratedmannerso as toproduceindividuals whoareintellectually, spiritually, emotionally andphysically balancedandharmonious, basedon afirm

beliefinGod.Suchaneffort isdesignedtoproduceMalaysiancitizenswhoare knowledgeable and competent,whopossess highmoralstandards, andwhoareresponsibleandcapableofachievinga

highlevelof


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personalwell-being as wellasbeingableto contribute to the betterment

ofthefamily, the society andthe nationatlarge.


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(vi)


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PREFACE

Scienceand technologyplays a

criticalrolein realisingMalaysiasaspirationtobecomea developednation. Sincemathematics is instrumentalin the developmentof scientific and

technological knowledge, theprovisionof qualitymathematics education fromanearlyage inthe education processisthusimportant.TheMalaysian

schoolcurriculumoffersthreemathematicseducation programs,namelyMathematics forprimaryschools,Mathematicsand Additional Mathematics for secondaryschools.

TheMalaysianschoolmathematicscurriculumaimstodevelopmathematical knowledge,competencyand

inculcate positive attitudestowardsmathematics among pupils. Mathematics for


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secondaryschoolsprovidesopportunitiesforpupilsto

acquiremathematical knowledgeandskills,and develophigher orderproblemsolving anddecisionmakingskillsto enable

pupils tocope withdailylife challenges.As withother subjects inthe secondaryschoolcurriculum, Mathematics aims toinculcate noblevaluesandlove for the nation

inthe developmentofaholisticperson, whointurnwill be abletocontributetothe harmonyand prosperityof the nationanditspeople.

Beginning 2003,Englishisusedasthe

mediumofinstruction


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for Scienceand Mathematicssubjects.The policyto changethemedium

of instructionfor ScienceandMathematics subjects follows aphased implementationschedule and isexpectedto becompletedby2008.

In the teachingandlearningofMathematics, the useof technologyespeciallyICTisgreatlyemphasised.Mathematics taught inEnglish,

coupledwiththe useof ICT,providegreater opportunitiesfor pupilstoimprovetheir knowledge andskillsin mathematicsbecauseofthe richnessof resources andrepositoriesof knowledge inEnglish.Pupilswillbe betterable tointeract withpupils

fromother countries,improve their proficiency


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in Englishandthusmake thelearningof mathematics

moreinterestingandexciting.

Thedevelopmentof this Mathematics syllabus isthe workofmanyindividuals

and experts inthe field. On behalf of theCurriculumDevelopmentCentre,I wouldliketo expressmuch gratitudeandappreciationto thosewho

have contributedinonewayoranother towardsthis initiative.

(MAHZANBINBAKARSMP,AMP)

DirectorCurriculumDevelopmentCentreMinistryof EducationMalaysia

(vii)


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(viii)


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INTRODUCTION

A well-informed and knowledgeablesocietywell versed

intheuse ofmathematicstocope withdailylife challengesis integral torealising thenations aspiration tobecome an industrialised nation.

Thus, efforts aretakentoensure a societythatassimilatesmathematics into theirdailylives. Pupilsare nurtured fromanearly age withthe skills to

solveproblemsandcommunicatemathematically, toenablethemtomakeeffectivedecisions.

Mathematicsisessential inpreparing aworkforcecapableofmeeting thedemandsofaprogressivenation.

Assuch,this


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fieldassumesitsroleasthedriving force behind

various developments in science andtechnology.In linewith thenations objective to createaknowledge-basedeconomy,the skillsof Research & Development inmathematics is

nurturedanddevelopedatschool level.

As a field of study,Mathematicstrains themindtothink logicallyand

systematicallyin solvingproblemsandmakingdecisions. Thisdisciplineencouragesmeaningfullearningandchallengesthe mind,and hencecontributes tothe holisticdevelopment of theindividual.To thisend,strategies tosolveproblemsare widelyused

intheteaching


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andlearningofmathematics.The developmentofmathematical reasoning is

believed tobecloselylinked to theintellectual development and communicationabilityofpupils.Hence,mathematicsreasoning skills are alsoincorporated

inthemathematicsactivities toenablepupilstorecognize, buildandevaluatemathematicsconjectures andstatements.

In keepingwiththe National EducationPhilosophy,the Mathematicscurriculumprovides opportunitiestopupilsfromvariousbackgrounds andlevelsof abilitiestoacquire mathematical skillsandknowledge. Pupils arethenableto seek relevant information,and be creative informulatingalternatives

and solutions whenfacedwith


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challenges.

The generalMathematicscurriculumhasoften been

seen to compriseofdiscrete areas relatedtocounting,measurement,geometry,algebra andsolvingproblems.Toavoid

theareas tobecontinuallyseen asseparateandpupilsacquiringconceptsand skillsinisolation,

mathematicsislinked toeverydaylifeand experiences in andout of school.Pupilswill havetheopportunitytoapplymathematics indifferentcontexts,and seetherelevanceof mathematicsin dailylife.

Ingiving

opinions andsolvingproblems


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either orallyorinwriting,pupilsareguided

inthe correct usage oflanguage and mathematicsregisters.Pupilsare trained toselect informationpresentedinmathematicalandnon-

mathematical language; interpretand represent information intables,graphs,diagrams, equations or inequalities;and subsequentlypresent informationclearlyandprecisely,withoutanydeviation from

theoriginal meaning.

Technologyin education supportsthemasteryandachievement of thedesiredlearningoutcomes.Technologyusedintheteachingand learningofMathematics,forexample calculators,areto beregarded

as tools to enhancethe teachingand


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learning processandnot to replaceteachers.

Importanceis

alsoplacedontheappreciation of theinherentbeautyofmathematics.Acquaintingpupils withthe life-history

ofwell-knownmathematiciansorevents, theinformationofwhich iseasilyavailable fromthe Internet for example, will goalong

wayinmotivating pupils toappreciatemathematics.

The intrinsic valuesofmathematics namelythinkingsystematically,accurately,thoroughly, diligentlyandwithconfidence,infused throughoutthe teaching and learningprocess;contributetothemouldingofcharacter

andthe inculcation ofpositive attitudes


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towards mathematics.Togetherwiththese,moral valuesarealso

introducedincontext throughoutthe teachingand learningofmathematics.

(ix)


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Assessment, intheformof testsandexaminations

helps to gauge pupilsachievements.The useofgood assessment data from avarietyofsourcesalsoprovides valuableinformationon

the developmentandprogressofpupils. On-goingassessmentbuiltinto the dailylessonsallows theidentification of pupils strengths andweaknesses,and effectiveness of the

instructional activities. Informationgained fromresponsestoquestions,group workresults, and homeworkhelps inimproving theteachingprocess,and hence enablesthe provision ofeffectivelyaimed lessons.

AIM

Themathematicscurriculum for secondaryschools aims todevelopindividualswho

areableto


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thinkmathematically, andapplymathematicalknowledgeeffectivelyand responsibly

in solvingproblemsandmakingdecisions; and face the challenges ineverydaylife brought about bytheadvancement ofscience andtechnology.

OBJECTIVES

The mathematicscurriculumforthesecondaryschool enables pupilsto:1understanddefinitions, concepts, laws,principles, and

theorems relatedtoNumber, Shape andSpace,and Relationship;2widen theuse ofbasic operationsofaddition, subtraction,multiplicationanddivisionrelatedtoNumber,Shape andSpace, andRelationship;3acquirebasicmathematical skills such as:

makingestimation and


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rounding;measuring andconstructing;collectingand

handlingdata;representing andinterpretingdata;recognising andrepresentingrelationshipmathematically;

using algorithmandrelationship;solving problems; andmakingdecisions.4communicatemathematically;5apply

knowledge andskills ofmathematicsinsolving problemsandmakingdecisions;6relate mathematicswithother areasofknowledge;7usesuitable technologiesinconceptbuilding, acquiringskills,solving

problemsand exploring the field

ofmathematics;8


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acquiremathematical knowledge anddevelopskillseffectivelyand usethem

responsibly;9inculcatea positive attitudetowardsmathematics;and

10appreciatetheimportance

andbeautyofmathematics.

CONTENT ORGANISATION

The MathematicsCurriculumcontentat the secondaryschoollevel

isorganised intothreemain areas, namely: Number;ShapeandSpace; andRelationship.Mathematical conceptsrelated to therespective area, arefurtherorganised intotopics. These topicsarearrangedin ahierarchicalmannersuchthat themorebasic and tangibleconceptsappear

firstand themore


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complex and abstractconceptsappear subsequently.

(x)


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TheLearningAreaoutlines the scope ofmathematical knowledge, abilitiesand

attitudes tobe developedinpupilswhen learning the subject.Theyaredeveloped according tothe appropriate learningobjectives and representedin fivecolumns, as follows:

Column1:Learning Objectives

Column2:SuggestedTeachingandLearning

Activities

Column 3:LearningOutcomes

Column4:Points ToNote;and

Column5:Vocabulary.

TheLearning Objectivesdefine clearlywhat shouldbetaught. They

coverall aspectsof


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the Mathematicscurriculumprogramme and are presented inadevelopmental sequencedesignedto

supportpupilsunderstanding of theconcepts and skill ofmathematics.

TheSuggested Teaching and LearningActivitieslists someexamples ofteaching

andlearningactivitiesincludingmethods, techniques, strategiesandresourcespertainingtothespecificconceptsor

skills. These are,however,not the onlyintended approachestobeusedinthe classrooms. Teachersareencouraged tolook forotherexamples, determineteaching and learningstrategiesmost suitablefor theirstudents andprovide appropriate teachingand learningmaterials. Teachersshouldalsomake cross-references

tootherresources such


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as the textbooksand theInternet.

TheLearning Outcomesdefine

specificallywhat pupilsshouldbe abletodo.Theyprescribethe knowledge, skills ormathematical processes andvalues that should be inculcated anddeveloped at

theappropriate level.These behaviouralobjectives aremeasurableinall aspects.

In thePointsToNotecolumn,

attentionisdrawntothe moresignificantaspects ofmathematicalconceptsandskills. These emphases are to betakenintoaccountsoas toensurethatthe conceptsand skills are taughtand learnteffectivelyas intended.

The

Vocabularyconsists ofstandard mathematical terms,


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instructionalwords orphrases whichare relevantin structuringactivities,in

askingquestionsorsetting tasks.It isimportanttopaycarefulattentionto the use ofcorrect terminology

andtheseneedtobesystematicallyintroducedtopupilsinvariouscontexts so as toenable them

tounderstandtheir meaningandlearn touse themappropriately.

EMPHASESINTEACHING ANDLEARNING

This MathematicsCurriculumisarrangedin such a waysoastogiveflexibilityto teachers toimplement an enjoyable, meaningful,

useful andchallenging teachingand learning


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environment. At thesametime, it isimportant toensurethatpupils

showprogression inacquiring themathematical concepts and skills.

Indeterminingthe change toanotherlearning areaortopic, the

followinghaveto betakenintoconsideration:

The skillsor conceptstobeacquired in

the learning areaorincertaintopics;Ensuring the hierarchyorrelationshipbetween learningareas ortopics hasbeenfollowed accordingly;andEnsuring the basiclearningareashavebeenacquiredfullybeforeprogressing

to moreabstractareas.


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The teachingand learningprocessesemphasise concept building and skillacquisition aswellas the inculcation

of goodand positivevalues.Besidesthese, thereareotherelementsthathavetobe taken into account

and infusedin theteaching and learningprocesses inthe classroom.Themainelementsfocusedinthe teachingandlearning

ofmathematics areasfollows:

(xi)


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1.Problem Solving in MathematicsProblemsolvingis themain

focus intheteaching andlearningofmathematics.Therefore the teaching and learningprocessmust includeproblemsolvingskills which are comprehensive and

coverthewholecurriculum.The development ofproblemsolvingskillsneed tobeemphasisedso that pupilsare

abletosolvevarious problemseffectively.Theskills involvedare:

Understanding theproblem;Devisinga plan;Carryingoutthe plan; andLookingbackat the solutions.Variousstrategies

andstepsare


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usedto solveproblems andthese areexpanded soasto

be applicablein other learningareas.Throughtheseactivities,pupilscan apply their conceptualunderstandingofmathematicsand

be confidentwhen facingneworcomplex situations. Amongtheproblem solvingstrategies that couldbe introduced are:

Tryinga

simplecase;Trial andimprovement;Drawing diagrams;Identifyingpatterns;Making atable,chartorsystematiclist;Simulation;Using analogies;Working backwards;Logical reasoning;

andUsing algebra.


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2.Communicationin MathematicsCommunicationis anessentialmeans

ofsharing ideas andclarifyingtheunderstanding ofMathematics. Through communication,mathematical ideasbecometheobjectofreflection, discussion

andmodification. Theprocessofanalyticalandsystematicreasoning helps pupils toreinforceand strengthentheirknowledge and understandingof

mathematics toa deeperlevel.Through effectivecommunication, pupilswill becomeefficient inproblemsolvingand beabletoexplaintheir conceptual understanding andmathematical skills to theirpeers and teachers.

Pupils whohavedeveloped theskillstocommunicatemathematicallywill

becomemoreinquisitive


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and,intheprocess,gainconfidence.Communication

skills inmathematicsincludereading andunderstandingproblems, interpretingdiagramsandgraphs, usingcorrect andconcisemathematical terms

during oralpresentationsandinwriting. The skillsshouldbe expandedtoinclude listening.

Communication in mathematics throughthe listeningprocess occurs when

individualsrespond towhattheyhear and this encouragesindividualstothinkusingtheir mathematicalknowledgein makingdecisions.

Communicationinmathematicsthroughthereading processtakesplacewhenanindividual collects

informationanddata


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and rearrangestherelationship betweenideas and concepts.

Communicationin

mathematicsthrough thevisualisation process takesplacewhenanindividual makes an observation, analyses,interpretsandsynthesisesdata andpresents them

intheformofgeometric board,picturesanddiagrams, tables andgraphs. Aneffectivecommunication environmentcanbe

createdbytakingintoconsideration the followingmethods:

Identifyingrelevant contexts associated withenvironmentandeveryday lifeexperienceofpupils;Identifyingpupilsinterests;(xii)


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Identifyingsuitableteaching materials;Ensuring

activelearning;Stimulating meta-cognitive skills;..........Inculcating positive attitudes;and

........Setting upconducivelearningenvironment.

Effective communication canbedevelopedthroughthefollowing methods:

Oralcommunication isaninteractiveprocess thatinvolvespsychomotoractivities like listening, touching,observing,tasting andsmelling.It isa two-wayinteraction that takesplace between teacherand pupils,pupilsandpupils,andpupils and object.

Some

ofthemore


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effective andmeaningfuloral communication techniquesin the learningofmathematics areas

follows:

Story-telling andquestionand answer sessionsusingones ownwords;Asking

andanswering questions;Structured andunstructured interviews;Discussionsduringforums, seminars, debates andbrainstormingsessions;and

Presentationoffindingsofassignments.Written communication istheprocess wherebymathematical ideas andinformation are disseminatedthrough writing. The writtenwork isusuallythe result ofdiscussion, input frompeople andbrainstormingactivitieswhenworking on assignments.Through writing,pupilswillbe encouragedto

think indepth aboutthe


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mathematicscontent andobserve the relationshipsbetween concepts. Examplesof writtencommunicationactivities that can

bedeveloped through assignmentsare:

Doing exercises;Keepingjournals;Keepingscrap

books;Keepingfolios;Keepingportfolios;Undertaking projects;andDoing writtentests.

Representationisa process of analysingamathematicalproblemandinterpreting itfromone modetoanother. Mathematical representationenables pupils tofindrelationshipsbetween mathematicalideas thatareinformal, intuitive andabstractusingeverydaylanguage. For example6xycan

be interpreted as a rectangular areawith sides 2xand 3y. This


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will makepupilsrealisethat some methods of representation aremoreeffective anduseful

if theyknow howto use theelements ofmathematical representation.

3.Reasoning in MathematicsLogical Reasoning orthinking is thebasis forunderstanding

and solvingmathematicalproblems. The developmentofmathematicalreasoningiscloselyrelatedtotheintellectualand

communicative developmentofpupils.Emphasisonlogical thinking,during mathematicalactivitiesopensuppupilsminds to acceptmathematics as apowerful toolin the worldtoday.

Pupils are encouragedto estimate,predict andmake intelligentguesses intheprocess

ofseeking solutions. Pupilsat


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all levelshavetobetrainedtoinvestigate their predictions

orguesses byusingconcrete material,calculators,computers,mathematicalrepresentationand others. Logicalreasoning hastobe

absorbedin the teaching ofmathematicssothat pupilscanrecognise, constructand evaluate predictions andmathematicalarguments.

(xiii)


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4.Mathematical ConnectionsIn themathematics curriculum,opportunities formaking

connectionsmustbe created sothat pupils can link conceptual toproceduralknowledge andrelate topicswithinmathematicsandother learning areas ingeneral.

Themathematics curriculumconsists ofseveralareassuchas arithmetic,geometry, algebra,measuresand problemsolving.Without connections

betweenthese areas,pupilswill havetolearnandmemorisetoomanyconceptsandskillsseparately. Bymakingconnections,pupilsare abletoseemathematicsasanintegratedwhole

ratherthana jumble


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ofunconnectedideas.Whenmathematical ideas and thecurriculumare connected

to reallifewithinoroutsidethe classroom,pupilswill becomemore conscious oftheimportance and significanceof

mathematics. Theywill also beable to usemathematicscontextuallyin different learningareasandinreallifesituations.

5.Applicationof TechnologyThe teachingandlearning ofmathematics shouldemploy thelatesttechnologytohelppupilsunderstand mathematicalconceptsindepth,meaningfullyand preciselyand enablethemtoexploremathematicalideas.

The use of calculators,computers,educational software, websites


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intheInternetandrelevantlearning packagescan

helpto upgrade thepedagogicalapproachandthus promotethe understandingofmathematical concepts.

The useof these teaching

resources will alsohelppupilsabsorbabstractideas, becreative, feelconfidentandbeable towork independentlyor

ingroups.Mostof theseresourcesare designedfor self-accesslearning.Through self-accesslearningpupilswill be able toaccess knowledgeorskills andinformations independentlyaccordingto their own pace. This will

serve to stimulatepupilsinterestsanddevelop asense

of responsibilitytowardstheir


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learningand understandingofmathematics.

Technologyhowever

does not replacetheneed forallpupils tolearnandmaster the basicmathematicalskills.Pupilsmust

beableto efficientlyadd,subtract,multiplyand dividewithout the use of calculators or otherelectronictools. Theuse of technologymust thereforeemphasise the

acquisitionofmathematical conceptsand knowledge rather thanmerelydoingcalculation.

APPROACHES INTEACHING AND LEARNING

The beliefonhow mathematicsislearntinfluence how mathematicalconcepts are tobetaught. Whatever beliefthe teacherssubscribe to, thefactremainsthat mathematical

concepts areabstract.The use


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ofteachingresources is therefore crucialinguiding pupils todevelopmatematical ideas.

Teachersshoulduse real or concretematerialstohelppupilsgainexperience,constructabstractideas,

make inventions,build self confidence,encourageindependenceand inculcate the spirit ofcooperation.

The teachingandlearning materials usedshouldcontainself

diagnosticelementsso thatpupilsknowhow far theyhave understood theconceptsandacquire the skills.

Inorder to assist pupils inhavingpositive attitudesand personalities,theintrinsic mathematical values ofaccuracy,confidenceand thinkingsystematicallyhave tobeinfused into

the teaching and learningprocess.Good


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moralvalues canbe cultivatedthroughsuitable contexts.Learningin

groups for example can helppupils todevelop social skills,encouragecooperation andbuildselfconfidence. The element of patriotismshouldalsobe inculcatedthrough

theteaching andlearningprocess in theclassroomusing certaintopics.

(xiv)


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Brief historical anecdotes related toaspectsofmathematicsandfamous

mathematicians associatedwiththe learningareasare alsoincorporatedintothe curriculum. It shouldbepresented atappropriate pointswhere it

providesstudents witha better understandingand appreciation of mathematics.

Variousteaching strategies andapproaches suchasdirect instruction,discoverylearning, investigation,guided

discoveryorothermethodsmustbeincorporated.Amongst the approachesthatcanbegivenconsiderationinclude thefollowing:

Pupils-centeredlearning that is interesting;Different learningabilitiesandstyles of pupils;

Usageofrelevant,


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suitableandeffective teachingmaterials; andFormative evaluationto

determine the effectiveness of teaching andlearning.Thechoice of anapproachthat issuitablewill stimulate the teaching andlearningenvironmentinsideor

outside theclassroom.Approachesthatareconsidered suitable include thefollowing:

Cooperativelearning;Contextual learning;

Masterylearning;Constructivism;Enquiry-discovery; andFuture studies.EVALUATION

Evaluationorassessmentispartofthe teachingand learningprocess toascertainthestrengthsandweaknesses of

pupils.It hasto


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beplannedandcarried outas part ofthe classroomactivities.

Different methods ofassessment can be conducted.Thesemaybein the formof assignments, oral

questioningandanswering,observationsand

interviews. Basedontheresponse, teachers can rectifypupilsmisconceptionsandweaknesses andalso improve theirownteaching skills.Teachers canthen

take subsequenteffectivemeasuresinconductingremedial andenrichmentactivities inupgradingpupils performances.

(xv)


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1LEARNING AREA:

Form 4

LEARNINGOBJECTIVESPupilswill betaughttoSUGGESTED TEACHINGANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupils

will be able to1

Understandand usetheconcept ofsignificantfigure.Discussthesignificance ofzeroin

anumber.(i)Roundoffpositive numberstoagiven number ofsignificantfigures whenthe numbersare:a)greaterthan1,b)lessthan 1.Discuss theuseofsignificant figuresin

everydaylife andother areas.


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(ii)Perform operations ofaddition,subtraction, multiplicationanddivision, involving afew

numbers and statetheanswerinspecific significant figures.(iii)Solve problemsinvolvingsignificant figures.2Understandand use

theconcept ofstandardformtosolveproblems.Useeverydaylife situationssuch asinhealth, technology,

industry,constructionandbusiness involvingnumbers instandardform.Use scientificcalculatortoexplorenumbers instandardform.(i)State positivenumbers instandardformwhenthenumbers are:a)greaterthan

orequalto 10,


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b)lessthan 1.(ii)Convert numbers instandardform

tosinglenumbers.(iii)Perform operations ofaddition,subtraction, multiplicationanddivision, involving anytwonumbers and statethe

answersinstandardform.(iv)Solve problemsinvolvingnumbers instandard form.POINTSTO NOTEVOCABULARYstandard

formsingle numberscientific notation

Roundednumbersare

onlyapproximates.Limit topositivenumbersonly.

Generally,roundingisdoneonthefinalanswer.

Another termforstandard form is


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scientific notation.

Includetwonumbersinstandard

form.

significancesignificant figurerelevantround offaccuracy

1


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2 2LEARNINGAREA:

Form 4

LEARNINGOBJECTIVESPupilswill betaughttoSUGGESTED TEACHINGANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupils

will be able toPOINTS

TO NOTEVOCABULARY1Understand the conceptof quadratic expression.Discuss the characteristicsofquadraticexpressions oftheform

02=++cbxax,wherea,bandcareconstants,a.0 andxisanunknown.(i)Identifyquadraticexpressions.(ii)

Formquadraticexpressions


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bymultiplyinganytwo linearexpressions.(iii)Form

quadraticexpressionsbasedon specific situations.2Factorise quadraticexpression.Discussthe various methodstoobtainthe desired

product.(i)Factorise quadraticexpressionsofthe formcbxax++2,b= 0

orc=0.(ii)Factorise quadraticexpressionsofthe formpx2-q,pandqareperfect squares.Begin withthe casea=1.Explore theuse ofgraphing calculator

to factorise quadraticexpressions.(iii)


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Factorise quadraticexpressionsofthe formax2+bx

+c, a,bandc not equal to zero.Includethecasewhenb= 0 and/or

c= 0.

Emphasisethatforthe termsx2andx, thecoefficients areunderstoodto

be 1.

Include everydaylifesituations.

1 isalsoaperfectsquare.

Factorisationmethodsthat canbeusedare:

cross method;inspection.quadratic expressionconstant

constant factor

unknown highest power


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expand

coefficienttermfactorisecommon factor

perfect square

cross methodinspectioncommon factorcomplete

factorisation

2


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2 2LEARNINGAREA:

Form 4

LEARNING OBJECTIVESPupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupilswill be able to

POINTS TONOTEVOCABULARY3Understandthe conceptof quadratic equation.Discuss the characteristicsofquadraticequations.(i)Identify

quadratic equationswith one unknown.quadratic equationgeneral form(ii)Writequadratic equations ingeneral form i.e. 02=++cbxax.(iii)Formquadraticequationsbasedon specific situations.Include everydaylifesituations.4Understand

and usetheconcept


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ofroots ofquadraticequationstosolveproblems.

(i)Determinewhether a givenvalueisa rootof aspecificquadratic equation.substituterootDiscuss the

numberof roots ofaquadratic equation.(ii)Determine the solutionsforquadratic equationsby:a)trial and errormethod,b)

factorisation.There are quadraticequations thatcannotbe solvedbyfactorisation.trial anderrormethodUseeverydaylife situations.(iii)Solve problemsinvolvingquadratic equations.Checktherationalityofthe solution.solution3


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3 3LEARNINGAREA:

Form

4

LEARNING OBJECTIVESPupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNING

OUTCOMESPupilswill be able to1Understandthe conceptof set.Useeverydaylife examplestointroduce theconcept of

set.(i)Sort givenobjectsintogroups.(ii)Define setsby:a)descriptions,b)using setnotation.(iii)Identifywhetheragiven objectis an elementofasetand usethe symbol

.or.


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.Discussthedifference betweentherepresentation ofelements and

thenumber ofelements inVenn diagrams.(iv)Represent setsbyusingVenndiagrams.POINTS TONOTE

Thewordsetreferstoanycollectionorgroup ofobjects.

The notation usedfor

setsisbraces, {}.The sameelements inaset neednotberepeated.

Sets areusuallydenoted bycapitalletters.

Thedefinition of setshas to beclearand

precise so thattheelements can


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beidentified.

The symbol.(epsilon) is read isan

elementof or isa memberof.

The symbol.is readis not an element oforis notamemberof.

VOCABULARYset elementdescriptionlabel set notationdenoteVenndiagram

emptyset4


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3 3LEARNINGAREA:

Form

4


OUTCOMESPupilswill be able toDiscusswhy{ 0 } and {} are notemptysets.(v)List the elementsand

statethenumberofelements ofaset.(vi)Determine whethera set is anemptyset.(vii)Determinewhether twosetsareequal.2Understandand usetheconcept ofsubset,universal

set and the complementofa


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set.Beginwitheverydaylifesituations.(i)

Determinewhether a givensetis a subsetofa specificsetanduse the symbol.or.

.(ii)RepresentsubsetusingVenndiagram.(iii)List the subsets for a specificset.Discuss the relationshipbetween setsand universal

sets.(iv)Illustratetherelationshipbetween set anduniversal setusing Venndiagram.(v)Determine the complement of agivenset.(vi)Determine the relationshipbetween set,subset,universalsetandthecomplementofaset.

POINTS TONOTEThe notation


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n(A)denotes the numberofelementsinset

A.

The symbol(phi)or{} denotes anemptyset.

An

emptysetisalso

called anull set.Anemptysetisasubset

ofanyset.

Everyset is asubsetofitself.

The symbol.denotes auniversalset.

The symbolA'denotes thecomplement of setA.


VOCABULARYequal sets


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subset universal setcomplement ofa set5


69/146

3 3LEARNINGAREA:

Form

4


OUTCOMESPupilswill be able to(iii)Statethe relationshipbetweena)AnBandA,

b)AnBandB.(iv)Determine the complement ofthe intersectionof sets;(v)Solve problemsinvolving theintersectionof sets.(vi)Determine the union of:a)twosets,b)three sets,and use the symbol..

(vii)Representthe union of sets


70/146

using Venndiagram.(viii)State the relationshipbetweena) A.

BandA,b)A.BandB.(ix)Determine the complement ofthe

unionof sets.Uerformoperations onsets:

the intersectionof sets,the unionof sets.POINTS TO

NOTEVOCABULARYintersectioncommonelementsDiscuss cases when:

AnB=,A.B.(i)Determine the intersectionof:a)twosets,b)

three sets,and use the symboln


71/146

.(ii)Representthe intersectionofsetsusing

Venndiagram.Include everydaylifesituations.


6


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3 3LEARNINGAREA:

Form

4



LEARNING OBJECTIVES

Pupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupilswill be able toPOINTS TONOTE

VOCABULARY(x)Solve problemsinvolving theunion ofsets.(xi)Determine the outcomeofcombinedoperationson sets.(xii)Solve problemsinvolvingcombinedoperationson sets.7


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4 4LEARNINGAREA:

Form 4

LEARNING OBJECTIVESPupilswill betaughtto1Understandthe conceptof statement.2Understand

the conceptof quantifiers all andsome.SUGGESTEDTEACHING ANDLEARNING ACTIVITIESIntroducethis topic using everydaylife situations.

Focus on mathematical sentences.

Discuss sentences

consistingof:

words only,numbers andwords,numbers andmathematicalsymbols.Start with everydaylife situations.

LEARNINGOUTCOMESPupilswill be able to(i)Determinewhether a givensentenceis astatement.

(ii)Determinewhether a given


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statement istrue orfalse.(iii)Construct true orfalsestatements using

givennumbers and mathematicalsymbols.(i)Constructstatements using thequantifier:a)all,b)some.

POINTS TONOTEStatements consistingof:

words only,e.g.Fiveis greaterthan

two;numbers andwords,e.g. 5isgreaterthan2;numbers andsymbols, e.g.5 > 2The following arenotstatements:

Is theplacevalueof digit

9in1928


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hundreds?4n-5m+

2sAddthetwonumbers.x+2 =8

Quantifiers suchasevery and anycan beintroducedbased oncontext.

VOCABULARYstatementtrue falsemathematical

sentencemathematicalstatementmathematicalsymbol quantifierall everyany8


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4 4LEARNINGAREA:

Form 4

LEARNING OBJECTIVESPupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupils

will be able toPOINTS TO

NOTEVOCABULARY(ii)Determine whethera statementthat containsthe quantifierall istrueorfalse.

someseveralone ofpart of(iii)Determine whethera statementcanbegeneralised tocoverallcases byusingthequantifierall.(iv)Construct a truestatementusing thequantifier allorsome

, given an object andaproperty.


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negatecontraryobjectExamples:

All squares are

four sidedfigures.Everysquareis afour sidedfigure.Anysquare is afour sided

figure.Other quantifierssuch as several,oneof andpartofcan be usedbasedoncontext.

Example:Object: Trapezium.Property: Twosides

areparallel toeach

other.Statement: Alltrapeziumshavetwoparallel sides.

Object: Even

numbers.Property: Divisibleby4.

Statement: Some


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evennumbersaredivisible by4.

9


79/146

4 4LEARNINGAREA:

Form 4

Perform operationsinvolvingthewordsnot orno, andand oronstatements.

LEARNING OBJECTIVESPupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupilswill be able toBegin

witheverydaylifesituations.(i)Change the truthvalueofagiven statementbyplacingtheword notinto theoriginalstatement.(ii)Identifytwo statements fromacompound statement thatcontains theword

and.POINTS TO


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NOTEVOCABULARYThe negation nocanbe usedwhereappropriate.

The symbol~ (tilde)

denotes negation.~pdenotes negationofp

whichmeans notpornop.

The truth table forpand~p

areasfollows:

p~pTrue FalseFalse TrueThe truthvalues for pandq are asfollows:

pq pandqTrueTrueTrueTrueFalse False

FalseTrueFalse


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FalseFalseFalse

negationnotp

nop truthtabletruth value

andcompoundstatement

10


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4 4LEARNINGAREA:

Form 4



NOTEVOCABULARY(iii)Forma compoundstatementbycombining two givenstatementsusingthe

wordand.(iv)Identifytwostatement fromacompound statement thatcontains thewordor.The truthvalues for porq are as follows:or(v)Forma compoundstatementbycombining two givenstatements

usingtheword


83/146

or.(vi)Determine the truthvalueof a

compound statement whichisthe combinationof twostatements with thewordand.(vii)Determine the truthvalueof acompound statement which

isthe combinationof twostatements with theword or.pq porqTrueTrueTrue

TrueFalseTrueFalseTrueTrueFalseFalseFalse

11


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4 4LEARNINGAREA:

Form 4


will be able to4

Understandthe conceptof implication.Start with everydaylife situations.(i)Identifythe antecedent andconsequentofan implication

ifp, thenq.(ii)Write twoimplications fromacompound statementcontaining if and onlyif.(iii)Construct mathematicalstatements inthe formofimplication:a)Ifp, thenq,b)pif and onlyifq.

(iv)Determine the converse ofa


85/146

given implication.(v)Determine whethertheconverseofan

implicationistrue orfalse.POINTS TONOTEVOCABULARYImplication ifp, then

q can be

writtenasp.q, and pifandonlyifqcan bewritten

asp.q, whichmeansp.qandq.p.

The converseofanimplicationis notnecessarilytrue.Example 1:Ifx< 3, then

x< 5 (true)Conversely:


86/146

Ifx< 5, thenx< 3 (false)

implicationantecedentconsequent

converse

12


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4 4LEARNINGAREA:

Form 4



NOTEVOCABULARY5Understandthe conceptofargument.Start with everydaylife situations.(i)Identify

thepremiseandconclusionofagivensimpleargument.(ii)Make aconclusionbasedontwogiven premisesfor:a)Argument FormI,b)Argument FormII,c)Argument Form

III.Example 2:If


88/146

PQRis a triangle, then

the sumofthe

interiorangles ofPQRis180(true)Conversely:If the sumof theinterior

angles ofPQRis180thenPQRis atriangle.

(true)Limit toarguments with

true premises.

Namesfor argumentforms,i.e.syllogism(Form I),modusponens(Form II) andmodustollens(Form III),neednot beintroduced.

argumentpremiseconclusion

13


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4 4LEARNINGAREA:

Form 4

Encourage students toproduce

(iii)Completean argument givenaarguments basedonprevious

premiseand the conclusion.

knowledge.

6Understandand usetheUse specific examples/activitiesto

(i)Determine whetheraconceptofdeduction andintroduce theconcept.

conclusionismadethrough:

inductiontosolveproblems.

a)reasoning bydeduction,b)reasoningby

induction.

LEARNING OBJECTIVES


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Pupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIES

LEARNINGOUTCOMESPupilswill be able toPOINTS TONOTESpecifythat these threeformsofarguments aredeductions based

ontwopremises only.

Argument Form I Premise 1: AllAareB.Premise 2:CisA.Conclusion:

CisB.Argument Form II:Premise 1: Ifp, thenq.Premise 2:pistrue.Conclusion:qis true. Argument Form III:Premise 1: Ifp, thenq.Premise 2: Notqis true.Conclusion: Notpis

true.

VOCABULARYreasoning


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deductioninductionpattern

14


92/146

4 4LEARNINGAREA:

Form 4


will be able to(ii)

Makea conclusionfor aspecific casebasedon agivengeneralstatement,bydeduction.

(iii)Makea generalizationbased onthe pattern ofanumericalsequence, byinduction.(iv)Use deduction andinduction inproblemsolving.POINTS TONOTEVOCABULARYLimit tocases whereformulae can beinduced.

Specifythat:

making conclusionbydeduction is


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definite;making conclusionby

induction isnotnecessarily

definite.

special

conclusiongeneralstatement

generalconclusion

specific case

numerical sequence

15


94/146

5 5LEARNINGAREA:

Form 4

Understandthe conceptofgradient ofa straightline.

LEARNING OBJECTIVESPupilswill betaught

toSUGGESTED

TEACHING ANDLEARNING ACTIVITIESUse technologysuchastheGeometers Sketchpad, graphingcalculators,graph boards,magneticboards

ortopo mapsasteachingaidswhere appropriate.

Beginwithconcreteexamples/dailysituationsto introduce theconceptofgradient.

.Verticaldistance

Horizontal distance

Discuss:

the relationshipbetween gradient


95/146

and tan.,thesteepnessof the straightline

withdifferentvalues ofgradient.Carryout activities tofindthe ratioofvertical distance tohorizontal distancefor

several pairsofpointsonastraightline toconclude that the ratioisconstant.

LEARNINGOUTCOMES

Pupilswill be able to(i)Determine thevertical andhorizontal distancesbetweentwo givenpoints on astraightline.(ii)Determine the ratioofverticaldistanceto horizontal distance.POINTS TONOTEVOCABULARYstraight line steepnesshorizontal distancevertical distancegradient ratio

16


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5 5LEARNINGAREA:

Form 4


will be able to2

Understandthe conceptofgradient ofa straightlinein Cartesian coordinates.Discuss thevalueofgradient

if:Pis chosenas (x1,y1) andQis(x2,y2),Pis chosenas (x2,y2) andQis(x1,y1).(i)Derive the formula for

thegradientof


97/146

a straightline.(ii)Calculate thegradient of astraightline passing

throughtwo points.(iii)Determine the relationshipbetween the valueofthegradientandthe:a)steepness,

b)direction ofinclinationofastraight line.3Understandthe conceptof intercept.(i)Determine thex-intercept and

they-interceptofa straightline.(ii)Derive the formula forthegradientofastraight line interms ofthex-intercept andthey-intercept.(iii)Performcalculationsinvolvinggradient,x-intercept andy-intercept.POINTS TO

NOTEVOCABULARYThe


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gradient ofastraightline passingthroughP(x1,

y1) and

Q(x2,y2) is:

y2-y1

m=

x2-x1

Emphasisethatx-interceptandy-intercept are notwritten inthe form

ofcoordinates.

acute angleobtuseangleinclinedupwards to

the right

inclineddownwardsto the rightundefined

x-intercepty-intercept

17


99/146

5 5LEARNINGAREA:

Form 4



NOTEVOCABULARY(iv)Determine the gradient andy-interceptofthe straight linewhichequationisof

theform:a)y=mx+c,b)ax+by=c.Understandand useequationof astraightline.

Discuss the changein theformof

thestraight line ifthe values


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ofmandcare changed.

Carry

out activitiesusing thegraphingcalculator, GeometersSketchpadorother teachingaids.

Verify that misthe gradient and

cisthey-interceptof a straightlinewithequationy=mx+c

.

(i)Drawthegraphgivenanequationofthe formy=mx+c.

(ii)Determinewhether a givenpointlies onaspecific straightline.

(iii)Write the equation ofthe


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straightline giventhe gradientandy-intercept.Emphasisethat the

graphobtained is astraightline.

If apointlies on astraightline,then thecoordinates of

thepoint satisfytheequationofthestraightline.

The equation

ax+

by=ccanbewritten inthe formy=mx+c.

linear equationgraph table of valuescoefficientconstantsatisfy

parallelpoint of intersectionsimultaneous

equations

18


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5 5LEARNINGAREA:

Form 4


will be able toDiscuss

andconclude thatthepointofintersectionis the onlypoint thatsatisfiesbothequations.

Use thegraphing calculatorandGeometers Sketchpad orotherteaching aidstofind the point ofintersection.(vi)Findthepoint of intersectionoftwo straight linesby:a)drawing the twostraightlines,b)solving simultaneousequations.5Understand

and usetheconcept of


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parallellines.Exploreproperties ofparallel linesusing thegraphing

calculator andGeometers Sketchpad orotherteaching aids.(i)Verifythattwo parallel lineshave the same gradientandviceversa.

parallel lines(ii)Determine fromthegivenequationswhether two straightlinesare parallel.(iii)Findthe equationof the

straightline whichpassesthrougha given pointandisparalleltoanother straightline.(iv)Solve problemsinvolvingequations ofstraightlines.POINTS TONOTEVOCABULARY19


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6 6LEARNINGAREA:

Form 4

1Understandthe conceptofclassinterval.(ii)Determine:a)theupper limit and

lowerlimit,b)the upper boundaryandlower boundaryofa classinagroupeddata.(iii)

Calculate thesizeofa classinterval.(iv)Determine the classinterval,given asetofdata and thenumberofclasses.Discusscriteria for suitableclassintervals.(v)Determine a suitable classinterval foragiven set ofdata.

(vi)Construct a frequencytable


105/146

foragivenset ofdata.2Understand

and usetheconceptofmodeandmeanofgrouped data.(i)Determine themodal

classfromthe frequencytable ofgroupeddata.(ii)Calculatethe midpoint of aclass.LEARNING OBJECTIVESPupilswill be

taughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupilswill be able toPOINTS TONOTEVOCABULARYUsedataobtained fromactivities andothersources suchasresearchstudiesto introduce theconcept of classinterval.

(i)Complete the class intervalfor


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a setofdatagivenoneofthe

class intervals.Sizeof class interval=[upper boundarylowerboundary]

Midpointofclass

=12(lower limit+upper limit)

statistics class intervaldata grouped data upper limitlower limit upper boundarylower boundarysize ofclass

interval frequencytable

modemodal class

meanmidpointof aclass

20


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6 6LEARNINGAREA:

Form 4


will be able to(iii)

Verifytheformula forthemeanofgroupeddata.(iv)Calculate themean

from thefrequencytable of groupeddata.(v)Discuss the effect of the sizeofclassinterval on the accuracyofthemean fora specificset ofgroupeddata.3Represent and interpretdata inhistogramswith classintervalsof the samesizeto

solveproblems.Discuss


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the difference betweenhistogramand bar chart.(i)Draw ahistogrambased on

thefrequencytable of agroupeddata.Usegraphingcalculatortoexploretheeffect

of different class intervalonhistogram.(ii)Interpret informationfrom agiven histogram.(iii)Solve problemsinvolvinghistograms.4Represent and interpret

data infrequencypolygonstosolveproblems.(i)Drawthe frequencypolygonbased on:a)a histogram,b)a frequencytable.(ii)Interpret informationfrom agivenfrequencypolygon.(iii)Solve problemsinvolving

frequencypolygon.POINTS TO


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NOTEVOCABULARYInclude everydaylife

situations.When

drawingafrequencypolygonadd aclass with 0frequencybefore thefirst classandafterthe last class.


uniformclassinterval histogram

vertical axishorizontal axis

frequency

polygon

21


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6 6LEARNINGAREA:

Form 4


will be able to5

Understandthe conceptof cumulative frequency.(i)Construct the cumulativefrequencytablefor:a)ungrouped data,b)

groupeddata.(ii)Drawtheogivefor:a)ungrouped data,b)groupeddata.6Understandand usetheconceptofmeasuresofdispersiontosolveproblems.Discuss

the meaning ofdispersionby


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comparinga few setsofdata.Graphingcalculator canbe

used forthis purpose.(i)Determine the rangeofa setofdata.(ii)Determine:a)the median,

b)the first quartile,c)the thirdquartile,d)the interquartilerange,fromtheogive.(iii)Interpret information

from anogive.POINTS TONOTEVOCABULARYWhen drawingogive:

usethe upperboundaries;adda class withzero frequencybefore the firstclass.Forgroupeddata:Range=[midpointof

the last classmidpoint


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ofthe firstclass]

cumulativefrequency ungrouped data ogive

range

measuresofdispersionmedian first quartile third quartile interquartile range

22


113/146

6 6LEARNINGAREA:

Form 4

Carryoutaproject/researchandanalyse as well as interpretthe data.Present the findingsof theproject/research.

Emphasisetheimportanceofhonestyand accuracyinmanagingstatisticalresearch.

LEARNING OBJECTIVES

Pupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupilswill be able toPOINTS TONOTEVOCABULARY23


114/146

7 7LEARNINGAREA:

Form 4

(ii)List allthepossibleoutcomesofanexperiment:a)fromactivities,

b)byreasoning.(iii)Determine the samplespace ofan experiment.(iv)Write the sample space byusingsetnotations.(i)

Identifytheelementsofasample spacewhich satisfygivenconditions.(ii)Listall the elements of asample spacewhich satisfycertainconditionsusingsetnotations.(iii)Determinewhether anevent ispossiblefor

a samplespace.(i)


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Findthe ratioofthenumber oftimesan

event occursto thenumber of trials.(ii)Findtheprobabilityof an eventfroma bigenoughnumber of

trials.LEARNING OBJECTIVESPupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupilswill be able to

POINTS TONOTEVOCABULARY1Understandthe conceptof samplespace.

2Understandthe conceptofevents.

3Understandanduse theconceptofprobabilityofanevent to

solve problems.

Use concrete


116/146

examplessuchasthrowingadieand tossing

a coin.

Discuss that an eventis asubsetofthe

sample space.Discuss also

impossibleeventsforasample space.

Discuss thatthesample spaceitself is

an event.Carryout activities tointroducetheconceptofprobability. The graphingcalculatorcan be usedtosimulate suchactivities.

(i)Determinewhether anoutcomeis a possibleoutcomeof anexperiment.An impossibleevent

isanempty


117/146

set.

Probabilityisobtainedfromactivities

andappropriate data.

sample spaceoutcome

experimentpossible outcome

eventelement subset emptyset

impossible event

probability

24


118/146

7 7LEARNINGAREA:

Form 4


will be able toDiscuss situation

whichresults in:

probabilityofevent =1.probabilityof

event =0.Emphasise that thevalueof

probabilityisbetween 0 and1.Predict possible eventswhich mightoccurin dailysituations.

(iii)Calculatethe expected numberoftimes an eventwill occur,giventhe probabilityof

theeventand number of


119/146

trials.(iv)Solve problemsinvolvingprobability.(v)Predict the

occurrenceofanoutcome andmakeadecisionbased onknown information.POINTS TONOTEVOCABULARY

25


120/146

8 8LEARNINGAREA:

Form 4


will be able to1

Understandanduse theconceptoftangents to acircle.(i)Identifytangentsto a

circle.Develop conceptsand abilitiesthroughactivities using technologysuchas theGeometers Sketchpad andgraphingcalculator.(ii)Make inference that the tangenttoa circleis astraightlineperpendicular to theradiusthatpasses through thecontactpoint.(iii)Construct the tangent to

acircle passing througha point:


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a)on the circumferenceofthecircle,b)outside the

circle.(iv)Determine the propertiesrelatedtotwo tangentstoacircle fromagivenpoint

outside the circle.POINTS TONOTEVOCABULARYcongruentCProperties of angleinsemicircles canbeused. Examplesofproperties of

twotangents toa circle:

A

BAC=BC.ACO=.BCO

O.AOC=.BOC.AOCand

.BOCarecongruent.


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tangent to a circlecircle

perpendicularradiuscircumference

semicircle

26


123/146

8 8LEARNINGAREA:

Form 4



NOTEVOCABULARY(v)Solve problemsinvolvingtangents toa circle.Relate toPythagorasTheorem2

Understandanduse thepropertiesof angle betweentangent and chordtosolveproblems.Explore thepropertyofangleinalternate segment using GeometersSketchpadorotherteaching aids.chords alternate segmentmajorsectorsubtended(ii)Verify

therelationship betweenthe angle


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formedbythetangent and the chordwith theangle in the alternatesegment

whichissubtendedbythechord..ABE=.BDE.

CBD=.BED(iii)Performcalculationsinvolvingtheangle inalternatesegment.(iv)

Solve problemsinvolvingtangent to acircleand angle inalternate segment.3Understandanduse theproperties ofcommontangents tosolve problems.Discussthemaximum numberofcommontangents for the three cases.(i)Determine the numberofcommon tangentswhich can be

drawn totwocircles


125/146

which:a)intersect attwopoints,b)intersect only

atone point,c)donot intersect.Emphasisethat thelengths ofcommon tangents are equal.common tangentsED

ABC27


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8 8LEARNINGAREA:

Form

4

Include dailysituations.

(ii)Determine the propertiesrelatedtothe common

tangenttotwocircles which:a)intersect attwopoints,b)intersect onlyatone point,c)

donot intersect.

(iii)Solve problemsinvolvingcommon tangents totwocircles.(iv)Solve problemsinvolvingtangents andcommon tangents.Include problemsinvolvingPythagorasTheorem.

LEARNING OBJECTIVESPupilswill betaughttoSUGGESTED

TEACHING ANDLEARNING ACTIVITIESLEARNING


127/146

OUTCOMESPupilswill be able toPOINTS TONOTEVOCABULARY28


128/146

9 9LEARNINGAREA:

Form 4

(ii)Determine:a)the valueofy-coordinate,b)the valueofx-coordinate,c)

the ratioofy-coordinatetox-coordinateofseveralpoints onthecircumference of the unitcircle.Beginwith

definitionsofsine,cosineandtangent ofanacute angle. yyOP PQ===1sin.xxOP OQ===1cos.x yOQ PQ==.tan(iii)

Verifythat,for an


129/146

angle inquadrantI oftheunit circle:a)sin

.=y-coordinate,b)cos.=x-coordinate,c)tan.=.(iv)Determine the values of:

a)sine,b)cosine,c)tangentofanangleinquadrantIof

theunit circle.0 yxP(x,y)y1xQy-coordinatex-coordinateLEARNING OBJECTIVESPupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESLEARNINGOUTCOMESPupilswill be able to

Understandand usethe


130/146

Explainthe meaning ofunit circle.

(i)Identifythe quadrants

andThe unitcircleistheconceptofthe values of

angles inthe unitcircle.

circleofradius1 with

itscentre at the

sin., cos.andtan

.(0=.

origin.

=360) to solveproblems.

POINTS TONOTEVOCABULARYquadrant

sine.cosine.tangent.

29


131/146


132/146

9 9LEARNINGAREA:

Form 4


will be able toExplain that the concept

sin.=y-coordinate ,cos.=x-coordinate,tan.=canbe

extendedtoanglesinquadrantII,III and IV.(vi)Determinewhether the valuesof:a)sine,b)cosine,c)tangent,ofan angle inaspecificquadrant ispositive ornegative.1

v245o


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160o 30o12v3y-coordinate

x-coordinateUsethe above triangles tofind thevaluesofsine,cosineand tangentfor30, 45

, 60.

Teaching can beexpandedthroughactivities such as reflection.

(v)Determine the values of:a)sin.,

b)cos.,c)tan.,for 90=.=360.(vii)Determine the values of sine,cosineand tangentfor specialangles.(viii)Determine the values of theangles inquadrantI whichcorrespond tothe values ofthe

anglesinother quadrants.


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POINTS TONOTEConsider specialangles such as0,30

, 45, 60, 90,180, 270, 360.

VOCABULARY30


135/146

9 9LEARNINGAREA:

Form 4


will be able toUse

the Geometers Sketchpadtoexplorethechange in thevalues ofsine,cosine andtangentrelative tothe

changeinangles.(ix)Statethe relationships betweenthe valuesof:a)sine,b)cosine, andc)tangentofanglesinquadrantII,III andIV with theirrespective valuesofthe correspondingangle

inquadrant I.(x)


136/146

Findthevalues of sine, cosineandtangentof the anglesbetween

90and 360.(xi)Findthe angles between0and360,giventhe valuesof

sine,cosine or tangent.Relate todailysituations.(xii)Solve problemsinvolving sine,cosine andtangent.POINTS TONOTEVOCABULARY

31


137/146

9 9LEARNINGAREA:

Form 4


will be able to2

Drawand use the graphsUse thegraphing calculatorandof sine, cosine andtangent.Geometers Sketchpad to explore thefeature ofthe graphsof

y= sin.,y=cos.,y=tan..

Discuss the featureof thegraphsofy= sin.,y=cos.,y=

tan..


138/146

Discuss the examples ofthesegraphsin other areas.

(i)Draw the graphs of

sine,cosineand tangent for angles between0and 360.(ii)Comparethe graphsofsine,cosine

and tangentfor anglesbetween0and360.(iii)Solve problemsinvolvinggraphs of sine,cosineand

tangent.POINTS TONOTEVOCABULARY32


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10 10LEARNINGAREA:

Form 4


will be able to(ii)

Represent a particularsituationinvolving:a)theangle ofelevation,b)the angleofdepression

using diagrams.(iii)Solve problemsinvolving theangleofelevation and theangle ofdepression.Understandand usetheconceptofangleofelevationandangleofdepressiontosolveproblems.

Usedailysituations to introduce the


140/146

concept.

(i)Identify:a)the horizontalline,

b)theangle ofelevation,c)the angleofdepressionfor a particularsituation.POINTS TONOTE

VOCABULARYangleofelevationangle of depressionhorizontal line

Include twoobservationsonthesamehorizontal

plane.

Involveactivitiesoutside theclassroom.

33


141/146

11 11LEARNINGAREA:

Form

4


OUTCOMESPupilswill be able to(iii)Sketchathreedimensionalshapeandidentifythespecific

planes.(iv)Identify:a)lines that lieona plane,b)lines that intersect withaplane.(v)Identifynormalstoa givenplane.Begin with3-dimensionalmodels.(vi)Determine the orthogonalprojectionof

alineon


142/146

aplane.(vii)Draw andnamethe orthogonalprojection

ofalineonaplane.(viii)Determine the anglebetween aline anda plane.Understand and use the

Carryout activitiesusingdaily

(i)Identifyplanes.horizontal planeconcept ofanglebetweensituations

and 3-dimensional models.

vertical planelines and planes to solve3-dimensional

problems.

normal toaplaneDifferentiate between 2-dimensional

(ii)Identifyhorizontalplanes,orthogonaland 3-dimensional shapes.Involveverticalplanes and inclined

projectionplanesfound in


143/146

natural surroundings.

planes.

space diagonal

Include lines

in3-dimensionalshapes.

Use3-dimensional models togive

(ix)Solve problems

involving theclearerpictures.anglebetweena line and aplane.

POINTS TONOTEVOCABULARY34


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11 11LEARNINGAREA:

Form 4

LEARNING OBJECTIVESPupilswill betaughttoSUGGESTEDTEACHING ANDLEARNING ACTIVITIESUnderstandand use the

concept ofanglebetweentwoplanes tosolveproblems.

Use3-dimensional models togiveclearerpictures.

LEARNING OUTCOMESPupilswill be able to(i)Identifythe line ofintersectionangle between twobetween two planes.planes

(ii)Drawa line oneach planewhichisperpendiculartotheline ofintersectionofthe two

planesat a point on the lineof


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intersection.(iii)Determine the anglebetweentwoplanes ona model and a

given diagram.(iv)Solve problemsinvolving linesand planesin3-dimensionalshapes.POINTS TONOTEVOCABULARY

35


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CS Maths F4