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COUNTDOWNSecond Edition

Teaching Guide

1

viii1

Introduction iv

Curriculum 1

• StrandsandBenchmarks

• SyllabusMatchingGrid

Teaching and Learning 13

• GuidingPrinciples

• MathematicalPractices

• LessonPlanning

• FeaturesoftheTeachingGuide

• SampleLessonPlans

Assessment 107

• SpecimenPaper

• MarkingScheme

Contents

iviv 1

Introduction

Welcome,usersoftheCountdownseries.CountdownhasbeenthechoiceofMathematicsteachersformanyyears.ThisTeachingGuidehasbeenspeciallydesignedtohelpthemteachmathematicsinthebestpossiblemanner.Itwillserveasareferencebooktostreamlinetheteachingandlearningexperienceintheclassroom.

Teachersareentrustedwiththetaskofprovidingsupportandmotivationtotheirstudents,especiallythosewhoareatthelowerendofthespectrumofabilities. Infact,theirsuccessisdeterminedbythelevelofunderstandingdemonstratedbytheleastablestudents.

Teachersregulatetheireffortsanddevelopateachingplanthatcorrespondstothepreviousknowledgeofthestudentsanddifficultyofthesubjectmatter.Themorewell-thoughtoutandcomprehensiveateachingplanis,themoreeffectiveitis.Thisteachingguidewillhelpteachersstreamlinethedevelopmentofalessonplanforeachtopicandguidetheteacheronthelevelofcomplexityandamountofpracticerequiredforeachtopic.Italsohelpstheteacherintroduceeffectivelearningtoolstothestudentstocompletetheirlearningprocess.

ShaziaAsad

v11

Strands and Benchmarks(National Curriculum for Mathematics 2006)

TheNationalCurriculumforMathematics2006isbasedonthesefivestrands:

Numbersand

Operations

Reasoningand

Logical Thinking

Information Handling

Algebra

Geometry and

Measurement

STRANDS

OF

MATHEMATICS

Curriculum

iv2 1Curriculum

Towards greater focus and coherence of a mathematical programme

Acomprehensiveandcoherentmathematicalprogrammeneedstoallocateproportionaltimetoallstrands.Acompositestrandcoversnumber,measurementandgeometry,algebra,andinformationhandling.

Eachstrandrequiresafocussedapproachtoavoidthepitfallofabroadgeneralapproach.If,say,analgebraicstrandisapproached,coherenceandintertwiningofconceptswithinthestrandatallgradelevelsisimperative.Theaimsandobjectivesofthegradesbelowandaboveshouldbekeptinmind.

“Whatandhowstudentsaretaughtshouldreflectnotonlythetopicsthatfallwithinacertainacademicdiscipline,butalsothekeyideasthatdeterminehowknowledgeisorganisedandgeneratedwithinthatdiscipline.”

WilliamSchmidtandRichardHouang(2002)

Strands and Benchmarks of the Pakistan National Curriculum (2006)

Strand 1: Numbers and Operations

Thestudentswillbeableto:• identifynumbers,waysofrepresentingnumbers,andeffectsofoperationsinvarioussitua-

tions;

• computefluentlywithfractions,decimals,andpercentages,and

• manipulatedifferenttypesofsequencesandapplyoperationsonmatrices.

Benchmarks

Grades VI, VII, VIII

• Identifydifferenttypesofsetswithnotations

• Verifycommutative,associative,distributive,andDeMorgan’slawswithrespecttounionandintersectionofsetsandillustratethemthroughVenndiagrams

• Identifyandcompareintegers,rational,andirrationalnumbers

• Applybasicoperationsonintegersandrationalnumbersandverifycommutative,associative,anddistributiveproperties

• Arrangeabsolutevaluesofintegersinascendinganddescendingorder

• FindHCFandLCMoftwoormorenumbersusingdivisionandprimefactorisation

• Convertnumbersfromdecimalsystemtonumberswithbases2,5,and8,andviceversa

• Add,subtract,andmultiplynumberswithbases2,5,and8

• Applythelawsofexponentstoevaluateexpressions

• Findsquareandsquareroot,cube,andcuberootofarealnumber

• Solveproblemsonratio,proportion,profit,loss,mark-up,leasing,zakat,ushr,taxes,insurance,andmoneyexchange

v31 Curriculum

Strand 2: Algebra

Thestudentswillbeableto:

• analysenumberpatternsandinterpretmathematicalsituationsbymanipulatingalgebraicexpressionsandrelations;

• modelandsolvecontextualisedproblems;and

• interpretfunctions,calculaterateofchangeoffunctions,integrateanalyticallyandnumerically,determineorthogonaltrajectoriesofafamilyofcurves,andsolvenon-linearequationsnumerically.

Benchmarks


• Identifyalgebraicexpressionsandbasicalgebraicformulae

• Applythefourbasicoperationsonpolynomials

• Manipulatealgebraicexpressionsusingformulae

• Formulatelinearequationsinoneandtwovariables

• Solvesimultaneouslinearequationsusingdifferenttechniques

Strand 3: Measurement and Geometry


• identifymeasurableattributesofobjects,andconstructanglesandtwodimensionalfigures;

• analysecharacteristicsandpropertiesandgeometricshapesanddevelopargumentsabouttheirgeometricrelationships;and

• recognisetrigonometricidentities,analyseconicsections,drawandinterpretgraphsoffunctions.

Benchmarks


• Drawandsubdividealinesegmentandanangle

• Constructatriangle(givenSSS,SAS,ASA,RHS),parallelogram,andsegmentsofacircle

• Applypropertiesoflines,angles,andtrianglestodevelopargumentsabouttheirgeometricrelationships

• Applyappropriateformulastocalculateperimeterandareaofquadrilateral,triangular,andcircularregions

• Determinesurfaceareaandvolumeofacube,cuboid,sphere,cylinder,andcone

• Findtrigonometricratiosofacuteanglesandusethemtosolveproblemsbasedon right-angledtriangles

iv4 1

Strand 4: Handling Information

Thestudentswillbeabletocollect,organise,analyse,display,andinterpretdata.

Benchmarks


• Read,display,andinterpretbarandpiegraphs

• Collectandorganisedata,constructfrequencytablesandhistogramstodisplaydata

• Findmeasuresofcentraltendency(mean,medianandmode)

Strand 5: Reasoning and Logical Thinking


• usepatterns,knownfacts,properties,andrelationshipstoanalysemathematicalsituations;

• examinereal-lifesituationsbyidentifyingmathematicallyvalidargumentsanddrawingconclusionstoenhancetheirmathematicalthinking.

Benchmarks


• Finddifferentwaysofapproachingaproblemtodeveloplogicalthinkingandexplaintheirreasoning

• Solveproblemsusingmathematicalrelationshipsandpresentresultsinanorganisedway

• Constructandcommunicateconvincingargumentsforgeometricsituations

Curriculum

v51

Syllabus Matching Grid

Unit 1: Operations on Sets

1.1 Setsi) Recognisesetof

•naturalnumbers(N),

•wholenumbers(W),

•integers(Z),

•rationalnumbers(Q),

•evennumbers(E),

•oddnumbers(0),

•primenumbers(P).

ii) Findasubsetofaset.

iii) Defineproper(⊂)andimproper(⊂)subsetsofaset.

iv) FindpowersetP(A)ofasetA. Chapter1

1.2 Operations on Sets

i) Verifycommutativeandassociativelawswithrespecttounionandintersection.

ii) Verifythedistributivelaws.

iii) StateandverifyDeMorgan’slaws.

1.3 Venn Diagram

i) DemonstrateunionandintersectionofthreeoverlappingsetsthroughVenndiagram.

ii) VerifyassociativeanddistributivelawsthroughVenndiagram.

Unit 2: Real Numbers

2.1 Irrational Numbers

i) Defineanirrationalnumber.

ii) Recogniserationalandirrationalnumbers.

iii) Definerealnumbers.

iv) Demonstratenon-terminating/non-repeating(ornon-periodic)decimals.Chapter2

2.2 Squares

i) Findperfectsquareofanumber.

ii) Establishpatternsforthesquaresofnaturalnumbers.

Curriculum

iv6 1

2.3 Square Roots

i) Findsquarerootof

• anaturalnumber,

• acommonfraction,

• adecimal, giveninperfectsquareform,byprimefactorisationanddivisionmethod.

ii) Findsquarerootofanumberwhichisnotaperfectsquare.

iii) Usethefollowingruletodeterminethenumberofdigitsinthesquarerootofaperfectsquare.Rule:Letnbethenumberofdigitsintheperfectsquarethenitssquarerootcontains

n2digitsifniseven,n+1

2digitsifnisodd.

iv) Solvereal-lifeproblemsinvolvingsquareroots.

Chapter2

2.4 Cubes and Cube Roots

i) Recognisecubesandperfectcubes.

ii) Findcuberootsofanumberwhichareperfectcubes.

iii) Recognisepropertiesofcubesofnumbers.

Chapter3

Unit 3: Numbers Systems

3.1 Number System

i) Recognisebaseofanumbersystem.

ii) Definenumbersystemwithbase2,5,8and10.

iii) Explain

• binarynumbersystem(systemwithbase2).

• numbersystemwithbase5,

• octalnumbersystem(systemwithbase8),

• decimalnumbersystem(systemwithbase10).Chapter4

3.2 Conversions

i) Convertanumberfromdecimalsystemtoasystemwithbases2,5and8,andviceversa.

ii) Add,subtractandmultiplynumberswithbases2,5and8.

iii) Add,subtractandmultiplynumberswithdifferentbases.

Unit 4 Financial Arithmetic

4.1 Compound Proportion

i) Definecompoundproportion.

ii) Solvereal-lifeproblemsinvolvingcompoundproportion,partnershipandinheritance.

Chapter8

Curriculum

v71

4.2 Banking

4.2.1 Types of a Bank Account

i) Definecommercialbankdeposits,typesofabankaccount(PLSsavingsbankaccount,currentdepositaccount,PLStermdepositaccountandforeigncurrencyaccount).

ii) Describenegotiableinstrumentslikecheque,demanddraftandpayorder.

4.2.2 On-line banking

iii) Explainon-linebanking,transactionsthroughATM(AutoTellerMachine),debitcardandcreditcard(VisaandMaster).

4.2.3 Conversion of Currencies

iv) ConvertPakistanicurrencytowell-knowninternationalcurrencies.

4.2.4 Profit/Markup

v) Calculate

• theprofit/markup,

• theprincipalamount,

• theprofit/markuprate,

• theperiod.

4.2.5 Types of Finance

vi) Explain

• Overdraft(OD),

• RunningFinance(RF),

• DemandFinance(DF),

• Leasing.

vii) Solvereal-lifeproblemsrelatedtobankingandfinance.

Chapter8

4.3 Percentage

4.3.1 Profit and Loss

i) Findpercentageprofitandpercentageloss.

4.3.2 Discount

ii) Findpercentagediscount.

iii) Solveproblemsinvolvingsuccessivetransactions.

Chapter7

4.4 Insurance

i) Defineinsurance.

ii) Solvereal-lifeproblemsregardinglifeandvehicleinsurance.

Curriculum

iv8 1

Unit 5 Polynomials

5.1 Algebraic Expression

i)Recallconstant,variable,literalandalgebraicexpression.

Chapter12

5.2 Polynomial

i) Define

• polynomial,

• degreeofapolynomial,

• coefficientsofapolynomial.

ii) Recognisepolynomialinone,twoandmorevariables.

iii) Recognisepolynomialsofvariousdegrees(e.g.,linear,quadratic,cubicandbiquadraticpolynomials).

5.3 Operations on Polynomials

i) Add,subtractandmultiplypolynomials.

ii) Divideapolynomialbyalinearpolynomial.

Unit 6 Factorisation, Simultaneous Equations

6.1 Basic Algebraic Formulas

Recalltheformulas:

• (a + b)2 = a2+2ab + b2,

• (a – b)2 = a2– 2ab + b2,

• a2 – b2=(a – b)(a + b), andapplythemtosolveproblems.

Chapter13,14,15,16

6.2 Factorisation

Factoriseexpressionsofthefollowingtypes:

• ka + kb + kc,

• ac + ad + bc + bd,

• a2±2ab + b2,

• a2 – b2,

• a2±2ab + b2–c2

6.3 Manipulation of Algebraic

Recognisetheformulas:

• (a + b)3 = a3+3a2b +3ab2 + b3,

• (a – b)3 = a3–3a2b +3ab2 – b3, andapplythemtosolvetheproblems.

Curriculum

v91

6.4 Simultaneous Linear Equations

i) Recognisesimultaneouslinearequationsinoneandtwovariables.

ii) Givetheconceptofformationoflinearequationintwovariables.

iii) Knowthat:

• asinglelinearequationintwounknownsissatisfiedbyasmanypairofvaluesasrequired.

• twolinearequationsintwounknownshaveonlyonesolution(i.e.,onepairofvalues).

Chapter17

6.5 Solution of Simultaneous Equations

i) Solvesimultaneouslinearequationsusinglinearequations

• methodofequatingthecoefficients,

• methodofeliminationbysubstitution,

• methodofcrossmultiplication.

ii) Solvereal-lifeproblemsinvolvingtwosimultaneouslinearequationsintwovariables.

6.6 Elimination

i) Eliminateavariablefromtwoequationsby:

• Substitution,

• applicationofformulae.

Unit 7 Fundamentals of Geometry

7.1 Parallel Lines

i) Defineparallellines.

ii) Demonstratethroughfiguresthefollowingpropertiesofparallellines.

• Twolineswhichareparalleltothesamegivenlineareparalleltoeachother.

• Ifthreeparallellinesareintersectedbytwotransversalsinsuchawaythatthetwointerceptsononetransversalareequaltoeachother,thetwointerceptsonthesecondtransversalarealsoequal.

• Alinethroughthemidpointofthesideofatriangleparalleltoanothersidebisectsthethirdside(anapplicationofaboveproperty).

iii) Drawatransversaltointersecttwoparallellinesanddemonstratecorrespondingangles,alternateinteriorangles,verticallyoppositeanglesandinterioranglesonthesamesideoftransversal.

iv) Describethefollowingrelationsbetweenthepairsofangleswhenatransversalintersectstwoparallellines.

• Pairsofcorrespondinganglesareequal.

• Pairsofalternateinterioranglesareequal.

• Pairofinterioranglesonthesamesideoftransversalissupplementary,anddemonstratethemthroughfigures.

*CoveredinBook7

Chapter12

Curriculum

iv10 1

7.2 Polygons

i) Defineapolygon.

ii) Demonstratethefollowingpropertiesofaparallelogram.

• Oppositesidesofaparallelogramareequal.

• Oppositeanglesofaparallelogramareequal.

• Diagonalsofaparallelogrambisecteachother.

iii) Defineregularpentagon,hexagonandoctagon.

Chapter19

7.3 Circle

i) Demonstrateapointlyingintheinteriorandexteriorofacircle.

ii) Describetheterms:sector,secantandchordofacircle,concyclicpoints,tangenttoacircleandconcentriccircles.

*CoveredinBook7

Chapter14

Unit 8 Practical Geometry

8.1 Construction of Quadrilaterals

i) Defineanddepicttwoconverging(non-parallel)linesandfindtheanglebetweenthemwithoutproducingthelines.

ii) Bisecttheanglebetweenthetwoconverginglineswithoutproducingthem.

iii) Constructasquare

• whenitsdiagonalisgiven, • whenthedifferencebetweenitsdiagonalandsideisgiven, • whenthesumofitsdiagonalandsideisgiven,iv) Constructarectangle • whentwosidesaregiven, • whenthediagonalandasidearegiven,v) Constructarhombus • whenonesideandthebaseanglearegiven, • whenonesideandadiagonalaregiven,vi) Constructaparallelogram • whentwodiagonalsandtheanglebetweenthemisgiven,vii) Constructakite • whentwounequalsidesandadiagonalaregiven.viii)Constructaregularpentagon • whenasideisgiven.ix) Constructaregularhexagon • whenasideisgiven

Chapter19

8.2 Construction of a Right-angled Triangle

Constructaright-angledtriangle

• whenhypotenuseandonesidearegiven,

• whenhypotenuseandtheverticalheightfromitsvertextothehypotenusearegiven.

Curriculum

v111

Unit 9 Areas and Volumes

9.1 Pythagoras Theorem

i) StatethePythagorastheoremandgiveitsinformalproof.

ii) Solveright-angledtrianglesusingPythagorastheorem.

Chapter22

9.2 Hero’s Formula

StateandapplyHero’sformulatofindtheareasoftriangularandquadrilateralregions.

Chapter219.3 Surface Area and Volume

i) Findthesurfaceareaandvolumeofasphere.

ii) Findthesurfaceareaandvolumeofacone.

iii) Solvereal-lifeproblemsinvolvingsurfaceareaandvolumeofsphereandcone.

Unit 10 Demonstrative Geometry

10.1 Demonstrative geometry

i) Definedemonstrativegeometry.

10.1.1 Reasoning

i) Describethebasicsofreasoning.

10.1.2 Axioms, Postulates and Theorem

i) Describethetypesofassumptions(axiomsandpostulates).

ii) Describepartsofaproposition.

iii) Describethemeaningsofageometricaltheorem,corollaryandconverseofatheorem.

Chapter20

10.2 Theorems

Provethefollowingtheoremsalongwithcorollariesandapplythemtosolveappropriateproblems.

i) Ifastraightlinestandsonanotherstraightline,thesumofmeasuresoftwoanglessoformedisequaltotworightangles.

ii) Ifthesumofmeasuresoftwoadjacentanglesisequaltotworightangles,theexternalarmsoftheanglesareinastraightline.

iii) Iftwolinesintersecteachother,thentheoppositeverticalanglesarecongruent.

iv) Inanycorrespondenceoftwotriangles,iftwosidesandincludedangleofonetrianglearecongruenttothecorrespondingsidesandincludedangleoftheother,thetwotrianglesarecongruent.

v) Iftwosidesofatrianglearecongruent,thentheanglesoppositetothesesidesarecongruent.

vi) Anexteriorangleofatriangleisgreaterinmeasurethaneitherofitsoppositeinteriorangles.

Curriculum

iv12 1

vii) Ifatransversalintersectstwolinessuchthatthepairofalternateanglesarecongruentthenthelinesareparallel.

viii)Ifatransversalintersectstwoparallellinesthealternateanglessoformedarecongruent.

ix) Thesumofmeasuresofthethreeanglesofatriangleis180º

Chapter20

Unit 11 Introduction to Trigonometry

11.1 Trigonometry

i) Definetrigonometry.

Chapter22

11.2 Trigonometric Ratios of Acute Angles

i) Definetrigonometricratiosofanacuteangle.

ii) Findtrigonometricratiosofacuteangles(30º,60ºand45º).

iii) Definetrigonometricratiosofcomplementaryangles.

iv) Solveright-angledtrianglesusingtrigonometricratios.

v) Solvereal-lifeproblemstofindheights

Unit 12 Information Handling

12.1 Frequency Distribution

i) Definefrequency,frequencydistribution.

ii) Constructfrequencytable.

iii) Constructahistogramrepresentingfrequencytable.

Chapter2312.2 Measures of Central Tendency

i) Describemeasuresofcentraltendency.

ii) Calculatemean(average),weightedmean,medianandmodeforungroupeddata.

iii) Solvereal-lifeproblemsinvolvingmean(average),weightedmean,medianandmode.

Curriculum

v131

Guiding Principles1. Studentsexploremathematicalideasinwaysthatmaintaintheirenjoymentofandcuriosity

aboutmathematics,helpthemdevelopdepthofunderstanding,andreflectreal-worldapplications.

2. Allstudentshaveaccesstohighqualitymathematicsprogrammes.

3. Mathematicslearningisalifelongprocessthatbeginsandcontinuesinthehomeandextendstoschool,communitysettings,andprofessionallife.

4. Mathematicsinstructionbothconnectswithotherdisciplinesandmovestowardintegrationofmathematicaldomains.

5. Workingtogetherinteamsandgroupsenhancesmathematicallearning,helpsstudentscommunicateeffectively,anddevelopssocialandmathematicalskills.

6. Mathematicsassessmentisamultifacetedtoolthatmonitorsstudentperformance,improvesinstruction,enhanceslearning,andencouragesstudentself-reflection.

Principle 1Studentsexploremathematicalideasinwaysthatmaintaintheirenjoymentofandcuriosityaboutmathematics,helpthemdevelopdepthofunderstanding,andreflectreal-worldapplications.

• Theunderstandingofmathematicalconceptsdependsnotonlyonwhatistaught,butalsohingesonthewaythetopicistaught.

• Inordertoplandevelopmentallyappropriatework,itisessentialforteacherstofamiliarisethemselveswitheachindividualstudent'smathematicalcapacity.

• Studentscanbeencouragedtomuseovertheirlearningandexpresstheirreasoningthroughquestionssuchas;

– How did you work through this problem? – Why did you choose this particular strategy to solve the problem? – Are there other ways? Can you think of them? – How can you be sure you have the correct solution? – Could there be more than one correct solution? – How can you convince me that your solution makes sense? • Foreffectivedevelopmentofmathematicalunderstandingstudentsshouldundertaketasksof

inquiry,reasoning,andproblemsolvingwhicharesimilartoreal-worldexperiences.

Teaching and Learning

iv14 1Teaching and Learning

• Learningismosteffectivewhenstudentsareabletoestablishaconnectionbetweentheactivitieswithintheclassroomandreal-worldexperiences.

• Activities,investigations,andprojectswhichfacilitateadeeperunderstandingofmathematicsshouldbestronglyencouragedastheypromoteinquiry,discovery,andmastery.

• Questionsforteacherstoconsiderwhenplanninganinvestigation:

– Have I identified and defined the mathematical content of the investigation, activity, or project? – Have I carefully compared the network of ideas included in the curriculum with the students’

knowledge? – Have I noted discrepancies, misunderstandings, and gaps in students’ knowledge as well as

evidence of learning?

Principle 2Allstudentshaveaccesstohighqualitymathematicsprogrammes.

• Everystudentshouldbefairlyrepresentedinaclassroomandbeensuredaccesstoresources.

• Studentsdevelopasenseofcontroloftheirfutureifateacherisattentivetoeachstudent’sideas.

Principle 3Mathematicslearningisalifelongprocessthatbeginsandcontinuesinthehomeandextendstoschool,communitysettings,andprofessionallife.

• Theformationofmathematicalideasisapartofanaturalprocessthataccompanies pre-kindergartenstudents'experienceofexploringtheworldandenvironmentaroundthem.Shape,size,position,andsymmetryareideasthatcanbeunderstoodbyplayingwithtoysthatcanbefoundinachild’splayroom,forexample,buildingblocks.

• Gatheringanditemisingobjectssuchasstones,shells,toycars,anderasers,leadstodiscoveryofpatternsandclassification.Atsecondarylevelresearchdatacollection,forexample,marketreviewsofthestockmarketandworldeconomy,isanintegralcontinuedlearningprocess.Withintheenvironsoftheclassroom,projectsandassignmentscanbesetwhichhelpstudentsrelatenewconceptstoreal-lifesituations.

Principle 4Mathematicsinstructionbothconnectswithotherdisciplinesandmovestowardintegrationofmathematicaldomains.

Anevaluationofmathstextbooksconsideredtwocriticalpoints.Thefirstwas,didthetextbookincludeavarietyofexamplesandapplicationsatdifferentlevelssothatstudentscouldproceedfromsimpletomorecomplexproblem-solvingsituations?

Andthesecondwaswhetheralgebraandgeometryweretrulyintegratedratherthanpresentedalternately.

• Itisimportanttounderstandthatstudentsarealwaysmakingconnectionsbetweentheirmathematicalunderstandingandotherdisciplinesinadditiontotheconnectionswiththeirworld.

• Anintegratedapproachtomathematicsmayincludeactivitieswhichcombinesorting,measurement,estimation,andgeometry.Suchactivitiesshouldbeintroducedatprimarylevel.

• Atsecondarylevel,connectionsbetweenalgebraandgeometry,ideasfromdiscretemathematics,statistics,andprobability,establishconnectionsbetweenmathematicsandlifeathome,atwork,andinthecommunity.

v151 Teaching and Learning

• Whatmakesintegrationeffortssuccessfulisopencommunicationbetweenteachers.Byobservingeachotheranddiscussingindividualstudentsteachersimprovethemathematicsprogrammeforstudentsandsupporttheirownprofessionalgrowth.

Principle 5Workingtogetherinteamsandgroupsenhancesmathematicallearning,helpsstudentscommunicateeffectively,anddevelopssocialandmathematicalskills.

• TheCommonCoreofLearningsuggeststhatteachers'develop,test,andevaluatepossiblesolutions'.

• Teamworkcanbebeneficialtostudentsinmanywaysasitencouragesthemtointeractwithothersandthusenhancesself-assessment,exposesthemtomultiplestrategies,andteachesthemtobemembersofacollectiveworkforce.

• Teachersshouldkeepinmindthefollowingconsiderationswhendealingwithagroupofstudents:

– High expectations and standards should be established for all students, including those with gaps in their knowledge bases.

– Students should be encouraged to achieve their highest potential in mathematics.

– Students learn mathematics at different rates, and the interest of different students’ in mathematics varies.

• Supportshouldbemadeavailabletostudentsbasedonindividualneeds.

• Levelsofmathematicsandexpectationsshouldbekepthighforallstudents.

Principle 6Mathematicsassessmentintheclassroomisamultifacetedtoolthatmonitorsstudentperformance,improvesinstruction,enhanceslearning,andencouragesstudentself-reflection.

• Anopen-endedassessmentfacilitatesmultipleapproachestoproblemsandcreativeexpressionofmathematicalideas.

• Portfolioassessmentsimplythatteachershaveworkedwithstudentstoestablishindividualcriteriaforselectingworkforplacementinaportfolioandjudgingitsmerit.

• Usingobservationforassessmentpurposesservesasareflectionofastudents’understandingofmathematics,andthestrategieshe/shecommonlyemploystosolveproblemsandhis/herlearningstyle.

Mathematical Practices1. Makesenseofproblemsandpersevereinsolvingthem.2. Reasonabstractlyandquantitatively.3. Constructviableargumentsandcritiquethereasoningofothers.4. Useappropriatetoolsstrategically.5. Attendtoprecisionandformat.6. Expressregularityinrepetitivereasoning.7. Analysemathematicalrelationshipsandusethemtosolveproblems.8. Applyandextendpreviousunderstandingofoperations.9. Usepropertiesofoperationstogenerateequivalentexpressions.10. Investigate,process,develop,andevaluatedata.

iv16 1

Lesson PlanningBeforestartinglessonplanning,itisimperativetoconsiderteachingandtheartofteaching.

FURLFirstUnderstandbyRelatingtoday-to-dayroutine,andthenLearn.Itisvitalforteacherstorelatefineteachingtoreal-lifesituationsandroutine.‘R’isre-teachingandrevising,whichofcoursefallsunderthesupplementary/continuitycategory.Effectiveteachingstemsfromengagingeverystudentintheclassroom.Thisisonlypossibleifyouhaveacomprehensivelessonplan.Therearethreeintegralfacetstolessonplanning:curriculum,instruction,andevaluation.

1. Curriculum Asyllabusshouldpertaintotheneedsofthestudentsandobjectivesoftheschool.Itshould

beneitherover-ambitious,norlacking.(Oneofthemajorpitfallsinschoolcurriculaarisesinplanningofmathematics.)

2. Instructions Anymethodofinstruction,forexampleverbalexplanation,materialaidedexplanation,or

teach-by-askingcanbeused.Themethodadoptedbytheteacherreflectshis/herskills.Experiencealonedoesnotwork,asthemostexperiencedteacherssometimeadoptashort-sightedapproach;thesamecouldbesaidforbeginnerteachers.Thebestteacheristheonewhoworksoutaplanthatiscustomisedtotheneedsofthestudents,andonlysuchaplancansucceedinachievingthedesiredobjectives.

3. Evaluation Theevaluationprocessshouldbetreatedasanintegralteachingtoolthattellstheteachers

howeffectivetheyhavebeenintheirattempttoteachthetopic.Noevaluationisjustatestofstudentlearning;italsoassesseshowwellateacherhastaught.

Evaluationhastobeanongoingprocess;duringthecourseofstudyformalteachingshouldbeinterspersedwiththought-provokingquestions,quizzes,assignments,andclasswork.

Long-term Lesson PlanAlong-termlessonplanextendsovertheentireterm.GenerallyschoolshavecoordinatorstoplanthebigpictureintheformofCoreSyllabusandUnitStudies.Coresyllabiarethetopicstobecoveredduringaterm.Twothingswhichareveryimportantduringplanningarethe‘TimeFrame’andthe‘Prerequisites’ofthestudents.Anexperiencedcoordinatorwillknowthedepthofthetopicandtheabilityofthestudentstograspitintheassignedtimeframe.

Suggested Unit Study Format

Weeks Dates Months Days Remarks

Short-term Lesson PlanningAshort-termplanisaday-to-daylessonplan,basedonthesub-topicschosenfromthelong-termplan.


v171

Features of the Teaching GuideTheTeachingGuidecontainsthefollowingfeatures.Theheadingsthroughwhichtheteacherswillbeledareexplainedasfollows.

Specific Learning Objectives Eachtopicisexplainedclearlybytheauthorinthetextbookwithdetailedexplanation,

supportedbyworkedexamples.Theguidewilldefineandhighlighttheobjectivesofthetopic.Itwillalsooutlinethelearningoutcomesandobjectives.

Suggested Time Frame Timingisimportantineachofthelessonplans.Theguidewillprovideasuggestedtime

frame.However,everylessonisimportantinshapingthebehaviouralandlearningpatternsofthestudents.Theteacherhasthediscretiontoeitherextendorshortenthetimeframeasrequired.

Prior Knowledge and Revision Itisimportanttohighlightanybackgroundknowledgeofthetopicinquestion.Theguidewill

identifyconceptstaughtearlieror,ineffect,revisethepriorknowledge.Revisionisessential,otherwisethestudentsmaynotunderstandthetopicfully.

Theinitialquestionwhenplanningforatopicshouldbehowmuchdothestudentsalreadyknowaboutthetopic?Ifitisanintroductorylesson,thenaprecedingtopiccouldbetouchedupon,whichcouldleadontothenewtopic.Inthelessonplan,theteachercannotewhatpriorknowledgethestudentshaveofthecurrenttopic.

Real-life Application and Activities Today'sstudentsareveryproactive.Thestudyofanytopic,ifnotrelatedtopracticalreal-life,

willnotexcitethem.Theirinterestcaneasilybestimulatedifwerelatethetopicathandtoreal-lifeexperiences.Activitiesandassignmentswillbesuggestedwhichwilldojustthat.

Flashcardsbasedontheconceptbeingtaughtwillhavemoreimpact.

Summary of Key Facts Factsandrulesmentionedinthetextarelistedforquickreference.

?OOPS

!

Frequently Made Mistakes Itisimportanttobeawareofstudents'commonmisunderstandingsofcertainconcepts.If

theteacherisawareofthesetheycanbeeasilyrectifiedduringthelessons.Suchtopicalmisconceptionsarementioned.


iv18 1

Lesson Plan

Sample Lesson Plan Planningyourworkandthenimplementingyourplanarethebuildingblocksofteaching.

Teachersadoptdifferentteachingmethods/approachestoatopic.

Asamplelessonplanisprovidedineverychapterasapreliminarystructurethatcanbefollowed.Atopicisselectedandalessonplanwrittenunderthefollowingheadings:

TopicThisisthemaintopic/sub-topic.

Specific Learning ObjectivesThisidentifiesthespecificlearningobjective/softhesub-topicbeingtaughtinthatparticularlesson.

Suggested DurationSuggesteddurationisthenumberofperiodsrequiredtocoverthetopic.Generally,classdynamicsvaryfromyeartoyear,soflexibilityisimportant.

Theteachershoulddrawhis/herownparameters,butcanadjusttheteachingtimedependingonthereceptivityoftheclasstothattopic.Notethatintroductiontoanewtopictakeslonger,butfamiliartopicstendtotakelesstime.

Key vocabularyListofmathematicalwordsandtermsrelatedtothetopicthatmayneedtobepre-taught.

Method and StrategyThissuggestshowyoucoulddemonstrate,discuss,andexplainatopic.

Theintroductiontothetopiccanbedonethroughstarteractivitiesandrecapofpreviousknowledgewhichcanbelinkedtothecurrenttopic.

Resources (Optional)Thissectionincludeseverydayobjectsandmodels,exercisesgiveninthechapter,worksheets,assignments,andprojects.

Written AssignmentsFinally,writtenassignmentscanbegivenforpractice.Itshouldbenotedthatclassworkshouldcomprisesumsofalllevelsofdifficulty,andoncetheteacherissurethatstudentsarecapableofindependentwork,homeworkshouldbehandedout.Forcontinuity,alternatesumsfromtheexercisesmaybedoneasclassworkandhomework.

Supplementary Work (Optional): Aprojectorassignmentcouldbegiven.Itcouldinvolvegroupworkorindividualresearchtocomplementandbuildonwhatstudentshavealreadylearntinclass.

Thestudentswilldotheworkathomeandmaypresenttheirfindingsinclass.

EvaluationAttheendofeachsub-topic,practiceexercisesshouldbedone.Forfurtherpractice,thestudentscanbegivenapracticeworksheetoracomprehensivemarkedassessment.


v19

Operations on Sets

1

11

Specific Learning Objectives

Inthisunitstudentswilllearn:

• howtodefineasubsetandapowerset.

• howtouseVenndiagramstodefineunionandintersectionofthreesets.

• howtoverifycommutativeandassociativepropertiesofsets.

• howtoverifydistributivepropertyofunionoverintersection.

• howtoverifyDeMorgan'slaws.

Suggested Time Frame

6to8periods.

Prior Knowledge and RevisionBeforestartingachapteritisessentialtorevisethebasicconceptsofthetopicthatwereintroducedinpreviousgrades.Therearevariouswaysofrevising.Theteachercaninitiateaclassdiscussionbyaskingstudentstodescribeandexplainconceptstheyalreadyknow.Asafurtherrecalltechniquetheycanbedividedintogroupsandtheboardcanbedividedintotwocolumns.Haveabrainstormingrevisionsessionbycallingstudentstotheboard.Theteach-by-askingmethodisaverysuccessfulwayofrevisingandalsoofintroducingorexplainingatopic.

Studentshavelearntaboutsetsinpreviousyearsandtheyareawareofthebasicconceptsofintersection,union,andsubsets.

Thestudentsshouldrevisecomplements,union,andintersection,aswellasthesignsandsymbolsofsubsets.Unionisthejoiningoftwosets,wherethecommonandtheuncommonelementsarewrittenonce.Aunionsetisdenotedby‘∪’,whichiseasytorememberasunionbeginswith‘∪’.

Intersectionofasetincludesthecommonelementsonly,whichisobviousfromthetermitself.Itisrepresentedby'∩'.

Thecomplementofaset(A´)containstheelementsthatarenotintheuniversalset.Forexample,A´wouldhavealltheelementsoftheuniversalsetthatarenotmembersofsetA.

iv20 1 Operations on Sets

Example:UniversalSet = {1,2,3,4,5,6,7,6,9,10}

SetA = {1,2,3,4}

ThenSetA = {5,6,7,8,9,10}(complementofSetA)

Aninterestingconcept:theintersectionofasetwithitscomplementwillalwaysbeanullset.

Intheaboveexample:

SetA = {1,2,3,4}

A = {5,6,7,8,9,10}

A∩A´= øor{}

Similarlytheunionofasetwithitscomplementwillgivetheuniversalset,providedthatsetistheonlysetoftheuniversalset.

SetA = {1,2,3,4}

A = {5,6,7,8,9,10}

A∩A´= {1,2,3,4,5,6,7,8,9,10}

A∪A´= 𝕌

Inthischaptertheconceptofthepowersetisintroduced.Ittellsushowtoascertainthenumberofsubsetsthatcanbemadefromagivenset.

Example:IfA={1,2,3,4},thenthenumberofsubsetsformedwillbe:

P(A)=2k(wherekisequaltothenumberofelementsinsetA). 24 =16Thus16subsetswillbeformedforSetA.

Exercise1containsproblemsrelatedtopowersetandcombinesallpreviouslylearntconceptsofsets.Itshouldnotbedifficultforstudentstosolvetheexerciseoncetheconceptshavebeenrevised.

Real-life Application and ActivitiesThischapteratthislevelbecomesquitetechnical.Thekeyterminologywillbeofsignificancetotheteacher.

Howevertheteachercangiveafunquizwherethestudentscanbedividedintotwogroups;onegrouppreparesquestionsrelatedtothetopicandtheothergroupsendsitsmemberstoanswerinturn.Thentheactivitiesofthegroupsareswapped.Thegroupwiththemostpointswins.Theteachercanplaytheroleofmoderator.

Summary of Key Facts• Asubsetisasetwhichisalsoapartofthebiggersetwherealltheelementsofthesubsetareinthemainset.

• Asubsetcanbeapropersubsetifthesupersethasatleastoneelementwhichisnotapartofthesubset.HencesubsetAcanneverbeequaltosubsetB.

v211 Operations on Sets

WhenA = B A ⊂ B

If A = B A ⊆ B

A✓

✓

✓

✓✓

B

A✓

✓

✓

B

• Thepowersetiscalculatedbytheformulagivenbelow: 2tothepowerofk,wherekisthenumberofelementsinsetA.

• UnionandintersectionofthreesetsarebestdonewiththehelpofVenndiagrams.

Example:

(A∪ B)′ ∩ C A∩ (B∪ C)A

BA B

C

• Thecommutativepropertyoftheunionofsetsstatesthattheorderdoesnotmatter.

• Theassociativepropertyoftheunionandintersectionofthreesetsstatesthattheorderdoesnotmatter.

• Thedistributivepropertyoftheunionofthreesetsstatesthattheorderofoperationsdoesnotaffecttheresult.

• DeMorgan’slawstatesthatthethecomplementoftheunionoftwosetsisequaltotheintersectionofthecomplementsofthetwosets.

?OOPS

!

Frequently Made MistakesStudentstendtogetconfusedwithsetnotationsinvolvingthreesets.Thefirstthingtoemphasiseisthatstudentsshouldbeconfidentinusingthesignsofunions,intersections,andcomplements.InordertosolveasetnotationtheyshoulduseaVenndiagram.

iv22 1Operations on Sets

Lesson Plan

Sample Lesson Plan

Topic

DeMorgan'sfirstlaw

Suggested DurationOneperiod

Key VocabularyUnion,Intersection,Complement

Method and StrategyDeMorgan'sfirstlawstates:(A∪B)′=A′ ∩B′

Example:𝕌 {2,3,4,5,6,7,8,9,10},A={2,3,4,6},B={2,4,5,7,8}

Provethat((A∪B)′ =A′∩B′

Toprovethat(A∪B)′=A′∩B′,theleft-handsidemustbeequaltotheright-handside.

(A∪B)′ = {9,10}

A

3

6

10 9

24

58

7

B

{9,10}

A′

5

7

𝕌 B

8

9 10

∩ =

910

B′

36

910

𝕌 B B𝕌A A A

ItisimportanttodothesesumsontheboardbydrawinghugeVenndiagrams.UsingVenndiagramstosolvesetsquestionsisveryhelpful.

Thestudentscanbeencouragedtomakecutoutsofthesequestionsonasheetofchartpaperanddisplaytheirworkonthesoftboardintheclassroom.Thiswillencouragegroupworkandinterestinthetopic.

Written AssignmentsExercise1Q4andQ5canbedoneorallybeforetheactivityofmakingcutoutsusingtheaboveexample.Latertheycanbedoneforhomework.

v231 Operations on Sets

EvaluationQuizzesatthebeginningofeachlessonwillbeveryhelpfulasnotonlywillthestudentsreviseandstrengthentheirunderstandingofkeyconcepts,butalsotheteachercanusethemasatooltoassesslearning.

Acomprehensiveassessmentbasedonmultiplechoicequestions,andadvancedconceptualquestionssimilartothesumsinExercise1canalsobegiven.

After completing this chapter, students should be able to:• defineasubsetandapowerset,

• useVenndiagramstodefineunionandintersectionofthreesets,

• verifythecommutativeandassociativepropertiesofsets,

• verifythedistributivepropertyofunionoverintersection,and

• verifyDeMorgan'slaws.

iv24

Squares and Square Roots

1

2


Inthisunit,studentswilllearn:

• thepropertiesofsquaresandperfectsquares.• howtofindsquarerootsofnaturalnumbers.• howtofindsquarerootsofdecimalnumbers.• howtofindsquarerootsofvulgarfractions.

Suggested Time Frame5to6periods

Prior Knowledge and RevisionSquaresandsquarerootshavebeenintroducedinpreviousyears.

Starter Activity

Studentscanrevisethistopicbyplayingagamewitheitherapackofcardswiththepicturecardsremoved,ortheteachercanmakesetsofflashcardsofnumbers1to9.

Studentscanworkinpairs.pairs.Theyshouldshuffletheirpackofcardsandpickoutfourcards.

Theythenhavetofindthesquareofthenumberoneachcardandaddthemupandrecordtheirresult.Thisactivitycanberepeatedthreetimes.

Thepairwiththehighesttotalofthe3attemptswins.Theactivityshouldnottakemorethanfiveminutes.

Itisaquickandenjoyableactivitywhichwillhelpthestudentstorevisethesquarestable.

Real-life Application and ActivitiesThistopicmaybedoneasaquizontheboard.Thestudentscouldbedividedintofourgroups.Eachgroupselectsoneofitsmemberstogototheboard.Promptingbyothermembersoftheteammaybeallowedinthefirsttworounds,butinthefinalround,theteammemberwillhavetoattemptthesumonhis/herown.

Thisisafunwaytolearnthestepsofmathematicalcomputationbyindirectpeerparticipationandinstructions.

v251 Squares and Square Roots

Summary of Key Facts• Irrationalnumberscannotbeplacedasfractionsastheyarenon-terminatingand non-recurringwhenindecimalform.

• Thesquareofanumberisrepresentedgeometricallyasa2Dshapeasasquare.Henceitistheproductofitself.

• Squarerootisdenotedbytheradicalsymbol√ .

• Perfectsquaresandsquarerootscanbefoundbytheprimefactorisationmethod.

Example

196225

2× 73×5

1415

2×2× 3×3×

7×75×5

= ==

[Findprimefactorpairsof196and225]

Thesquarerootof196225 is1415 .

• Thesquarerootsofnumbersthatarenotsquarenumbersarefoundbythelongdivisionmethod.Findingthesquarerootofanon-squarenumberisdonebymakingpairsandbringingthemdown.Oncethisisdonethecalculationisthesame.Inthedivisionmethod,thefirstdivisorandthenewdivisorareputtogetherandthenusedfordivision.

Example 18. 471 +1 –128 241+ 8 –224 364 1714+4 – 14563687 25809 – 10192 15617

Therulesofthelongdivisionmethodshouldberevisedorallyinclasssothatstudentsknowthe steps.Displaying these stepsonapin-boardduring thedurationofthechapterwouldbenefitthestudents.

• Tofindthesquarerootsofdecimalsthesameprocessisfollowedbutifthedecimalsarenotevenlypairedinitially,zeroisaddedtocompletethepairs.

Decimalscanalsobeconvertedtofractionsandthesquarerootsofthenumeratoranddenominatorfoundseparately(eitherbyprimefactorisationorlongdivision)andthenconvertedbacktodecimalsastheanswer.

Whenfindingthesquarerootofavulgarfraction,thedenominatorisconvertedtoaperfectsquarebymultiplyingboththenumeratoranddenominatorbyacommonfactor.Findthesquarerootofthenumeratorbylongdivision.Thequotientofthesquarerootsofthenumeratorandthedenominatoristheanswer.

?OOPS

!

Frequently Made MistakesWhendoinglongdivision,studentsgenerallyfaceproblemsinfindingtheclueforthenextstep.Theyshouldwritethestepsintheirnotebooksandtheteachershouldgiveencouragementandhelpwhereverneeded.

341.14 09ˆ

iv26 1Squares and Square Roots

Lesson Plan

Sample Lesson Plan

Topic

Squaresandsquareroots

Specific Learning ObjectivesSolvingwordproblemsinvolvingsquaresandsquareroots


Key VocabularySquare,Squareroot,Radical,Longdivision,Primefactorisation

Method and StrategySolvingwordproblemswherethestudentshavetoapplytheirmathematicalskillstoidentifywhichtechniquetoapply,isdifficultateverylevel.Thestudentswillhavetodecidewhethertofindthesquareorsquarerootandthendecidewhichmethodtouse.

Theexerciseforthisunitisrelativelyeasy.Thestudentscanattemptitontheirownbyfollowingthestepscarefully.Thewordproblemsmayhavetobeexplainedbytheteacher.

Example:

Thesidesoftwosquaresare2.4mand1.8mrespectively.Findthesideofanewsquareequalinareatothecombinedareaofthetwosquares.

Eachsideofsquare1=2.4m

Eachsideofsquare2=1.8m

Thestudentswillhavetoperformmultiplecomputations.Firsttheywillhavetofindtheradicalwhosesquarerootisrequired,andthenthesquareroot.

Areaofsquare1=5.76m2

Areaofsquare2=3.24m2

Totalarea=9m2

Thesideofnewsquare=√9 =3meach

Written AssignmentThesumsonpage31canbediscussedinclassandthestudentsshouldthenbeencouragedtodothemintheirnotebooks.

EvaluationAcomprehensiveassessmentattheendofthechapterisnecessary.Amentalmathsquizcanalsobegivenforfindingsquaresandsquarerootsofnumbers.

After completing this chapter, students should be able to:• applythepropertiesofsquaresandperfectsquares,• findsquarerootsofnaturalnumbers,• findsquarerootsofdecimalnumbers,and• findsquarerootsofvulgarfractions.

v271

3 Cubes and Cube Roots

Specific Learning ObjectivesInthisunitstudentswilllearn:

• thepropertiesofcubesandperfectcubes.

• howtofindcuberootsofperfectcubes.

• howtofindcubesofnegativeintegers.

• howtofindcubesofrationalnumbers.


Prior Knowledge and RevisionThischapterisanextensionofthepreviouschapteronsquareroots.Thestudentshavelearntaboutsquareandcubenumbersinearliergrades.Asquarenumberisobtainedbymultiplyingthenumberbyitself;cubenumbersareobtainedwhenthenumberismultipliedbyitselfthreetimes.

Anoralquizofthefirsttencubenumberscanbedone.Flashcardscanbemadeandthestudentscancomeonebyone,pickupacard,andgiveitscube.

13 = 1

23 = 8

33 = 27

43 = 64

53 = 125

63 = 216

73 = 343

83 = 512

93 = 729

103 = 1000

iv28 1Cubes and Cube Roots

Real-life Application and Activities

Activity

Thebestwaytoexplaincubenumbersiswiththeuseofa3Dobject.Asquareisa2Dshapewithequaldimensions;acubehasthreedimensionswhichareequal.

l

l Asquarehastwobasicdimensions:l × b = l × l = l2

l

l

l Acubehas3basicdimensions:l × b × h = l × l × l = l3

ExampleAcubewitheachside3cmhasavolumeof:3×3×3=27cm3

However,ifasolidhasavolumeof28cubiccentimetres,thenitcannotbeacubebecause28isnotacubicnumber.Itcanbeavolumeofacuboid:28=2×2×7.

Theteachershouldbringcubesandcuboidsofdifferentdimensionstothelessontoexplainthedifference.

Thestudentscanmaketheirownpresentationofcard-boardcutoutsofcubenumbers.Thestudentscanbeassessedonpresentationandunderstanding.Thiscanbeagrouporindividualmarkedassignment.

Activity

RecallthatthestudentslearnedtofindLCMbytheprimefactorisationmethodinGrade6.Findingcubesandcuberootsinvolvesasimilarprocedure.Nowalltheyhavetodoistomakegroupsofthreesinsteadofgroupsoftwosastheydidforsquarenumbers.

ExampleCuberootof729

3 729

3 243

3 81

3 27

3 9

3 3

1

√3×3×3×3×3×3 = √3×3=9

Thus9isthecuberootof729.

v291 Cubes and Cube Roots

Theteachershouldalsoexplainthatalthoughtherearenonegativesquarenumbers,cubenumberscanbenegativebecausewhenthreenegativesignsaremultipliedtheresultisnegative.

Example–8isacubenumber.

=(–2)×(–2)×(–2)=–8.Itisacubeof(–2)3.

Summary of Key Facts• Perfectcubesarefoundwhenaproducthasthreeidenticalfactors.

• Thecubeofapositivenumberispositive.

• Thecubeofanegativenumberisnegative.

• Thecubesofoddnumbersareoddandthoseofevennumbersareeven.

• Multipleofthreescubesarealsoamultipleof27.

• Thecubeofanumber,onemorethanthreetimesitself(3n+1),isanumberofthesameform.

• Thecubeofanumber,twomorethanthreetimesitself(3n+2),isanumberofthesameform.

• Foranynumbern, n3 iscalledaradical,nistheradicand,and3istheindexoftheradical.

• Tofindthecuberootofpositiveandnegativenumberstheprimeindexnotationmethodisusedwherebygroupsofthreearemadeandtheirproductisthecuberoot.

• Ifpq isarationalnumber,then(

pq )

3 = p3

q3 .Thisisknownasthedistributivepropertyunderdivision.

?OOPS

!

Frequently Made MistakesThisisarelativelyeasychapterandstudentsgenerallyenjoyfindingcubesandcuberootsofnumbers.Otherthanmathematicalcomputationerror,studentsgenerallydonotmakeanymistakes.

Lesson Plan

Sample Lesson Plan

Topic

Cubesandcuberoots

Specific Learning ObjectivesMakingperfectcubes


Key Vocabulary WordsSmallestpossibleinteger,Product,Perfectcube

iv30 1Cubes and Cube Roots

Method and Strategy"Whatisthesmallestpossiblevalueof24n,sothatitbecomesaperfectcube?’’

Theteachercanwritethisquestionontheboard.He/shecanexplainthisconceptusingthe'teach-by-asking'method.

1. Whatisaperfectcube?

2. Howthendoweapproachthisquestion?

3. Dowebreak24intoitsprimefactors?

4. Whichnumber/numbersarenotinsetsofthree?

5 Whatismissingtomakesetsoffactorsofthree?

6. Isthisthevalueofn?

7. Isthisthesmallestpossiblevalueofn,whereby24whenmultipliedbyitmakesthenumberaperfectcube?

Iftheteachingmethodisenjoyed,thestudentsshouldeasilygraspthisconceptualquestion.

Written AssignmentAfterdoingacoupleofexamplesontheboard(sumsonpage41ofthetextbook),studentscanbegivenquestionstodoindependentlyintheirnotebooks.

Whatisthesmallestinteger‘n’bywhichthefollowingnumberswillbecomeaperfectcube?

Answers

1. 54n 4

2. 84n 882

3. 108n 2

4. 324n 18

5. 50n 20

EvaluationGiveacombinedassessmentofsquaresandcubesasthisisquiteafunchapter.QuestionsfromExercises2and3canbeusedforthetest.Wordproblemsshouldalsobegivenintheassessmentasthestudentsshouldnottreatthischapterinisolationbutapplyittoreal-lifesituations.

After completing this chapter, students should be able to:• applythepropertiesofcubesandperfectcubes,

• findcuberootsofperfectcubes,

• findcubesofnegativeintegers,and

• findcubesofrationalnumbers.

v311

4 Number Systems

Specific Learning ObjectivesInthisunitstudentswilllearn:• aboutthebinarysystem.• howtoconvert: ▪ base2numberstobase10numbersandviceversa. ▪ base5numberstobase10numbersandviceversa. ▪ base8numberstobase10numbersandviceversa.• howtoaddandsubtractbases2,5,and8numbers.

• howtomultiplybases2,5,and8numbers.


Prior Knowledge and RevisionThisisanewtopic.Thewaytointroduceanewlanguageofmathematicsistorelateittoothernotations.Theteachercanpointoutthat 14 canberepresentedas0.25indecimals,25%inpercentages,and1:4asaratio.

Inthedecimalnumbersystem,weusedigits0to9.Allothernumbersareformedbycombiningthesetendigits.Computersarebasedonthebinarynumbersystemwhichconsistsofjusttwonumbers:0and1.

Real-life Application and ActivitiesSincethisisanewconcept,anactivitywillbeimportantforstudents'understanding.

Activity

Materialsneeded:

A4sizeflashcards

blackmarkers

Beforethegamebegins,dividestudentsintogroupsanddemonstratethisconceptwiththehelpofflashcards.

iv32 1Number Systems

A B C D E

16 8 4 2 1

Teachbyasking.

“HowmanydotsareoncardA?”

“Whatdoyounoticeaboutthesequenceofthecards?”

“Whatwouldbethenextcard?”

Ifnecessary,pointoutthepattern,thatiseachcardishalfoftheoneonitsleft.CardEishalfofcardD,cardDishalfofcardC,andsoon.

Oncetheyrecognisethepattern,askthestudentstomakethefollowingsetsofflashcards.

6 (4dotsand2dotscards)

15 (8,4,2and1dotcards)

21 (16,4and1dotcards)

Askavolunteertoflipcards

Whenabinarynumberisnotshowingitisrepresentedbyzero,andwhenabinarynumbershows,itisrepresentedby1.

Inthebinarysystem9canbeshownasgivenbelow:

0 1 0 0 1

9=1001

Repeatthisactivitytillthestudentsareconfidentandthenletthempracticeingroups.

Summary of Key Facts• Thebinarynumbersystemisanumbersystembasedon2.

• Othernumbersystemsarethedecimalnumbersystem,8(octal),and5(penta).

• Inthebinarysystemtheplacevaluesofthedigitsareintermsofpowerof2.

v331 Number Systems

• Toconvertanumberfromthebinarysystemtothedecimalsystem,wefollowthetablebelow:

Base2number Base10number

011011100101110111100010011010

012345678910

02=01012=110102=210112=3101002=4101012=5101102=6101112=71010002=81010012=91010102=1010

• Toconvertanumberfromthedecimaltothebinarynumbersystem:

Example

Express2010asabinarynumber.

2 20 remainder

2 10–0

2 5–0

2 2–1

2 1–0

1 0 1 0 0(binarynumber)

Wecanusetheexpansionmethodtoconvertanumberinanyotherbasetothebase10system.

• Intheadditionofbinarynumberswefollowthesamemethodasinthedecimalsystem,exceptthatwecarrya2nota10.

• Similarly,insubtractionweborrowa2insteadofa10.

• Inmultiplicationwefollowtheruleofmultiplicationwhichsaysthattheproductofzeroandanynumberincludingitself,iszero.

?OOPS

!

Frequently Made MistakesThisisacompletelynewconceptbutifexplainedwellitisquiteeasytograsp.

iv34 1Number Systems

Lesson Plan

Sample Lesson PlanTopicBinarysystem

Specific Learning ObjectivesAdditioninthebinarysystem


Key VocabularyBinarynumbersystem,Baseten

Method and StrategyThiscanalsobetaughtasanactivity.Theteachercanfirstdemonstrateontheboard,thendividethestudentsintopairs,andpractisebyplay.

ActivityFormulateasetoftenquestionsandwritethemonflashcards.Thestudentswillpickaquestionandsolveitintheirnotebook.Thepairthatfinishesfirstwithallanswerscorrectgainsapoint.

Fordemonstration,theteachercandothefollowingsumsontheboard.

Example 1

Addthefollowing:(i)110and110(ii)12and12.

Solution:

12+12102

110+110210

(i) (ii)

Written AssignmentsQuestion5ofExercise4canthenbegivenforhomework.Thestudentswouldhavehadampleclasspracticewhenworkinginpairs.Thisalsoenablesthestudentstolearnfrompeers.

EvaluationAcomprehensiveassessmentbasedonfillintheblanksandmultiple-choicequestionscanbegiven.

After completing this chapter, students should be able to:• explainthebinarysystem.

• convert:

– base2numbertobase10numberandviceversa, – base5numbertobase10numberandviceversa, – base8numbertobase10numberandviceversa,• addandsubtractbases2,5,and8numbers,and

• multiplybases2,5,and8numbers.

v351

5Exponents and Radicals

Specific Learning ObjectivesInthisunitstudentswilllearn:• thelawsofindices.• aboutnumberswithrationalexponents.• howtoexpressrationalnumbersinradicalform.• howtoexpressradicalsasrationalnumbers.• howtoaddandsubtractradicals.• aboutsurds.

• howtoapplythefouroperationsonexponentsandradicals.


4to5periods

Prior Knowledge and RevisionStudentsarefamiliarwithpowersandbases.Radicalformandsurdswillbenewforthem.Henceitisimportantthattheteacherrevisestheconceptof‘thepowerof’.

Examplean

Whereaisthebaseand'n'thepower.

Thepowerorexponentwilltellthenumber(n)oftimesthebase(a)willbemultipliedbyitself.

Real-life Application and Activities

Activity

Theconceptofexponentialnotationcanbere-introducedwithafunactivity.

Youneed‘x’packsofcardswiththepicturecardsremoved.

Dividethestudentsintogroupsoffour.

Askonestudentfromeachgrouptodealthecards.

Eachstudentthenorganiseshis/hercardsintheexponentialform.

iv36 1Exponents and Radicals

ExampleIfthestudenthas3fiveshewillwriteitas:53

Nextaskthestudentstofindtheproductoftheirexponentiallist.

Example53=5×5×5=125,andsoon.

Nowaskeachstudenttoaddalltheproducts.

Whoevergetsthehighestscoreisthewinner.Theactivitymaybetimed;thewinneristheonewhofinishesfirst.

Thisactivitynotonlydevelopsthestudents’abilitytoorganisedatabutisalsoanindirectwayofexplainingwhatexponents/indices/poweractuallysignify.

Summary of Key Facts• Thebasenumberisraisedtoapowerorexponent.

• Thelawsofrationalexponentsinindicesareasfollows:

(i) Forapositiveintegern,n√ab = n√a × n√b ,

(ii) Foranytworationalnumbersm andn,

am×an = am + n,a ∙ o (powersaddedwhenbasesaresameinmultiplication)

(iii) Forallrationalvaluesofmandn,

am ÷ an = am – n,a ∙ o(powerssubtractedindivisionofsamebases)

(iv) Foranytworationalnumbersmandn,

(am)n = amn,a ∙ o(powersmultipliedwheninbrackets)

(v) a0=1(anynumberotherthanzerotothepowerzeroisI)

(vi) Forallrationalvaluesofn,

(ab)n = an×bn,aandbarenon-zeronumbers.

(vii) Forallrationalvaluesofn,

ab

n

= an

bn,aandbarenon-zero.

• Theradicaloftheformn√a,whichcannotbereducedtoarationalnumber,iscalledasurdofordern.

?OOPS

!

Frequently Made MistakesStudentstendtomakemistakesasthereisrepetitivemultiplication.Theiroral/mentalmathsshouldbesharpenedwithlotsoforalquizzes.

Lesson Plan

Sample Lesson Plan

Topic

Exponentsandradicals

Specific Learning ObjectiveSolvingsumsinofradicalandindexform

v371 Exponents and Radicals


Key VocabularyRadicals,Radicands

Method and StrategyThe teacher shouldreinforcetheradicalsandindexformconceptsclearlyontheboard.Thestudentsshouldunderstandthatevaluationcanalsobedoneforbothradicalandindexform.

Example3 27 actuallymeans27tothepowerof 3

1 .Oncetheradicalformissolved,thestudentsareabletoproceedwiththeoperation.

Example

273 + 42 = 33 × 31 + 2 ×2 1

2 = 3 + 2 = 5

Similarly,itisimportanttopointoutthatanynegativefractionalpowerbecomesapositivefractionalpoweronlywhenthebasevariableofthenumeratorbecomesthedenominator,orviceversa.

Example–

5 21

^ h × –

5 23

^ h

51 2

1

d n × 51 2

3

d n

51

×21 1

2 ×

51

×21

23

5114 ×

5143

514 4

31 + =

51

44 = 5

1

Written AssignmentsThefollowingsumscanbedoneinclass.

Evaluate:

Answers

1. (5)2 + ∙ 65 × (5)2 ∙ – 153 6

2. 8+(64)56 40

3. 20+43 65

4. 2713 +32 12

5. 223 × 23 2

iv38 1Exponents and Radicals

EvaluationThisisanimportanttopic.Regular5minutequizzesshouldbegivenandfinallystudentsshouldbe testedwithsumsthatcontainallconceptstaught.

After completing this chapter, students should be able to:• applythelawsofindices,

• expressrationalnumbersinradicalform,

• expressradicalsasrationalnumbers,

• addandsubtractradicals,

• simplifysurds,and

• applyfouroperationsonexponentsandradicals.

v391

Logarithms

Specific Learning ObjectivesInthisunitstudentswilllearn:• toexpressnumbersinstandardformandscientificnotation.

• aboutlogarithms.

• howtoexpressnumbersinlogarithmform.

• thelawsoflogarithms.

• howtosolveproblemsbasedonlogarithms.


4to5periods

Prior Knowledge and RevisionThisisanewwayofrepresentingnumbers.

Thebestwaytointroducelogarithmsistoreviseindicesandscientificnotation.Logarithmsisanaturalextensiontotheseconcepts.

Indicesandscientificnotationcanberevisedthroughatimedworksheet.Beforehandingouttherevisionworksheet,thebasicrulesofthesetopicsshouldberevised.Thestudentswhofinishearlywithcorrectanswersgainapoint.

Real-life Application and ActivitiesActivity

Understandingthekeytermssuchasbase,exponent,andthepowerofbaseten,iscriticaltotheuseoflogarithms.

Tofamiliarisestudentswiththeseterms,playthefollowinggamewiththem.

Materialsneeded:markersandA4sizedflashcards.

iv40 1Logarithms

Youneedtomaketwosetsofcards.

Onesetwillhavesumsoflogswrittenonthem(atleast10sums).

Theothersetwillhaveonlythreecards,withbase,exponent,andanswerwrittenonthem.

Set 1. Set 1. Set 2. Set 2. Set 2.

log2 8 = 3 log2 64 = 6

Base Exponent Answer

Thestudentsaredividedintopairs.Onestudentchoosesacardfromset1andpointsatanyofthevalueonthecard.Theotherstudentwillhavetoraisethecardwiththecorrectterm.

Forexample,ifonestudentpicksup log2 64 = 6 andpointsto2,thentheother

studentmustraisethecardwith'Base'writtenonit.

Theyplayandswapmultipletimes;theonewiththemostcorrectanswerswins.

Summary of Key Facts• Inindices,numbersaredenotedbyabaseraisedtoapower.Forexample,1000isdenotedby10raisedtothepower3,i.e.103.

• Anumberexpressedinscientificnotationiswrittenintheform: A×10n,where1≤ A< 10and'n'isaninteger.

• Inlogarithmicnotation,log264=6,2iscalledthebase,64iscalledtheargument,and6istheanswer.

• Thelogarithmoftheproductoftwonumbersaandbisequaltothesumoftheirindividuallogarithms.

logx ab=log

x a+log,b

• Thelogarithmofthequotientoftwonumbersxandyisequaltothedifferencebetweentheirindividuallogarithms.

logx a ÷ b=log

x a–logx b

• Thelogarithmofanumberaraisedtoapowerbisequaltotheproductofthepowerbandthelogarithmofthenumbera.

logxab = blog

xa

?OOPS

!

Frequently Made MistakesSincenewtermsareinvolvedinthistopic,stressshouldbelaiduponnotationthroughlotsofpracticesumsandSnapgameswithflashcards.

v411 Logarithms

Lesson Plan

Sample Lesson Plan

Topic

Logarithms

Specific learning ObjectiveIntroducingtheconceptofreadinglogs

Suggested Duration1period

Key VocabularyBase,Exponent,Argument

Method and StrategyThebestwaytointroducelogsistoaskstudentstowritethemethodintheirnotebooks.Thefollowingpointsshouldbeshownontheboardandcopiedinnotebooks.logbasebofaisxwillbewrittenas:

Logba = x

Whatexponentisrequiredtoformabaseofb toreachthevalueofa?

Exponentialform: 26=64

base exponent answer

2 6 64

Logarithmicform: log264=6

log base answer exponent

log2 2 64 6

Activity

Theteachercanteachlogswiththetriedandtested‘looptrick’,whichisasfollows:

The loop trick

Alwaysdrawyourloopanti-clockwise

Logb a = x bx = a

Log264 = x 2x = 64

Activity

Anothergamethatcanbeplayedis‘logwar’.

Youneedtomaketwoidenticalsetsofcardsoftwentysumsoflogsandshufflethem.Thestudentsaredividedintopairsandthetimersetfor5minutes.ThestudentsplaySnapbyflippingthecardsandquicklyworkingoutthevalues.Theonewhofinishesfirstorpicksoutthelastcardandworksoutthesumfirstcallsout'Snap!'.

logx9=2 logwar log10x =2

iv42 1Logarithms

Written AssignmentsTheteachershoulddoafewsumsorallyandthenaskthestudentstodoquestions1and7ofExercise6forhomework.

EvaluationThestudentscanbeassessedwhiledoingactivitiesandmarksassignedaccordingtoresults.Theactivitiescanbedoneperiodicallyduringtheweek.Acomprehensiveassessmentoflearningcanalsobedoneaftercompletingtheunit.

After completing this chapter, students should be able to:• expressnumbersinstandardformandscientificnotation,

• expressnumbersinlogarithmform,

• applylawsoflogarithms,and

• solveproblemsbasedonlogarithms.

v431

7 Profit, Loss, Insurance, and Taxes

Specific Learning ObjectivesInthisunitstudentswilllearn:• howtocalculatepercentageprofit,percentageloss,andpercentagediscount.

• aboutvarioustypesofinsurance:lifeinsurance,vehicleinsurance.

• aboutincometax.


Prior Knowledge and RevisionStudentshaveagoodknowledgeofpercentages.Theyhavedonesumsonpercentagesinearliergradesandalsousedtheminearlierchapters.Arevisionactivitycanbedone.

Activity

Thestudentscanbedividedintogroupsoffour.Theycanallmaketheirownsumsonpercentagesandprepareatestpaper.Eachshouldcontainfoursumsandthetestpapersshouldbeswappedbetweenthegroups.Thetestshouldtaketenminutesandthecompletedanswersshouldbereturnedtotheexaminergroup.Theythencheckthetestandgradethegroupandhanditbacktothem.Thisentireactivityshouldnottakemorethan25minutes.

Itisafunrevisionactivitywherethestudentsroleplayteacher,examiner,andstudents.Itisimperativethatpercentagesarerevisedontheboardinitially.

Role Play Ativity

Theteachercancreateamake-believebusinessandexplaintheprocessinthefollowingmanner.She/hecanevenpresentitonaPowerPointslideshow.

‘How business work’

RaheemBrothersbuyT-shirtsfromthesuppliers.Thecostofbuyinggoodsiscalledthecost price.

TheythenselltheT-shirtstotheconsumeratahigherpriceknownastheselling price.

Thedifferencebetweenthecostpriceandthesellingpriceistheprofit.

Lossisincurredifthesellersellsatalowerpricethanthecostprice.

Tofindtheprofit%orloss%dividetheprofitorlossbythecostprice,andthenmultiplyby100%.

iv44 1Profit, Loss, Insurance, and Taxes

ExampleAshopkeeperbuystoycarsinboxesof50.Hepurchases10boxesforRs6000.EachtoycaristhensoldforRs15.Findhisprofitorlosspercentage.

Find

• numberoftoycarsbought

• costpriceofeachtoycar

• profitmade

• profitpercentage

Costprice =6000500

=Rs12

Profit =Rs15–Rs12=Rs3

Profit%= 312

×100%=25%

Real-life Application and ActivitiesPercentagescanbeexplainedwithreal-lifeexamples.

Arrangeanactivityasfollows:

Introducetheideaoforganisingaschoolconcert.

Thestudentsrole-playorganisers.Theyhavetoworkoutthecostoftheeventandthentheprofit.Tocalculatethecostprice:

calculatethewagesoftheperformersandthesupportstaff

calculatethetotalcostofputtingontheevente.g:sound,stage,lighting,refreshments,etc.

profitprojectionwillbebasedonthenumberofticketssold.

Dividestudentsintogroupsandaskthemtoworkoutthetotalcostofputtingontheconcert.

Groupsarethenaskedtoworkoutprofit projectionbasedonthreecasescenarios.

50%ofthestudentsattendtheconcert.



Foreachcasetheycalculatethepercentageprofitusingtheircostprojectionsandtheticketprice.

Lastly,theyareaskedtocalculatetheminimumnumberofticketstheyneedtoselltobreakeven(nolossisincurredandthecostiscovered).

Thisisaninterestingproject/assignmentandthestudentscanbegiventwodaystocompleteit.Eachgroupcanbemarkedonthisassignment.

Summary of Key Facts• Percentageprofitorlossiscalculatedbydividingtheprofitorloss bythecostpriceandmultiplyingby100%.

• Discountisareductiononthecostpriceoftheproducttobesold.Itspercentagecanbefoundbydividingbytheoriginalcostpriceandmultiplyingby100%.

v451 Profit, Loss, Insurance, and Taxes

• Insuranceisacontractwherebythepersonisinsuredagainsttheriskoflosstohis/herproperty.Acertainpremiumispaidtotheinsurancecompanymonthlyoryearlyandinexchangehe/sheisinsuredfromanylossordamagetohis/herproperty.

• Depreciationisthelossofactualvalueofaproductoveraperiodoftime.

• Incometaxisimposedbythegovernment.Itsetstheamounttobetaxedandtherateatwhichtheincomeistaxed.

• Exemptincomeistheamountuponwhichnotaxispaid.

• Taxableincomeistheamountofincomeonwhichtaxispaid.

• Taxesarealsochargedonagriculturaloutputandbusinessprofits.

?OOPS

!

Frequently Made MistakesThistopichasreal-lifeapplications.Studentstendtomisunderstandthetermsandareunabletoapplythecorrectformula.

Lesson Plan

Sample Lesson Plan

Topic

Taxation

Specific Learning ObjectivesTocalculatechargeableincomeandthetaxpayableonitatagivenrate


Key VocabularyExemptincome,Chargeableincome,Taxation

Method and Strategy

Explainthisconcepttostudentwithroleplay.

Selectastudenttobethebreadwinnerofthefamilyroleplay,awife,twokids,andamother.Chargeableincomeisthencalculatedbyanaccountreliefgivenforeachmemberofthefamily.

Taxisthencalculatedwiththetermsgivenontheboard.

Thisactivitywilltake3-5minutesbutthestudentswillunderstandtheconceptbetter.

Setupsumswithvariableincomesandreliefsandafixedrateoftaxation.

Example

Relief: 2childrenattherateofRs10,000each

MemberattherateofRs15,000

CharityofRs5000

Income: Rs200,000

Tax: 10%overandaboveRs50,000

iv46 1Profit, Loss, Insurance, and Taxes

Written AssignmentQuestion11,12,and13ofexercise7canbedoneasclasswork.

EvaluationThischapterinvolvesalotofmathematicaltermsandformulae.Thestudentscanbeassessedonthedefinitionsandformulaeina10minutequiz.AcomprehensiveassessmentbasedonquestionsfromExercise7canbeconductedtoassesslearning.

After completing this chapter, students should be able to:• calculatepercentageprofit,percentageloss,andpercentagediscount,

• calculatevarioustypesofinsurance,and

• calculatevarioustypesoftaxes.

v471

8Compound Proportion and Banking

Specific Learning ObjectivesInthisunitstudentswilllearn:• todefinecompoundproportion.

• howtosolvereal-lifeproblemsinvolvingcompoundproportion,partnership,andinheritance

• aboutdifferenttypesofbankaccounts:currentdepositaccount,PLSsavingsaccount,PLStermdeposit,fixeddepositaccount,recurringdepositaccount,foreigncurrencyaccount.

• howtocalculatesimpleandcompoundinterest.

• howtoidentifyvariousbankinginstruments:cheque,demanddraft,payorder.

• aboutbankservices:ATM,debitcard,creditcard.

• aboutfinancialservices:runningfinance,demandfinance,lease.

• howtoconvertonecurrencytoanother.


6to8periods

Prior Knowledge and RevisionStudentsarefamiliarwithsimpleinterest.Thischapteramalgamatespercentagesandarithmeticintoaformalteachingofbankingprocesses.

Theteachercanexplainthattheinterweavingofallconceptstaughtearliersuchaspercentagesandratioswordproblemswillbeappliedinthereal-world.Thisreal-lifeapplicationwillexciteandchallengethestudents.

Asawarmup,storysumsofpercentagesandsimpleinterestcanbedoneontheboardandtheteachercanencouragestudentparticipation.

Real-life Application and ActivitiesCompoundinterestisanextensionoftheconceptofsimpleinterest.Thisisappliedbymostbanks.

Thebasicconceptofcompoundinterestisthatwhenthetenureisofmorethanayear,theprincipalchangesasthesimpleinterestisadded.

iv48 1Compound Proportion and Banking

Theinvestorgainsmore'interest'astheprincipalamountincreaseseveryyear.Insimpleinterest,nomatternowlongthetimeperiodis,theprincipalandtheinterestremainfixed.

Activity

Thistopiccanbereinforcedbyorganisinganeducationaltriptoabank.Arepresentativeofthebankcanexplainhowthebankingsystemworks.Thedifferenceandcorrelationbetweensimpleandcompoundinterest,variousbankingterms,transactions,andhowdifferentdepartmentsworkcanbeexplained.

Ifatripcannotbeorganised,abankmanagercouldbeinvitedtogiveapresentationinclass.Thiswillhelpthestudentstounderstand:

bankingterminology

departmentalfunctions

bankingcalculations

ThestudentsshouldalsobeencouragedtodoresearchandmakeapresentationonchartpaperaboutthevariousbanksinPakistan.

Summary of Key Facts• Therearevarioustypesofbankaccount.

• Acurrentaccountisusedwhendepositsandwithdrawalsaremadefrequently.

• InaPLSsavingsaccount,theprofitandlosssharingisnotusedfrequently.

• Inafixeddepositaccounttheamountisfixedforacertainlengthoftime.

• Theformulaforcompoundinterestisgivenbelow: C.I=A–P

A=P 100r1

t

+b l

where:

A istheamount

P istheprincipal

R istherate

T isthetimeinyears

[Itmustbeexplainedthattherateortimecanalsobecalculatedforagivenamountandtheprincipal.]

• Achequeisawritteninstructionforacertainamounttobepaidtothebeneficiary.

• Ademanddraftisissuedbythebankonbehalfofanaccountholder.Thiscanbeusedinanybranchofthebank.

• Apayorderissimilartoademanddraftbutcanonlybeusedtowithdrawmoneyfromthesamebranch.

• Therearevariousbankingservices.

• Anautomatedtellermachine(ATM)isamachinewhichallowstheaccountholdertowithdrawcashafterbankinghoursandfromwhereveronehasbeeninstalled.

v491 Compound Proportion and Banking

• Debitandcreditcardsareplasticcardsissuedtoaccountholderstomakepaymentswithoutusingcash.Onceenabled,thecardtransfersmoneyfromtheowner'saccounttotherecipientaccountwithoutanyexchangeofcash.

• Anoverdraftfacilityenablestheaccountholdertomakepaymentsregardlessofinsufficientfundsintheaccount.

• Runningfinanceissimilartoanoverdraftfacilitybutitcanusedforalongerperiodoftimeandthebankchargesinterestontheamountborrowed.

• Demandfinanceisanamountloanedtotheaccountholderbythebank.Thetenurecanbeofanylengthbutthebankcandemandtheloantobepaidbackatanytime.

• Aleaseisanarrangementbywhichthebankpaysanamountequaltothevalueofanaccountholder'sasset(s)forhis/heruse.He/shewillthenrepaytheamountloanedandtheinterestininstallmentsoveragivenperiodoftime.

• Thetimeframeiscalledtheterm,thebankisthelessor,andthecustomer,thelessee.

• Inordertomakepaymentsabroad,thecurrencyexchangerateissuedbytheregulatorybankisused.Thisrateistheequivalenceratiobetweentwocurrencies.

?OOPS

!

Frequently Made MistakesThestudentsconfusethevariousbankingterms.Ifallthetermsandtheirdefinitionsaredisplayedonasheetofchartpaperintheclassroomitwillbeveryhelpfultothestudents.Thestudentsshouldbeaskedtonoteallofthemintheirnotebooks.

Lesson Plan

Sample Lesson PlanTopic

Currencyconversion

Specific Learning ObjectivesCurrencyconversion


Key VocabularyPounds,USdollars,Rupee,Yen,Riyal,DirhamCurrency,Exchangerate

Method and StrategyItwillbeexcitingforthestudentstoinvestigateandpresentvariouscurrenciesoftheworld.Theequivalenceratescanbefoundonthebusinesspagesofnewspapers,orontheinternet.

Oncethestudentsarefamiliarwiththecurrenciesofdifferentcountries,giveashortquizwheretheteachernamesacountryandthestudentsnameitscurrency.

Pointoutthefactthatdirectproportionisinvolvedincurrencyconversion.

iv50 1Compound Proportion and Banking

Currencytableforafewcountries:

China:yuan

USA:dollar

Canada:Canadiandollar

Australia:Australiandollar

Japan:yen

Pakistan:rupee

India:rupee

Bangladesh:taka

Thailand:bhat

Thelistislongandtheteachercanaddtoit.

Written AssignmentsThestudentscancopythelistandtheteachercangivefivesumsalongwiththeexchangeratestobecalculated.

Evaluation Anobjectivetestbasedontrueorfalsestatements,fillintheblanks,andmultiplechoicequestionscanbegivenonvariousbankingterms.

Acomprehensivesubjectivetypeoftestcanalsobegivenattheendofthetopic.

After completing this chapter, students should be able to:• definecompoundproportion,

• solvereal-lifeproblemsinvolvingcompoundproportion,partnership,andinheritance,

• differentiatebetweendifferenttypesofbankaccounts:currentaccount,depositaccount,PLSsavingsaccount,PLStermdepositfixeddepositaccount,recurringdepositaccount,foreigncurrencyaccount,

• calculatesimpleandcompoundinterest,

• identifyvariousbankinginstruments:cheque,demanddraft,payorder,

• identifyvariousbankservices:ATM,debitcard,creditcard,

• differentiatebetweenfinancialservicesprovidedbythebanks:runningfinance,demandfinance,lease,and

• convertonecurrencytoanother.

v511

Stocks and Shares


Inthisunitstudentswilllearn:

• aboutstocks,shares,anddividends.

• whatnominalvalueandmarketvalueofsharesmeans.

• whatbrokerageisintermsofbuyingandsellingshares.


5to6periods

Prior Knowledge and RevisionThestudentswillhavenopriorknowledgeofthistopic.Infact,mostofthematthisagewouldhavenoideaideahowthestockexchangeworks,northeterminologyusedthere.

Thistopicwillintroducethemtotheworldofeconomicsandwillhavetoberelatedtotheworkingoftheircountry'sstockexchange.

Thestudentsshouldbeencouragedtoreadthebusinesspagesofthenewspaperpriortostartingthistopicsothattheyhavesomeunderstandingoftheeconomy.

Real-life Application and ActivitiesSimulationofthestockexchangecanbecreatedintheclassroom.

Activity 1

Thestudentsrole-playandeachcouldbeassignedaparticularrole.

Therecanbeabuyer,seller,andabroker.

ThemainsettingcanbetheKarachiStockExchange,withnewspaperclippingsofstocksandsharesdisplayedonasheetofchartpaper.

Thestudentscanplaywithagivenamountandcalculatethesharesneededandtheirprices.Thebrokercancalculatehiscommissionandthebuyercancreatehisportfoliowithpredictionsofhisgainsbycalculatingthedividendrate.

iv52 1Stocks and Shares

Activity 2

• EachgroupcanbeginwithRs10,000.

• Eachgroupmustinvestinatleast5stocksortheteachercansetalimitofaminimumof3stocksandmaximumof6stocks.

• Acommissioncanbesetforeverystocktransaction,e.g.,3%commission.

• Thetimeframeforthisactivitycouldbeaweek.

• Eachgroupwillmaintainarecordoftheiractivitiesandideas.Theirtransactionhistoryshouldalsoberecorded

• Onthelastdayoftheactivityeachgroupwillselltheirstocksandreportonthevalueoftheirportfolio.

• Thegroupwiththehighestgainwillwin.

Ithastobeexplainedthatalthoughrealstocksandvalueswillbeused,noactualbuyingorsellingwilltakeplace.

Thissimulationisaperfectwayofhelpingthestudentsunderstandtheworkingsofastockmarket.

Summary of Key Facts• Adividendisashareofacompany'searnings,decidedbyitsboardofdirectors,tobepaidtobegiventotheshareholders.

• Aholderofstockhasaclaimonthecompany'sassets,whichispredeterminedasacertainvalue.

• Asharecertificateindicatesthenumberofsharesoneholdsinaparticularcompany.

• Asharehasacertainmarketvalueandanominalfacevalue.Themarketvaluefluctuatesdependingontheperformanceofthecompanyandthecountry'seconomy.

• Abrokerdealswiththetransactionsofthebuyerandthesellerandchargesacertainpercentageofthevalueofthetransactionascommission.

?OOPS

!

Frequently Made MistakesStudentsmaynotunderstandthebroaderpictureoftheworkingsofthestockexchange.Role-playandnewspaperarticleswillhelpthestudentstounderstandthisbetter.

Lesson Plan


Stocksandshares

Specific learning ObjectivesCalculatingthedividendandbrokeragecommission


v531 Stocks and Shares

Key VocabularyDividend,Shares,Commission,Brokerage,Percentage

Method and StrategyMr.Pervezpurchased300sharesoffacevalueRs100atRs800pershare.Ifthecompanypaysadividendof40%,findhisearningpercentonthisinvestment.

Thisquestiontakesintoaccountthefactthatthestudentshouldunderstandthetermsused.Theteachershouldfirsthighlightthekeywords:purchase,facevalue,dividend,andpercentage.Thedefinitionsandcontextinwhichtheyareusedshouldbediscussed.

Siftingtheinformationandthenidentifyingtheproblemarekeytothistopic.

Percentageswillbecalculatedaspreviouslytaught.Thestudentsshouldknowthatwhencalculatingthedividendthefacevalueisusedandnotthemarketvalue.Thedividendcalculatedisthenmultipliedbythetotalnumberofshares.

EachquestionisacontinuationofthechapteronpercentagestaughtinGrade7.Thestudentsshouldbeencouragedtousetheirpriorknowledgeinreal-lifesituations.

Written AssignmentsDoquestions10to14ofExercise9inclass.

EvaluationAcomprehensiveassessmentcanbegivenattheendofthechapter.Thestudentscanalsobeassessedontheirunderstandingoftheworkingofthestockexchangemarketandtheirdecision-makingskillsduringtherole-playactivity.

After completing this chapter, students should be able to:• explainthetermsstocks,shares,anddividends,

• differentiatebetweenthenominalvalueandmarketvalueofshare,and

• explainwhatwhatbrokerageisintermsofbuyingandsellingshares.

iv54 1

10 Averages

Specific Learning ObjectivesInthisunitstudentswilllearn:• abouttheconceptofaverages:simpleaverage,weightedaverage,andaveragespeed.

– howtofindtheaverage.

– howtocalculatetheweightedaverage.

– howtofindtheaveragespeed.


3to4classes

Prior Knowledge and RevisionThisisaninterestingtopicwherethestudentscollectactualdatafromvarioussources.Averagesisatopicthatstudentsmusthavealreadycomeacross.Theclassaverageisoftendiscussedwhenatestresultisannounced.Themarksofallthestudentsstudentsareaddedanddividedbythetotalnumberofstudentstogettheaveragemark.

Real-life Application and ActivitiesAverageisactuallyaneverydayEnglishword,butthestudentsshouldunderstandthemathematicalcomputationsofaverages.

Asimpleformulatocalculateaverage:

sumofthequantities/numberofquantities

Thestudentscanapplythisformulaeasily.Itisadvisablenottomentionfrequencyinplaceofnumberatthisstage.

However,theteachershouldbeawarethataveragesleadtomean,median,andmode.Thedifferencebetweenthesumofthequantitiesandthetotalnumberofthevaluesshouldbemadeclear.

v551 Averages

Activity

Thestudentsshouldbeaskedtocollecttendifferentitemsofdatawhoseaveragecanbefound.

Forexample,cricketscores,heightsofstudents,temperatureofacityinthelastsevendays,stockmarketvalueofacertaincompanyinamonth,currencyexchangerateofdollartotherupeeforaweek,etc.

Afunactivitycanbedonewhereeachgroupisassignedtocollecttherelevantdataandbringittoclass.Oncethedataiscollected,askstudentstofindtheaveragesofeachsetintheirnotebooks.Theteachershouldremindthestudentstobecarefulaboutthedivisor.Incaseoftemperatureofthelastsevendays,thestudentswilldivideby7andalsointhecaseofthestockmarketvaluedivideby30.

Summary of Key Facts• Thesimpleaverageisthesumofthequantitiesdividedbythenumberofthequantities.

• Theweightedaverageisthesumofthenumbersinadatasetmultipliedbytheweights,dividedbysumoftheweights.

• Averagespeedisthetotaldistancecoveredduringajourneydividedbythetotaltimeofthejourney.

?OOPS

!

Frequently Made MistakesStudentstendtomakeminormistakeswhileaddinganddividing.

Lesson Plan


Averages

Specific Learning ObjectivesCalculatingaveragespeed

Suggested Duration Oneperiod

Key VocabularyTimeDistance,Unitsofspeed,km/hr,M/sec

Method and StrategyPervaizdrivestohisofficeataspeedof50km/hrandcomesbackhomeataspeedof40km/hr.Ifthedistancebetweenhishomeandtheofficeis20km,calculatehisaveragespeedfortheentirejourney.

iv56 1Averages

Averages

Journeyfromhometooffice:

Distance = 20km

Speed = 50km/hr

S=DS

∴ T=DS

Time = 2050

= 0.4hours

Journey:fromofficetohome

Disance = 20km

Speed = 40km

Time = 2040

= 0.5hours

Totaldistance = 20+20=40km

Totaltime = 0.4+0.5=0.9hours

Averagespeed= Totaldistance Totaltime

= 400.9

= 44.4km/hr

Similarsumscanbedoneinclassusingreal-lifecharacterssuchasparentsandbusdriversandtheirjourneys.

Written AssignmentsStudentscandoQuestions20,21,and22ofExercise10intheirnotebooks.

EvaluationAcomprehensiveassessmentbasedonfillintheblanks,multiplechoicequestions,andwordproblemscanbegiventoassesslearning.

After completing this chapter, students should be able to:• explaintheconceptofaverages:simpleaverage,weightedaverage,andaveragespeed.

– findtheaverage,

– calculatetheweightedaverage,and

– findtheaveragespeed.

v571

11 Matrices

Specific Learning ObjectivesInthisunitstudentswilllearn:• whatamatrixis.

• abouttypesofmatrix:squarematrix,rectanglematrix,rowmatrix,columnmatrix,squarematrix,diagonalmatrix,scalermatrix,nullmatrix,unitmatrix.

• howtotransposeamatrix.

• theconditionsfortwomatricestobeequal.

• howtoaddandsubtractmatrices.


5to6periods

Prior Knowledge and RevisionThisisanentirelynewtopic.Amatrixisisyetanotherwayofrepresentingnumbersandauniquewayofapplyingmathematicaloperations.Wordproblemscanalsobesolvedwithmatrices.

Real-life Application and ActivitiesAmatrixisanarrayofnumbersarrangedinrowsandcolumns.Theconceptofmatricescanbeexplainedinafunway.

Amatrixisusuallydenotedbyacapitalletter,forexample,AorB.

Theelementsareplacedwithinbox-bracketsin‘rows’and‘columns’.Remember,rowsarehorizontalandcolumnsvertical.

A= ∙a11 a12a21 a22 ∙

AmatrixorderiswrittenasArc

iv58 1Matrices

Summary of Key Facts• Thenumberofrowsandcolumnsofamatrixiscalledtheorderofthematrix.

• Thelocationofanelementofamatrixisdenotedbywritingitsrowandcolumnposition.

Example

B=6 4 163 9 8

b11 =6(elementatrow1,column1)

b12 = 4(elementatrow1,column2)

b21 =3(elementatrow2,column1)

Thereare8typeofmatrix.

• A‘squarematrix’hasanequalnumberofrowsandcolumns.

• A‘rectanglematrix’hasanunequalnumberofrowsandcolumns.

• A‘rowmatrix’hasonlyonerowandcanhavemanycolumns.

• A‘columnmatrix’hasonlyonecolumnandcanhavemanyrows.

• A‘diagonalmatrix’hasallelementsexcepttheleadingdiagonal,aszero.

• A‘scalarmatrix’hasalltheelementsintheleadingdiagonalequal.

• Allelementsofa‘nullmatrix’arezero.

• A‘unitmatrix’has‘1’initsleadingdiagonal.

• Inordertotransposeamatrix,weswaptherowswiththecolumns.

Example:Totransposeamatrixweswaptherowsandcolumns.

Weputa'T'intoprighthandcorner.

6 3 25 6 7

T

= 6 53 62 7

RowRow

column

• Twomatricesareequalwhentwoconditionsaremet:theorderandthecorrespondingelementsareequal.

• Additionandsubtractionofmatricesisdonebymakingsuretheorderofthematricesarethesame.

?OOPS

!

Frequently Made MistakesThisisarelativelyeasychapter;however,studentstendtomakecarelesserrorswhileaddingorsubtracting.Lotsoforalandpracticeexercisesshouldbedonetoavoidcarelessmistakes.

v591 Matrices

Lesson Plan


Matrices:additionandsubtractionofmatrices

Specific Learning ObjectiveStudentswilllearntoaddandsubtractmatrices

Suggested DurationOnetotwoperiods

Key VocabularyRow,Column,Order

Method and StrategyThistopiccanbedoneinafunway.Theteachercanassignashapetoeachelement.Beforetheconceptofaddingandsubtractingistaught,thefactthattheorderofmatriceshastobethesame,shouldbeemphasised.

Addition of matrices:

Addthenumbersinthesamepositions.

+ = 3 2

4 5

4 0

1 9

7 2

5 14

Thesearethecalculations

3 + 4 = 7 2 + 0 = 2

4 + 1 = 5 5 + 9 = 14

Thetwomatricesmustbeofthesameorder.

Subtraction of matrices:

Subtractthenumbersinthesamepositions.

– = 3 2

4 5

4 0

1 9

–1 2

3 –4

Thesearethecalculations

3 – 4 = –1 2 – 0 = 2

4 – 1 = 3 5 – 9 = 4

iv60 1Matrices

Written AssignmentSumscanbedoneontheboard,andthenaskstudentstodothemintheirnotebooks.

Answers

3 7

–2 5

4 3

5 2

7 10

3 71) +

3 0

5 0

0 7

6 8

3 –7

–1 –82) –

8 0

2 1

5 3

2 –3

13 3

4 –23) +

5 3

2 1

–3 2

4 –7

8 1

–2 84) –

Evaluation Acomprehensiveassessmentbasedonmultiplechoicequestions,fillintheblanks,andwordproblemscanbegiven.FollowthepatternofExercise11.

After completing this chapter, students should be able to:• explainwhatamatrixis,

• identifytypesofmatrix:squarematrix,rectanglematrix,rowmatrix,columnmatrix,squarematrix,diagonalmatrix,scalermatrix,nullmatrix,unitmatrix,

• transposeamatrix,

• applytheconditionsfortwomatricestobeequal,and

• addandsubtractmatrices.

v611

12 Operations on Polynomials



• howtomultiplypolynomials.• howtodividepolynomials.


4to5classes

Prior Knowledge and RevisionInGrade7studentsweretaughttheorder,classification,addition,andsubtractionofpolynomials.Itshouldbepointedoutthatunliketermscannotbeaddedorsubtracted.Theseoperationscanbeappliedonliketermsonly.Rulesofsignswerealsoexplainedwiththehelpofanumberline.

Arevisionworksheetcanbegiven.Oncecompletedtheteachercansolvethesumsontheboardandthestudentscanchecktheirpartners'work.Itisimportantthateachsumissolvedontheboardasitwillbeeasierforthestudentstodoanycorrections.Generallytheymakeamistakewithsignsortheytendtoaddthepowerswhenadding.Itshouldbepointedoutthatthepowersdeterminewhetherthetermscanbeaddedorsubtracted.

Followingsumscanbegivenasrevision. Answers

1. Add2x2 + 3xyand4x2 – 6xy. 1.6x2–3xy

2. Whatisthedifferencebetween10wz–7xzand6wz–5xz? 2.4x–2xz

3. Whatmustbeaddedto 3x + 5ytoget 7x – 3y? 3.4x–8y

4. Subtract2x2+3y – zfrom–5x2+4y–2z. 4.–7x2 + y–3z

5. Findthesumof 2x + 3y + zand –5x – 7y – 3z. 5.–3x–4y–2z

iv62 1Operations on Polynomials

Real-life Application and ActivitiesExplaintheoperationsofmultiplicationanddivision.Theserequireconcreterules,steps,andmethod.Additionandsubtractionhavedifferentrulesforsignsfromtomultiplicationanddivision.

Thefollowingsignrulesareappliedinmultiplicationanddivision:

(+)×(+)=(+) (+)÷(+)=(+)

(+)×(–)=(–) (+)÷(–)=(–)

(–)×(–)=(+) (–)÷(–)=(+)

Threemethodsofmultiplicationaretaughtinthischapter:

1. verticalmultiplication

2. horizontalmultiplication

3. theFOILmethod

Eachmethodshouldbeexplainedseparately.Itisadvisablethatthesamesumissolvedbyallthreemethodsforcomparison.Studentswilldeterminethedifferencesintheapproachesbyrememberingthestepsinvolvedineachmethodgivenonpages148and152.

Activity

Make20cardsof'+'signand'–'signeach.AsktwostudentstocomeupandplaySnap.Theteacherwillcalloutanyoperation,e.g.multiplicationordivision.

Forexample,'multiplytwopositivesigns'.

Thestudentswillstartlookingthroughthecardsonebyone.

Ifhe/sheget+sign,he/shewillcallout'Snap'.Similarlytheteachermaysay'divide+signand–sign'.Thestudentwhofindsthecardwitha–signwillsays'Snap'.

Playthisgamefortwominutes,thestudentwithmorecardswins.

Thisactivitycanbedoneinpairstillthewholeclassunderstandstoconceptofsign.

Summary of Key Facts• Multiplicationofpolynomialscanbedonewithunliketerms.

• Therulesofsignsformultiplicationanddivisionarethesame.

• Intheverticalmethodofmultiplicationthetermsarealignedverticallyandmultiplicationiscarriedoutinsteps.

• Inhorizontalmultiplication,thefirstpolynomialissplitintotermsandtheneachisindividuallymultipliedbythesecondterm.Liketermsarethensimplified.

• IntheFOILmethodtheoutertermsaremultiplied,andthentheinnerterms,andthenthelastterms.Theorderneedstobefollowedinthismethod.

• Divisionrequiresthatthefirstdivisortermdividesthefirsttermofthedividendandthenthequotientmultipliestheentiredivisorterm.Thetermsarethensubtractedandtheprocessrepeateduntilthereisnoremainder.

v631 Operations on Polynomials

?OOPS

!

Frequently Made MistakesStudentsgenerallygetconfusedwiththerulesofthesignsastheyaredifferentfromtheadditionandsubtractionrules.Stepsandtheformatformultiplicationanddivisionshouldbewrittenforbetterunderstanding.

Lesson Plan

Sample Lesson PlanTopicOperationsonpolynomials

Specific Learning ObjectivesFOILmethodofmultiplication


Key VocabularyFOIL,Horizontalmultiplication,Polynomial

Method and StrategyInthislessonstudentswilllearnanewmethodofmultiplyingabinomialbyapolynomial.

Theteacherwillwritetheacronym‘FOIL’ontheboard.

First

Outer

Inner

Last

Itisadvisablethattheteacherusescolouredchalkorboardmarkerswhenexplainingthismethod.TheteachercanuseadifferentcolourforeachletterofFOIL.

Written AssignmentAlternatesumsfromExercise12acanbedoneforclassworkandhomework.Studentscanalsosolveasumbyusingtwodifferentmethodsforpractice.

EvaluationTherevisionexercisecanbeusedasaformatforcomprehensivetest.Multiplicationanddivisionshouldbetestedbyspecifyingtherequiredmethodinthequestion.Quizzesshouldbegivenatleastthreetimesaweekinalternatelessons.This5-minuteactivityhelpsthestudentandteacheridentifyanyproblemsandthenworkonthem.

After completing this chapter, students should be able to:• multiplypolynomials,and• dividepolynomials.

iv64 1

13 Algebraic Identities



• howtofindcubesofthesumoftwoterms.

• howtofindcubesofthedifferenceoftwoterms.

• howtofindspecialproductsusingalgebraicidentities.


6to8periods

Prior Knowledge and RevisionStudentshavelearntthethreeidentitiesintheearliergrade.Aquickrevisionoftheseidentitiesisextremelyimportantascubicidentitiesdependonthem.Thestudentsgetconfusedoverwhentoapplytheperfectsquareorthedifferencebetweentwosquares.Inmostcasesitshouldbestressedthatthesumsrequireinitialfactorisationandthentheapplicationofidentities.

Thefollowingsumscanbegivenasrevision.

Answers

1. (2x +2)2 1.4x2+8x+4

2. (4x – y)2 2.16x2–8xy + y2

3. 81x2–1 3.(9x +1)(9x –1)

4. (x + 12

y)2 4.x2 + xy + y2

5. 16x4 – y4 5.(4x2 + y2)(2x – y)(2x + y)

Explainthatperfectsquaresbreakupintotwosquares,andtheproductofthetwotermsismultipliedby2.

Thedifferencebetweentwosquaresisidentifiedeasilyastheyareindividuallysquared.Theyarefactorisedbyfindingthesquarerootofthetermsandthenasthesumanddifference.

v651 Algebraic Identities

Real-life Application and ActivitiesRefertopage160oftextbookwherethediagrammatic/geometricexplanationofthecubicidentityisgiven.

Wecanconvertittoanactivityinclass.

Materialsrequired:

Chartpaperwithcutoutofthenetdiagrams:

1cubeofsidea ×a ×a

a

a

1cubeofsideb ×b ×b

b b

b

3cuboidsofsidea ×b ×b

a

b

b

b

iv66 1Algebraic Identities

3cuboidsofsidea ×a ×b

a

b

Oncethesenetdiagramshavebeenmade,jointhemtogethertomakecubesandcuboids.

Askstudentstowritetheirindividualvolumesoneachmodelandcombine.

Nextstepwouldbetoglueallthemodelsandcreateabigcubeofsides(a+b)3.

Thisbranchofalgebrawherethederivationisdonegeometricallyiscalledgeo-algebra.

Thistopicisconceptualandlotsofexamplesshouldbeexplainedontheboard,andgiventostudentstosolveintheirnotebooks.

Theconceptsofcubicidentitiescanbeintroducedbyexplainingthatthisidentityisanextensionfromlinearandquadraticfunctions.

Summary of Key Facts• Cubesofthesumoftwoterms,(a + b)3 = a3+ b3+3ab(a + b)

• Cubesofthedifferenceoftwoterms,(a – b)3 = a3– b3–3ab(a – b)

• Numericevaluationofcubicfunctioncanbedonebysubstitutingnumericvalues.

• Identitiescanbeprovedusingcubicfunctions.

• Cubicidentitycanbeusedtofindthesumofthreetofourterms.

?OOPS

!

Frequently Made MistakesThesumsinthistopicwillrequirealotofconcentrationastheworkingofeachsumisquiteextensiveandinvolvesmanysteps.Whendoingthesumsontheboard,theteachershouldenunciateeachstepslowlyandcarefully.Itiscommonforteacherstokeepdoingthesumsontheboardwithoutexplaining.Ifworkedcarefully,studentswillmemorise,understandtheprocess,andnotmissoutsteps.

v671 Algebraic Identities

Lesson Plan


Algebraicidentities

Specific Learning ObjectiveResolvingcubictermsintofactors


Key VocabularyCubicfunctions,Factorisation

Method and Strategy

Studentsarealreadyawareofthederivationoftheidentity,thereforeresolvingintofactorswouldbeeasy.

Alsostudentsatatimetocomeontheboardandsimultaneouslydothreesums.Therestofthestudentswillpromptanymistakesandalsocopytheworkingdown.Thiswayeachstudentwillgetaturnontheboardandpearlearningwillbeachieved.

Written AssignmentsExercise13a,question5(parts1toV).

EvaluationAlternatesumsfromExercises13aand13bcanbegivenasatesttoassessstudents'abilitytousethecorrectidentity.

After completing this chapter, students should be able to:• findthecubesofthesumoftwoterms,

• findthecubesofthedifferencebetweentwoterms,and

• findspecialproductsusingbetweenalgebraicidentities.

iv68 1

14Factorisation of Algebraic Expressions

Specific Learning ObjectivesInthisunit,studentswilllearn:• howtoresolveexpressionsintofactors.

• howtofactoriseexpressionsoftheformax2 + bx + c.


5to6periods

Prior Knowledge and RevisionThestudentshavelearntaboutthecubicfunctionfactorisationintheearlierchapter.Thischaptertakesfactorisationforward.Norevisionisrequiredasitisacontinuationofpreviouslearning.

Real-life Application and ActivitiesByresolvingintofactorswederivenewalgebraicidentities.

a3 + b3=(a + b)(a2 – ab + b2)

a3 – b3=(a – b)(a2 + ab + b2)

Middletermbreakup!

Inquadraticfunctionsofax2 + bx + cwemultiplythenumericvaluesof‘a’and‘c’andselectfactorsthatwillbreakupthemiddleterm‘bx’.Henceresolvingintofactorsbymiddletermbreakup.

Activity

Specificlearningobjective:Factorisingtrinomialsbyusinggrouping/breakupmethod.

Example:

Theteachershouldwritethetrinomial6x2 + x–2ontheboard.

Teacher:Isthistrinomialintheformofax2 + bx + c?

If'yes'thenletusfactorise.

v691 Factorisation of Algebraic Expressions

Steps:

1. Writethevaluesofa,b,andc.

a=6 b=+1 c=–2

2. Find‘ac’.

6x2 + x–2

ac=6×2=12

3. Findtwointegerswhoseproductis‘ac’andsumis‘b’.

Factorsof12:(1×12),(2×6),(3×4).

Choosethepair(3,4)becausetheirproductis12andtheirdifferenceis1.

4. Rewritethemiddleterm'bx'asthesumoftwotermswhosecoefficientsaretheintegersfoundinstep3.

Hence,

6x2 + x–2

6x2+4x–3x–2

5. Nowletusfactorisebygrouping,amethodthatwaslearntearlier.

6x2+4x–3x–2 2x(3x+2)–1(3x+2) (3x +2)(2x–1)

∴ Thefactorsof6x2 + x–2are(3x+2)(2x–1).

Nowsolvethese. Answers

1. x2+7x+10 1.(x+2)(x+5)

2. x2–9x+14 2.(x–2)(x–7)

3. x2 + x–20 3.(x–4)(x+5)

4. x2 – x–12 4.(x+3)(x–4)

5. 2x2+7x+3 5.(2x+1)(x+3)

6. 3x2+7x–6 6.(3x–2)(x+3)

7. –x2–4x+32 7.(–x+4)(x+8)

8. 4x2–6 8.2(2x+1)(x–2)

Summary of Key FactsThefollowingidentitiesarecoveredinthischapter:

• a3 + b3=(a + b)(a2 – ab + b2)

• a3 – b3=(a – b)(a2 + ab + b2)

• ax2 + bx + c

?OOPS

!

Frequently Made MistakesToavoidmistakes,studentsneedtoidentifyandthenapplythecorrectformattofindfactors.Repetitiveexercisesandpracticeisamust.

iv70 1Factorisation of Algebraic Expressions

Lesson Plan


Factorisationofalgebraicexpressions

Specific Learning ObjectivesResolvecubicexpressionsintofactors


Key VocabularyCubic,Trinomial,Factoring,Grouping,Identity

Methodology and StrategyExplanationofsolvingcubicfunctionscanbedonebythe‘teach-by-askingmethod’andbyengagingstudentsindiscussionanddoingsumsonboard.

Derivationofalgebraicidentitybygeometryiscalled'geo-algebra'.

Derivingthea3+b3identitybyshowingacubeofsides'a'andanothercubeofsides'b'andcalculatingtheirvolumesisessentiallythegeometricaspectofthisalgebraicidentity.

Thestudentswillhaveabetterunderstandingofthisconcept.

a3 +b3

+

a3

b3

Refertopage168ofthetextbookforfurtherexplanationoftheconcept.

Written AssignmentsTheteachercangivethesumsfromExercise14aasclassworkandforhomework.

Peercheckingwillhelpthemunderstandtheirmistakesbetter.Followupofhomeworkthenextdaywithpeercheckingisrecommended.

EvaluationFrequentquizzesandshorttestsbasedonalgebraicidentitieswillhelpstudentstounderstandtheconceptbetter.

After completing this chapter, students should be able to:• resolveexpressionsintofactors,and

• factoriseexpressionsoftheformax2 + bx + c.

v711

15 Basic Operations on Algebraic Fractions



• howtomultiplyalgebraicfractions.

• howtodividealgebraicfractions.


4to5classes

Prior Knowledge and RevisionAlgebraicfractionsinvolvethesameconceptsasnumericfractions.Itisadvisabletoreviseafewadditionandsubtractionfractionsumswithnumericalvalues.

Real-life Application and Activities• Thischapterisaculminationoffactorisation.Dividestudentsintogroupsoffourorfiveandaskthemtodesigntheirowntestpaper.Thetestpaperformatshouldbeclearlyspecifiedasfollows:

• fivesums,

• thefirstquestionshouldbeofsimplereductionofalgebraicfractionbyfactorisation,

• thesecondquestioncanbeofcomplexindicesfractionswherethelawsofindicesareapplied,

• thethirdquestioncanbeofalgebraicfractionswithmultiplicationanddivision,

• thefourthandfifthquestionsshouldcomprisealgebraicfractionswithadditionandsubtractionrespectively.

Givestudentstenminutestosetthetestpaper.

Thesetestsshouldthenbehandedtoadifferentgroup.Studentswillalltakethetestsandthenafter15minutes,handthembacktothethegroupwhosetthepaper.

Thetestsshouldthenbecheckedandmarkedbythegroup.

iv72 1Basic Operations on Algebraic Fractions

Thisexerciseisaninteractiveactivitywhichcanbeadaptedforanychapter.

TheteachershoulddivideExercise15intothreesections.

Thefirstsectionwillinvolvesimplefactorisationofthenumeratorandthedenominatorandthetermssimplified.Theteachershouldemphasisethethreeidentitiesandmiddletermbreakup.Thekeytothereductionisthatthestudentsfirstfactoriseandthenreducethebrackets.Oncefactorised,thetermneedtobesimplified.

Beforestartingthelessonhighlighttherulesgivenonpage176.Thestudentscanwritethemintheirnotebooksforreference.

Answers

1) ∙1 – 254x2∙ ÷ ∙1 – 52x∙ 1) 1 + 52x

2) x2+2xx2+ x –2

– 3xx +1 2) 2x(2–x)

(x –1)(x+1)

3) aa –3

– 8a =2 3) –6,4

4) 56x

+ 67x

– 914x

=14 4) 11147

5) 52x – 1

– 44x – 2

– 36x – 3

=1 5) 32

or1 12

6) a+12a – 8

– a + z12x – 3a

6) = 5a + 76(a–4)

7) 3x + 1

– 12x + 2

=5 7) _ 12

8) 14e + 2f

– 1f – 2e

8) 32(2e – f)

Summary of Key Facts• Tofactoriseapolynomialthethreeidentitiesarehighlighted.

• Iftheidentitiesdonotapply,thensimplyfactorise.

• Thenextstepistocheckifmiddletermbreakupisapplicable.

• Oncethetermsarefactorised,theyaresimplifiediftheoperationismultiplicationordivision.

• Inthecaseofadditionandsubtraction,thedenominatorsareconvertedtoacommontermbyfindingtheLCM.

• TherulesforfindingtheLCMaretotakethecommontermonceandtheuncommontermsalso.

• TheLCMisthendividedbythedenominators.

• Thetermisthenmultipliedbythenumeratorandsimplified.

v731 Basic Operations on Algebraic Fractions

?OOPS

!

Frequently Made MistakesThekeytosolvingthesesumsistofactoriseandfindthecorrectLCM.Asstudentssometimesgetconfused,thethreeidentitiesneedtoberevisedregularly.

Lesson Plan


Basicoperationsonalgebraicfractions

Specific Learning ObjectiveMultiplyanddividealgebraicfractions


Key VocabularyFactorisation,LowestCommonMultiple,Identity,Middleterm,Simplification

Method and StrategyThischapterinvolvesanadvancedlevelofknowledgeofalgebra,involvingfractions.So,ifcon-ceptsofalgebrataughtearlierarestillunclear,thestudentswillfacedifficultyinhandlingthesesums.

Exercise15shouldbedividedintothreeparts:questions1to6,questions7to16,andquestions17to25.Eachsectionwilltakealotoftimeandlotsofpracticeisrequiredtograspalltheconcepts.

Itisadvisablethatalternatesumsaredoneontheboardbytheteacher,andtheremainingsumsaredonebythestudents.

Onlywhenthestudentsareabletotackleeasiersums(questions1to6),withoutanydifficulty,shouldtheteacherproceedtoquestions7to16.Sums16onwardsarequitelengthyandmustbedoneverycarefully.

Example

a ba 7ab 12b

–2 2

2 2+ + ÷ 4a b

b–

a +

Writethereciprocalofthesecondfraction.

a ba 7ab 12b

–2 2

2 2+ + ((a b) a b)a 4ab b b

(a 4b)(a b)×–

–3 122 2

++ + +

+

Factoriseeachtermandsimplify.

((a b) a b)a 4ab b b

(a 4b)(a b)×–

–3 122 2

++ + +

+

((

(a b) (a b) a b)a b)(a 4b) (a 3b)

×––4+ +

+ +

(a b)(a 3b)

++

Answer

iv74 1Basic Operations on Algebraic Fractions

Example

a2–5a+6a2–8a+16

= a2–3a–2a+6a2–4a–4a+16

= a(a–3)–2(a–3)a(a–4)–4(a–4)

= (a–3)(a–2)(a–4)(a–4)

Factorisethenumeratoranddenominatorindividuallyandthenreducedtosimplestform.

Boththeexpressionsareoftheformax2 + bx + c,thereforeapplytherulesforresolvingquadraticexpressionsintofractions,taughtinthepreviouschapter.

Example

(x+2)2

x2–4 = (x+2)(x+2)(x+2)(x–2)

= (x+2)(x–2)

Written AssignmentsAlternatesumsofExercise15shouldbedoneasclassworkandhomework.

EvaluationFive-minutequizzesshouldbegivenforeachsectiontocheckunderstandingofconcepts.Generallythesequizzesactasadiagnostictooltoidentifystudents'weaknessesbeforetheyproceedtoamorecomplexstage.Assessmentoflearningisanimportanttoolinteachingandlearningmathematicalconcepts.

AcomprehensivetestcanbegivenalongthelinesofExercise15toassessstudents'performance.

After completing this chapter, students should be able to:• multiplyalgebraicfractions,and

• dividealgebraicfractions.

v751

1More Simple Equations



• howtosolveequationsinvolvingalgebraicfractions.

• howtosolvereal-lifeproblemsinvolvingsimpleequations.


3to4periods

Prior Knowledge and RevisionStudentsshouldbeabletotransposevaluesinalgebraicequations,wheretheinverseof:

additionissubtraction,

subtractionisaddition,

multiplicationisdivision,and

divisionismultiplication.

Theyknowhowtoworkoutthevalueofanunknownvariable‘x’.

Arevisionexercisecanbedoneontheboard,involvingstudentsinsolvingquestionsstepbystep.

Answers

1. 6x+4=7 1. 12

2. 12x+4=3x+2 2. –29

3. 5x+7=2 3. –1

4. 22–x=4x–3 4. 5

5. 35=12x–1 5. 3

iv76 1More Simple Equations

Real-life Application and ActivitiesThischaptertakesalgebraicequationstoahigherlevel.Therulesoftransposing,learntearlier,appliesinalgebraicequations.ThetermwhosevalueistobefoundistakentotheLHSoftheequation.Theothertermsaretransposed,thepositivetermasnegative,andthemultiplicandasdivisor,andviceversa.

Thebasicruleisthatfractionsontheeithersideoftheequationaretobecross-multipliedtogetasimplealgebraicequation.

Iftherearemorethanonefractiononeithersideoftheequation,thensimplifythembyaddingorsubtractingasstatedinthequestion.

Example

17x–5 = 312x 2

Multiplytheterm(17x–5)by2,and12x by3,toformasingleequation:2(17x–5)=3(12x)

Example

3x –1

+ 1

x +1 =

4x

3(x +1)+1(x–1)= 4x (x–1)(x+1)

3x +3 + x–1= 4x (x–1)(x+1)

4x +2 = 4x x2–1

x (4x+2)=4(x2–1)

4x2+2x=4x2–4

4x2–4x2+2x=–4

2x=–4

x=–2Answer

Therulesoftransposingrevisedearlierwillplayanimportantroleinsolvingtheequation.

Exercise16a,questions1to5canbedoneinoneperiod.

Inthenextlesson,pointoutthatdecimalpointscanberemovedeitherbyconvertingthemtofractionsorbymultiplyingtheentireequationbymultiplesof10.Questions11to17canbedonealternatelyinclassandforhomework.

Summary of Key Facts• Theunknownvariableisplacedonthelefthandside.

• Eithertransposeapartfromthelefthandsidetotherighthandsidebychangingitssignifitisadditionorsubtraction,ordivideormultiply.

v771 More Simple Equations

• Problemsinvolvingreal-lifesituationsrequiredatarepresentation,wheretherequiredvalueisassignedthevariable‘x’.

• Theunknownquantityandthegivenconditionareexpressedinanequation.

• Theequationisthensolvedbytransposing.

?OOPS

!

Frequently Made MistakesStudentsgenerallymakemistakesintransposing.Theyforgettochangethesignwhentakingapositivetermtoeithersideoftheequation.

Lesson Plan


Algebraicequations

Specific Learning ObjectiveToformequationsundergivenconditionsandsolvethem


Key VocabularyTranspose,RHS,LHS,Variable

Method and StrategyFormingtheequationsforawordprobleminvolvesthinkingskills.Anumberofsumscouldbedoneontheboard.Askthestudentstobreakupthestatementintophrases,andthenformtheequation.Roleplaycanalsobedonewherethestudentscanenactaworkproblem.

ThesumsinExercises16aand16bareprogressive.Oncethestudentshaveunderstoodhowtosolvethem,theycaneasilybedoneinclassandalsobegivenashomework.

Example

Afatheristwiceasoldashissonnow.20yearsagothefatherwas4timesasoldashisson.Findtheirpresentages.

Wecanwritethiswordproblemalgebraically.

Supposetheson'spresentageis'x'

∴Fatherpresentageis2x

20yearsagoson'sage=x–20

20yearsagofather'sage=2x–20

20yearsagofather'sagewas4timeshissonsage.

∴2x–20=4(x–20)

Thisalgebraicequationcannowbesolved.

Itisimportantthatthestudentsshouldhavedevelopedtheabilitytoconvertawordstatementsintoamathematicalstatement.

iv78 1More Simple Equations

Written AssignmentsForquestions1to11ofExercise16b,theequationscanbediscussedandformedinclass.Findingthesolutioncanbegivenforhomework.Thesolutionoftheequationsinthisexerciseareverysimpleandonlybasictransposingisrequired.

EvaluationAcomprehensivetestof20markscanbegivenwhere4algebraicequationsof4markseachcanbegivenandonesumof4markscouldbeawordproblem.

After completing this chapter, students should be able to:• manipulateequationsinvolvingalgebraicfractions,and

• solvereal-lifeproblems.

v791

17 Simultaneous Equations



• aboutsimultaneouslinearequations.

• howtosolveapairofsimultaneouslinearequationsbyeliminationandbythesubstitutionmethod.

• howtosolvereal-lifeproblemsinvolvingsimplelinearequations.


4to5periods

Prior Knowledge and RevisionTheteachershouldpointouttothestudentsthattheyhavebeensolvinglinearequationswithoneunknownvariable.Itistimetomoveonandsolvemorecomplexsumswherethelinearequationscontaintwounknownvariables.

Generallyifthefirstvariableis‘x’,thenthesecondunknownvariableistakentobe’y’.

Real-life Application and ActivitiesThisconceptcanbetaughtbythe'teach-by-asking'method.

Writes6x+4y=7ontheboard.

Teacher:'Whatisnewaboutthisequation?'

Students:Therearetwounknownvariables.

Teacher:Good.

Tosolvetwounknowns,weneedtwoequationstosolvesimultaneously.

So,letuswriteanotherequation.

6x+4y=7 (i)

3x + y=2 (ii)

Whatshouldwedonow?

iv80 1Simultaneous Equations

Students:Letuseliminateoneofthevariable.

Teacher:How?

Students:Bycancelling'y'.

Teacher:Correct,buthowshouldwedothat?

Toeliminate'y'weneedtomake:

coefficients thesame

signs opposite.

Howdowemakecoefficientsof'y'thesame?

Students:Bymaking'y'ofequation(ii)become4y.

Teacher:'Great.Wecandothatbymultiplyingtheentireequationby4andthenmultiplyingby–1tochangethesigns.'

Hence:

6x+4y=7 (i)

3x + y=2 (ii)

Multiplyequation(ii)by–4andaddbothequations.

6x+4y=7

–12x–4y=–8

–6x=–1

x = 16

Substitutevalueofxinequation(i).

6 ∙ 16∙ + 4y =7

1+4y =7

4y =6

y = 32

Answer ∙16 , 32∙

Summary of Key Facts• Theeliminationmethodrequiresoneofthevariablestobecancelledout.

• Thiscanbedonebymultiplyingeitheroneorbothequationsbyanumbertogetthesamecoefficients.

• Thesignshavetobeoppositetocancelthecoefficient.Thisispossiblebymultiplyingtheentireequationby–1.

• Thesubstitutionmethodrequiresmakingonevariablethesubjectoftheequationandexpressingitintermsoftheothervariable.

• Wenowreplacethetermandhencetheentireequationiscomprisedofonevariable.

• Oncewegettheanswerofavariablebyeithermethodwesubstitutethevalueinanyoneequationtogetthevalueoftheothervariable.

v811 Simultaneous Equations

?OOPS

!

Frequently Made MistakesWhenusingeliminationmethodthestudentstendtoforgettomultiplytheentireequationandtheconstant.TheteachershouldexplainthatallthetermsonbothLHSandRHSoftheequationhavetobemultiplied.

Lesson Plan

Sample Lesson Plan

Topic

Simultaneouslinearequations

Specific Learning ObjectiveApplyingreal-lifeconditionsandformingsimultaneousequations


Key VocabularyVariable,Simultaneously

Method and StrategyWhenintroducingreal-lifewordproblems,datarepresentationisextremelycritical.

Howdoyouassignthevariables‘x’and‘y’.?

Anentirelessonshouldbeplannedonformulatingsimultaneouslinearequationsandnotsolvingthem.Thenextlessoncanbebasedonsolutions.

Activity

Dividethestudentsintogroupsoffourandwritedownasumforeachgroupontheboard.

Astudentfromeachgroupwillformtheequationsontheboard.

Thefollowingsumscanbewrittenontheboard.

1)Findtheothertwoanglesofaright-angledtriangleifoneangleis13 oftheother.

x + y =90º (i)(given)

x = 13 y (given)

∴ 3x – y =0 (ii)

Solveequation(i)and(ii)simultaneously

x + y=90º

3x – y=0

4x=90º

∴ x=22.5º

iv82 1Simultaneous Equations

Substitutethevalueofx inequation(i)

x + y =90º

22.5º+y=90º

∴ y=67.5º

2)Thesumoftwonumbersis48andtheirdifferenceis16.Findthetwonumbers.

x + y = 48

x – y = 16

– + –

x=32

Substitutethevalueof'x'inequation(i)

32+y = 48.

y = 48–32

y = 16

3) Ifauntis4timesasoldashernephewandthesumoftheirpresentagesis50,findtheirpresentages.

Supposeaunt'sageisxandnephew'sageisy

Then x=4y

x –4y=0 (i)

and x + y=50 (ii)

x–4y=0

x + y=50 – – –

–5y=–50

y=10

Substitutethevalueof'y'inequation(i)

x=4(10)

x=40

Theauntis40yearsoldandthenephew10yearsold.

Written AssignmentsTheequationsinQuestions5to8ofExercises7canbeformedinclassandsolutionsdoneas homework.

v831 Simultaneous Equations

EvaluationTwo5-minutequizzescanbedoneduringthislesson.Thefirstquizmaycontaintwosumstobesolvedbytheeliminationmethodandthenexttwosumsbythesubstitutionmethod.

Thesequizzeswillensurethatthestudentshaveunderstoodthisnewconcept.Thischaptercanbetaughtinthreestages:findingasolutionbytheeliminationmethod,findingasolutionbythesubstitutionmethod,andsolvingreal-lifewordproblems.

Acomprehensivetestcanbegivenattheendofthechapter.Thetestshouldcontainsumstosolvedbytheeliminationandsubstitutionmethodsandwordproblems.Donotspecifyanymethodtosolvethethewordproblems.Thestudentsshouldbeencouragedtochoosetheirownmethodtosolvethem.Generally,studentsprefertheeliminationmethodtothesubstitutionmethod.

After completing this chapter, students should be able to:• solvesimultaneousequationsbyeliminationandsubstitutionmethod,and

• solvereal-lifeproblemsinvolvingtwounknownvariables.

iv84 1

18 Symmetry

Specific Learning ObjectivesInthisunit,studentswilllearn:

• linearsymmetry.

• propertiesofsymmetricalfigures.

• propertiesofsymmetryaboutabisector.


2periods

Prior Knowledge and RevisionStudentsintheirkindergartenandprimaryclasseshavedonesimpleactivitiesofcuttingshapesintotwoequalpartsandofcompletingafigurebyreplicatingtheotherhalf.TheMathsFlashonpage202canbedoneasrevisionorintroductiontosymmetry.Symmetrycanalsobelinkedtocongruency.Congruencyhasbeencoveredearlierwheretheconceptofexactreplicaisexplained.Symmetryalsoinvolvescongruentshapesoneithersidesofalineofsymmetry.

Theteachercanmakeaworksheetofincompleteshapesandaskstudentstocompletethesymmetricalshapes.

Real-life Application and ActivitiesTheteachercanbringamirrortothelessonandaskastudenttostandinfrontofit.Explainthatthe‘mirrorimage’isactuallycongruencyandthemirroritselfactsasalineofsymmetry,separatingtwocongruentoridenticalobjectsorimages.

Studentscanbeencouragedtobringpicturesofsymmetricalobjectsandbuildingsandthenaskthemtodrawthelineofsymmetry.

v851 Symmetry

Summary of Key Facts• Symmetryiswhencongruentshapesareseeneithersideofthelineofsymmetry.

• Verticalsymmetryiswhenaverticallinecutsanyshapeintotwosymmetricalhalves.

• Thehorizontallineofsymmetryisahorizontallinethatcutsashapeintotwosymmetricalhalves.

• Adiagonallineofsymmetryisalinethatcutashapediagonallyintotwosymmetricalhalves.

• Thelineofsymmetryisalsoknownasaxisofsymmetryofanyshape.

• Pointsofsymmetryareequidistantfromtheaxisofsymmetry.

• Aperpendicularbisectorisalineofsymmetryofaline.

• Ananglebisectorisequidistantfromtwolinesformingtheangleandalsoactsasalineofsymmetry,containingpointsequidistantfromthetwolines.

• Twocongruentshapescanbeseparatedbyalineofsymmetry.Aperpendicularbisectorisdrawnbetweenthetwoshapes.

?OOPS

!

Frequently Made MistakesSymmetryisasimpleconceptthathasbeentaughtinpreviousgradestoo.Studentsenjoydoing activitiesrelatedtosymmetry.

Lesson Plan

Sample Lesson PlanTopicSymmetry

Specific Learning ObjectivePropertiesofsymmetryofshapes


Key VocabularyLineofsymmetry,Equidistant

Method and StrategyThisisafuntopicandtheconceptsoflineofsymmetryandpointsymmetrywillbetaughtbyanactivity.

Activity

Youwillneed:A4paper,scissors,markers,andrulers.

Askthestudentstocutoutthefollowingshapes:

rhombus

square

rectangle

parallelogram

isoscelestriangle

iv86 1Symmetry

Tellthemtofoldthecutoutsinhalfasmanytimesaspossible.Unfoldtheshapeandcountthelinesofsymmetry.

Studentswillthenrecordtheirfindingsintheirnotebooks.

Shape LinesofSymmetry

rhombus 2

square 4

rectangle 2

parallelogram 0

isoscelestriangle 1

Studentscanfindlinesofsymmetryinlettersalso.

Forexample,

i) Theletter‘Z’hasnolineofsymmetry.

ii) Theletter‘A’hasonelineofsymmetry.

iii) Aparallelogramhasnolineofsymmetry.

Written AssignmentsQuestion6ofExercise18canbedoneasclasswork.

Evaluation

Ashort,simpletestcanbegiventoevaluatelearning.

After completing this chapter, students should be able to:• identifythelineofsymmetry,

• usepropertiesofsymmetricalfigurestoprovetheorems,and

• drawlinesegmentsandanglebisectors.

v871

19 Practical Geometry

Specific Learning ObjectivesInthisunit,studentswilllearn:

• aboutlinesandangles.

• todefinepolygonsandquadrilaterals.

• howto:

– drawparallellinesandtwonon-parallellinesandfindtheanglebetweenthem.

– dividealinesegmentingivenpartsinthegivenratio.

– constructquadrilateralswithvariousconditions.

– constructatriangleandapentagon.

– drawtangentstoacircleatagivenpointonthecircumferenceandfromapointoutsideit.


4to5periods

Prior Knowledge and RevisionStudentslearnedtheconstructionofaperpendicularandanglebisectorsinthepreviousgrade.Itwouldbeadvisabletorevisethestepsontheboardandthenaskthestudentstoconstructtwoofeachtypeintheirnotebooks.Revisionofconstructionoftriangleswithgivenconditionscanalsodone.

Theteachershoulddotheconstructionontheboardusinggeometricinstruments,ascorrecthandlingoftheprotractorandcompassesisimportant.

Real-life Application and ActivitiesThischapterrequiresstudentstodevelopskillsinhandlinggeometricinstrumentsandrememberingthestepsofconstruction.

Oncetheyknowalltheconstructionmethodsaclassactivitycanbedonebydividingstudentsintogroups.Allthedifferentconstructionscanbewrittenonflashcards.EachgroupcanpickacardandstartdoingtheconstructiononA4paper.Thegroupthatattemptsthemostconstructionsinoneperiodwins.

iv88 1Practical Geometry

Summary of Key Facts• Thedistancebetweentwoparallellinesremainsconstantatdifferentpointsthroughtheentirelengthofthelines.Twoparallellinescaneasilybedrawnwithafixeddistancebetweenthem.

• Alinesegmentofanylengthcanbedividedintoanygivennumberofparts.

• Alinesegmentofanylengthinanyratio.

• Theanglebetweentwonon-parallellinescanbefoundwithoutextendingthelines.

• Theanglebetweentwoconverginglinescanbebisectedwithoutproducingthem.

• Propertiesofquadrilaterals:

–Asquarehasallsidesequal;eachangleofasquareequals90°.

–Arhombushasallfoursidesequal.

–Arectanglehastwopairsofoppositesidesparallelandequal,andeachangleis90°.

–Aparallelogramhasoppositesidesparallelandequal.

–Akitehastwopairsofadjacentsidesequal.

–Atrapeziumhasonepairofoppositesidesparallel.

• Aquadrilateralwithtwopairsofoppositesidesparallelandequaliscalledaparallelogram.

• Propertiesofaparallelogram:

–Oppositesidesofaparallelogramareequal. –Oppositeanglesofaparallelogramareequal. –Thediagonalsofaparallelogrambisecteachother.

• Aquadrilateralcanbeconstructedwhen: –thelengthsofitsfoursidesandthemeasureofoneoftheanglesaregiven. –thelengthsofthreeofitssidesandthemeasuresoftwoincludedanglesaregiven. –thelengthsofitsfoursidesandadiagonalaregiven. –thelengthsofthreesidesandtwodiagonalsaregiven.

• Asquarecanbeconstructedwhen: –thelengthofeachofitssideisgiven.

–thediagonalisgiven.

–thedifferencebetweenthelengthofthesideanddiagonalisgiven.

–thesumofitsdiagonalandsideisgiven.

• Arectanglecanbeconstructedwhen: –thelengthoftwoadjacentsidesisgiven.

–thelengthofonesideandadiagonalaregiven.

• Aparallelogramcanbeconstructedwhen: –thelengthsoftheadjacentsidesaregiven,andthemeasureoftheincludedangleisknown.

–thediagonalsandtheanglebetweenthemaregiven.

v891 Practical Geometry

• Arhombuscanbeconstructedwhen: –thelengthofeachsideandthesizeofoneangleisgiven

–thelengthofeachsideandonediagonalaregiven

• Akitecanbeconstructedwhenlengthsoftwosideandadiagonalaregiven.

• Aregularpentagonandhexagoncanbeconstructedwhenthelengthsofonesideisgiven.

• Aright-angledtrianglecanbeconstructedwhenthelengthofthehypotenuseandtheverticalheightfromthevertextothehypotenusearegiven.

Remember:atriangleconstructedonthediameterofacircle,withinasemi-circle,isalsoaright-angledtriangle.

• Thein-circleandcircum-circleofatrianglecanalsobedrawn.

• Acircleisasetofpoints,equidistantfromafixedpointonthesameplane.

• Atangenttoacircleisalinewhichmeetsthecircleatonlyonepoint. –Thedistanceofthetangenttoacirclefromthecentreisequaltotheradiusofthecircle.

–Atangenttoacircleisperpendiculartotheradiusdrawnthroughthepointofcontact.

• Atangenttoacirclecanbedrawn: –atapointgivenonthecircumference.

–fromapointoutsidethecircle.

. ?OOP

S!

Frequently Made Mistakes

Thischapterismainlyskills-based,anddoesnotrequiremuchmathematicalreasoning.However,emphasisshouldbeplacedonneatnessandaccuracy.Pencilsshouldbewell-sharp,andgoodcompassesshouldbeusedwhichareneithertoostiffnortooloose.Studentsaresometimesunabletodrawperfectcircum-circlesandtangents.Theteachershouldsortoutstudents’concernsindividually.

Lesson Plan


Geometricconstructions

Specific Learning ObjectiveRevisionofpropertiesofallshapes


Key vocabularyBisector,Quadrilateral,Polygon,andCircles

iv90 1Practical Geometry

Method and Strategy

Parallellines,bisectors,andthelinesegmentsareafewkeytermsthatthestudentswillbedealingwithinthischapter.Thestudentswillalsoneedtorevisethepropertiesofquadrilaterals(rectangle,square,andrhombus).Sincetheyhavetodoconstruction,theyshouldbeawareoftheseproperties:

Square:allsidesareequalandallanglesareof90°

Rectangle:lengthsandbreadthsareequal;alltheanglesareof90°.

Rhombus:allsidesareequalandtheadjacentanglesaresupplementary.

Thischapternotonlydevelopsthestudents’skillsinconstruction,butalsoencouragesthemtolinkmathematicalproofsandcomputation.Thestudentsshouldbeinvolvedina‘thinkingprocess’whiledividingalinesegmentintoratiosorwhiledrawinganin-circleoracircum-circle.Explainthatin-circleanglebisectorswillgivethecentreoftheangle.Similarly,perpendicularbisectorswillneedtobeconstructedtolocatethecentreofacircum-circle.

Written AssignmentsThislessonisprimarilyarevisionclassofthepropertiesofshapesandspecificconstructioncases.Thestudentsshouldbeencouragedtowriteallthekeypointsintheirnotebooksandmakeatableoffacts.

EvaluationThischapterisaskills-basedchapterwhereabilitytoconstructaclearandpreciseshapeistested.Mathematicalreasoningisalsoinvolvedasthepropertiesareemployedtounderstandthegivenconstruction.Thereshouldberegularassessmentoflearningandunderstandingandtheirabilitytodrawshapesandusemathematicsinstruments.Exercise19constructionscanbegivenasatest.Forexample,questions8,16,17,and20canbegivenasonetest.Thesecondtestcanbeofpolygonsandcircles.

After completing this chapter, students should be able to:• definepolygonsandquadrilaterals,

• drawparallellines,

• dividelinesegmentsintogivenpartsandratios,

• drawangleandperpendicularbisectors,

• constructquadrilateralswithdifferentconditions,

• constructtrianglesandpentagons,and

• drawtangents.

v911

Axioms, Postulates, and Propositions20



• todefinedemonstrativegeometryandreasoning.

• howtodescribeaxiomsandpostulates.

• howtodescribepropositionsortheorems.

• howtoprovetheoremsbasedonlinesandangles.

• howtoprovetheoremsbasedonparallellines.

• howtoprovetheoremsbasedontriangles.


6to8periods

Prior Knowledge and RevisionThestudentshavelearnedaboutthepropertiesofparallelandintersectinglinesandtheanglesformedbythem.Thischapterdealswiththemathematicalreasoningandproofsofthesefacts.Itisimportantthattheteacherfirstdoesabriefrecapofthebasicpropertiesofperpendicular,intersecting,andparallellines.

Thestudentsshouldrecallthattwointersectinglinesformequalverticallyoppositeangles.

Twoparallellinescreatecorrespondingandalternateangles.

Interioranglesarenotequalandaddupto180º.

Real-life Application and ActivitiesThereareninetheoremsinthischapterandeachtheoremrequiresknowledgeofaccuratemathematicalfactsbackedbymathematicalreasoning.

Eachtheoremshouldbedoneontheboardbytheteacherandexplainedindetail.Thestudentsshouldthencopythemintotheirnotebooks.

iv92 1Axioms, Postulates, and Propositions

Activity

Studentsshouldenjoyedthispracticalactivitywhichwillsupporttheirlearning.

Youwillneed:

2cmdiameterbamboosticks,markers,protractors,A4paper,andtape.

Askthestudentstoplacethe2bamboosticksonthepaperintheformofacross,andtolabeltheangleswiththemarker.

Keeponchangingthesizeoftheangles,andaskthemtoprovethattheverticallyoppositeanglesareequalbymeasuringtheangles.

Ifaboardprotractorisnotavailable,thestudentscanmeasuretheanglesbytracingtheanglesmadebythesticksontochartpaperandthenmeasuringthem.

Thisactivitywillenablethestudentstodemonstratetheirgeometricalknowledgewithmathematicalprecision.

Summary of Key Facts• Geometricdemonstrationistheprovingofmathematicalfactswiththehelpoflogicandmeasurement.

• Deductivereasoningisdrawingaconclusiondrawnbasedongivenfacts.

• Anaxiomisafundamentalstatementrelatedtonumbers.

• Apostulateisafundamentalstatementrelatedtoageometricfigure.

• Atheoremorpropositionisbasedonaxiomsandpostulates.

• Acorollaryisastatementthatisdeducedfromatheorem.

• Theoremscanbeprovedbymathematicalandgeometricreasoning.

• Twoperpendicularlinesformtworightangles.

• Twointersectinglineswillformanglesaddingupto180º.

• Twointersectinglinesformtwoequal,verticallyoppositeangles.

• Iftwosidesofatriangleandtheincludedanglesarecongruent,thenthetwotrianglesarecongruent.

• Thetwobaseanglesofanisoscelestriangleareequaltoeachother.

• Theexteriorangleofatriangleisalwaysgreaterthanthesumoftheoppositeinteriorangles.

• Thetwoalternateanglesareequal,thelinesformingtheseanglesareparallel.

• Whentwoparallellinesareintersectedbyatransversal,thealternateanglesformedareequal.

• Thesumofthesizesofallthreeanglesofatriangleaddupto180º.

?OOPS

!

Frequently Made MistakesThenamingoflines,angles,andtrianglesshouldbeaccurate.

Whenwritingmathematicalproofs,geometricdiagramsandreasoningshouldbegivenalotofimportance.

v931 Axioms, Postulates, and Propositions

Lesson Plan


Axiomsandpostulates

Specific Learning ObjectiveToprovethattwointersectinglinesformsupplementaryangles

Suggested Duration1periodKey Vocabulary

Intersecting,Perpendicular,Supplementary

Method and StrategyThestudentsaregivenasetofdatawiththehelpofwhichtheyhavetoproveafact.Thisiscalledmathematicalproof.Derivationsofformulasandprovingtheoremsaredonesimultaneously.Thestudentslearntoapplymathematicalproofssystematically,improvingtheirlogicalthinkingskills.Theteachermustensurethatthestudentsdothederivationinaverysystematicandorganisedmanner.

Thetheoremsgiveninthetextbookshouldbeexplainedindetailintheclassroom.

Theyshouldbeexplainedwithdiagramsontheboard,andputupontheclasssoft-boardforreference.Asafurtherpracticetool,askthestudentstonotethemdownintheirnotebooks.

Itisimportantthattheteacherstressesthestepsofmathematicalproof,labelling,andrecognisingtheangles.Itisalsoimportanttoexplainthereasonsforthemathematicproof.

Example

Iftwostraightlines,ABandCD,intersecteachotheratthepointOandm∠AOD=90°,provethattheremainingangleswillalsoberightangles.

If ∠AOD = 90°

then ∠BOC = 90° (vertically opposite)

∠ BOC + ∠AOC = 180° (angles on a straight line/supplementary)

Since ∠BOC = 90° A B

O

90°

D

C

∴ ∠AOC = 90°

∠AOC = ∠BOD (vertically opposite angle)

Similarly ∠BOD = 90°

∠AOD = ∠BOC = ∠AOC = ∠BOD = 90° (Proved)

∠AOD + ∠BOC + ∠AOC + ∠BOD = 4(90°) = 360°

Whenprovingthatallanglesareequalto90°,itisimportanttomentionsupplementaryandverticallyoppositeanglesasreasons.

iv94 1Axioms, Postulates, and Propositions

Written AssignmentQuestions1to4ofExercise20canbedoneinclass.

EvaluationAcomprehensiveassessmentcanbegiventoevaluatestudents'understandingoftheoremsandtheirproofs.Sincethisisnewterritory,itwillbeadvisabletogivetests.Thefirsttestcancoverthefirstfourtheoremsandthesecondcancovertheorems5to9.

After completing this chapter, students should be able to:

• definedemonstrativegeometryandreasoning,

• defineaxiomandpostulate,

• definepropositionandpostulate,

• provetheoremsrelatingtolinesandangles,

• provetheoremsrelatedtoparallellines,and

• provetheoremsrelatedtotriangles.

v951

21 Area and Volume21



• howtouseHero'sformulatofindtheareaofatriangleintermsofitssides.

• howtofindvolumeandsurfaceareaofarightcircularcone.

• howtofindthevolumeandsurfaceareaofasphere.


4to5periods.

Prior Knowledge and RevisionInearlierclassesstudentshavelearntabouttheareaofshapes,volume,andsurfaceareaofcubes,cuboids,andcylinders.Itwillbeimportanttorevisealltheformulasofeachshapeontheboard.Aquick’recallquizcanbeconductedandthewinninggroupcanbeawardedmarksthatcanbeaddedtotheiraggregatemarksasabonus.

Real-life Application and ActivitiesThestudentsarefamiliarwiththeconceptsofvolumeandsurfacearea.Theteachershouldrevisetheconceptsandexplainthelinkbetween2Dand3Dfigures.Itisimportanttore-teachtheconceptofheight/depthtodifferentiatebetween2Dand3Dfigures.

Cubesandcuboidswerestudiedingrade7;theformulascouldberevisedandthestudentscouldbegivenaquiz.Thiswillhelptheteachertoassesswhetherornottomoveonwiththetopic.

Inthischapter,cylinders,cones,andsphereshavebeendiscussed.Beforetheformulasareintroduced,itisimperativethattheteacherexplainsthedimensionsofeachshape.

Cylinder:thisshapehasradiusandheight.

Volume=πr2h

Curvedsurfacearea=2πrh+πr2 + πr2

Totalsurfacearea=2πrh+2πr2

iv96 1Area and Volume

Cone:thisshapehasaradiusandheight,butalsoslantheight‘l’.Volume:13 πr2h

Curvedsurfacearea=πrlTotalsurfacearea=πrl + πr2

Sphere:thisshapehasaradiusandacurvedsurface.

Volume: 43

πr3

Surfacearea:4πr2

Ifthestudentsunderstandtheformulasandcanidentifythedimensions,itshouldnotposeaproblemforthem.Continuousrevisionofformulasisthereforeessentials.

Activity

Thestudentswillunderstandbetteriftheshapesareexplainedwiththehelpofnetdiagrams.

Tomakenetdiagramsyouwillneed:A4paper,geometryinstrumentsandmarkers.

Cylinder:thisshapeisformedbyfoldingarectangleintoaroll.Thebreadthoftherectanglebecomestheheightofthecylinder,andthelength,thecircumferenceofitsbase.

Netdiagramofacylinder

‘a’istheheightofthecylinder.

‘b’isthecircumferenceofthebaseofthecylinder.

Thisactivitycanbeusedasarecallinordertounderstandothershapesbetter.

a

b

Netdiagramofcone

‘a’isthecircumferenceofthebaseofthecone.

‘b’istheslantheightofthecone.

b

a

a

Cones:thisshapeisformedbyfoldingasemicircleoranyfractionalpartofacircle(sector),andacircleasitsbase.

Thestudentsshouldbeencouragedtofirstcutoutthe2Dfigureandthenfoldittoformthe3Dfigure.

v971 Area and Volume

Summary of Key Facts• Hero’sformulacalculatestheareaofatrianglebyfirstfindingthevalueof‘s’,whichisthesumofallthreesidesdividedbytwo.Thevalueof's'istheninsertedintheformula:

A=√s(s − a)(s − b)(s − c)

wheres is thesemi-perimeterofatriangle:

s = a + b + c2

.

Thisformulaisappliedwhentheperpendiculardistanceisnotgiven.• Thesurfaceareaofaconehastwoparts,thecurvedsurfaceareaandthetotalsurfacearea.

Areaofthecircularbase=πr2

Totalsurfaceareaofarightcircularcone=Areaofthecurvedsurface+Areaofthebase

= πrl + πr2

= πr(l + r)

Volumeofaconeisaderivativeofthecylinder;itisonethirdthevolumeofacylinder.

• Thevolumeofaright-circularcone = 13areaofbase×height

= 13

πr2h

• Surfaceareaofasphere=4πr2,whereristheradiusofthesphere.

• Volumeofasphere= 43

πr3,whereristheradiusofthesphere.

Lesson Plan

Sample Lesson Plan

Topic

Area

Specific Learning ObjectiveFindingtheareaofshadedregionsofcompositeshapeandHero’sformula

Suggested Duration

Oneperiod

Key Vocabulary Hero’sformula,Perpendicularheight

Method and StrategyPerpendicularheightsareslightlydifficultforstudentstoidentifyastheyaresometimedrawnoutsidethetriangle.Similarly,areasofshadedregionsarefoundbyfirstfindingthetotalarea;then,theunshadedareaisfound,andsubtractedfromthetotalarea.

iv98 1Area and Volume

Areaoftheshadedregion=Areaoftriangle–Areaofparallelogram

3 cm

6 cm

10 cm

5 cm

= 10 6× ×21b l – (5 × 3)

= 30 – 15= 15 cm2

Thischapterisanextensionoftheworkalreadytaughtingrade7.Theteachernowhastodevelopthestudents’abilitytovisualisewordproblemsandthensketchtheminordertoworkoutthevaluesandformulas.Justasinwordproblems,datarepresentationisthekeytofindingthesolution.Similarly,diagrammaticrepresentationisveryimportantinchaptersrelatedtoareaandvolume.

Studentsshouldunderstandthatacircle’srevolutioncovers,linearly,adistanceequaltoitscircumference:

onerevolution=circumferenceofthecircle.

revolution.

circumference

Withthisactivity-basedmethod,thewholeclasswillparticipateenthusiasticallyandtheleast-ablestudentswillunderstandtheconceptsbetter.

AlternatelytheuseofHero’sformulashouldbeencouragedwhentheperpendicularheightisnotgiven.becausetheydonotneedtousePythagoras'theoremtofindit.Allthethreesidesareaddedandthevalueof‘s’calculatedandsubstitutedintheformula.

Written AssignmentsQuestion3ofExercise21canbedoneintheirnotebooks.

EvaluationQuestions8to13ofExercise21canbedoneinclasswiththeteacherandsimilarsumscanbegivenasatest.Questions1to8ofExercise21canbegivenasarevisiontest.Theteachershouldmakesuretheconceptsarerevisedandthenthesumsaregivenasamarkedassignment.

After completing this chapter, students should be able to:• useHero’sformulatocalculatetheareaofatriangle,

• findthesurfaceareaandvolumeofacone,and

• findthesurfaceareaandvolumeofasphere.

v991

22 Trigonometry


Inthisunit,thestudentswilllearn:

• howtocalculatethesidesofaright-angledtriangleusingPythagoras'theorem.

• howtodefinethetrignometricratiosofanacuteangle.

• howtofindthetrignometricratiosof30°,45°,and60°.

• howtofindthetrignometricratiosofcomplementaryangles.

• howtosolvearight-angledtriangleusingthemeasuresofoneangleandlengthofoneside.

• howtosolvearight-angledtriangleusingthemeasuresoftwosides.

• howtosolvereal-lifeproblemsrelatedtotrignometry.


6to8periods

Prior Knowledge and RevisionStudentsknowaboutright-angledtrianglesandtheirproperties.Theyknowthattheremainingtwoanglesarecomplementary.

Real-life Application and ActivitiesPythagoras’Theoremcanbeprovedpracticallythroughanactivity.

Activity

YouwillneedA4paper,ruler,markers,gluestick,andpencils.

Theactivityonpage252ofthetextbookcanbecarriedoutinclass.Askthestudentstobringthelistedmaterialtothelesson.

Thestudentscanmeasureandcutouttheshapeandglueitintheirnotebooks.

Themathematicalproofcanbedoneonthecut-outsandthefinaltheoremwrittenonthebaseofthecut-outintheirnotebooks.

iv100 1Trigonometry

Activity

Similarlyoncethetrigonometricratioshavebeendonethoroughly,examples4,5,and6onpages254and255canbedonepractically.

Forexample4,ifaladderisavailable,inschoolitcanbeproppedupoutsidetheclassroomandthestudentscangooutandinvestigate.Alternately,ifaladderisnotavailable,thestudentscanstandunderastaircaseandthesamesumcanbevisualisedanddone.

Forexample5,iftherearetwobuildingsintheschool,thestudentscanbetakenoutsideandtheactivitycarriedout.However,itwouldbeadvisabletoasktheschoolcaretakertohelpbyusingaropeandchalktodrawright-angledtrianglemarkingsonthefield.

Fortheactivityinexample6,theteacherwillberequiredtobringasmallsuitcaseoranycuboidboxwilldo.Aright-angledtrianglecanbemarkedusingawhiteboardmarker.Thestudentscanmeasurethedimensionsofthesuitcaseandcarryouttheworkingintheirnotebooks.

Summary of Key Facts• Pythagoras'theoremstatesthatinaright-angledtriangle,thesquareofthehypotenuseisequaltothesumofthesquaresoftheothertwosides.

c 2 = a2 + b2

• Theratiosofaright-angledtrianglehavebeenderivedandassignednamesasfollows:

sinθ = perpendicularhypotenuse

= ac (Ratio

BCAB

)

cosθ = basehypotenuse

= bc (Ratio

ACAB

) a

b

c

θ

B

C A

tanθ = perpendicularbase

= ab (Ratio

BCAC

)

• Thetrigonometricratiosof30ºanglesareasfollows:

sin30°= 12

cos30°=√32

tan30°= 1√3

v1011 Trigonometry


sin60°=√32

cos60°= 12

tan60°=√31

= √3


sin45°= 1√2

cos45°= 1√2

tan45°=1

• Thetrigonometricratiosofcomplementaryanglesareasfollows:

sin(90°− θ)=cosθ.

cos(90°–θ)=sinθ.

tan(90°–θ)=cotθ.

?OOPS

!

Frequently Made MistakesEmphasisethattheidentificationofthedimensionsareveryimportant.Thepositionoftheangledeterminestheperpendicularandthebase.Theperpendicularisalwaysoppositetotheangleandthebaseisadjacenttotheangle.Thehypotenuseisthelongestside,oppositetherightangle.


Trigonometry

Specific Learning ObjectiveReal-lifewordproblemsusingtrigonometricratios

Suggested Duration1to2periods

Key VocabularySin,Cos,Tan,Hypotenuse,Perpendicular,Base

Method and StrategyQuestions4,7,8,9,10,11,and13ofExercise22arebasedonreal-lifeapplications.

Eachsumcanbevisualisedanddiscussed.Studentscangiveexamplesofplacesinschoolandoutsidetheirhomes.

iv102 1

Ifinternetisavailable,real-lifevideosofplacesandobjectscanbeshownorfiveminutePowerPointpresentationbytheteacherwouldhelpstudents'understanding.

Oncevisualisedthestudentscanthendrawthediagramsofthewordproblemsandapplythetrigonometricratio.

Written AssignmentsWheneachsumhasbeendiscussedindetail,studentsshoulddothecalculationsintheirnotebooks.

EvaluationAcomprehensiveassessmentshouldbecompleted.Timecanbeassignedtodoanycorrectionsasthisisalsoalearningprocess.

After completing this chapter, students should be able to:• calculatethesidesofaright-angledtriangleusingPythagoras'theorem,

• definethetrignometricratiosofanacuteangle,

• findthetrignometricratiosof30°,45°,and60°angles,

• findthetrignometricratiosofcomplementaryangles,

• solvearight-angledtriangleusingthesizeofoneangleandlengthofoneside,

• solvearight-angledtriangleusingthelengthsoftwosides,and

• solvereal-lifeproblemsrelatedtotrignometry.

v1031

23Information Handling


Inthisunit,thestudentswilllearn:

• howtodefineprimaryandsecondarydata.

• aboutvariousmethodsofcollectingprimarydata.

• howtoclassifyandtabulatedata.

• aboutrangeandclassinterval.

• howtopresentdatagraphically:bargraphsandhistograms.

• howtofindmean,weightmean,median,andmode.


4to5periods

Prior Knowledge and Revision Studentsarefamiliarwiththisstrandofmathematics.Statisticaldatacanberepresentedinvariousways:pictograms,linegraphs,piecharts,andbargraphs.

Aquickreviewofthisrepresentationcanbedoneontheboardwhereafrequencydistributiontableisgivenandthestudentsareaskedtorepresentthedatainanytwowaysoftheirchoice.

Real-life Application and ActivitiesActivity

Onpage265thereisalistofquestionnairesforstatisticaldatacollectionandinvestigation.

Forexample,studentscanbegivenapassagetoreadandcounteachofthevowelsa, e, i, o, u.

Theycanthenprepareaquestionnaireandcollectthedatafromtheirfriends.

Oncethedataiscollected,thenextstepistoclassifyandtabulate.

Thestudentscanbetaughtthe‘x’and‘y’valuesasthesubjectanditsfrequencyrespectively.Thelaststageisthepresentationwhichcanbedonebothasabargraphandahistogram.

Thisactivitytakesaweekandattheendtheirpresentationscanbeputuponthedisplayboard.

iv104 1Information Handling

Thestudentshavelearntaboutstatistics.Itisbasedonthecollection,organisation,andrepresentationofdata.Theteachershouldgivethemarevisionworksheetofbargraphswheretheyinterpret,readanddrawbargraphs.

Inthischapter,notonlyarebargraphsrevised,buthistogramsarealsointroduced.Interestingly,histogramsofgroupeddataarealsointroduced.Classintervalsorclasswidths,frequency,andfrequencydistributionaretermsthatthestudentswillcomeacrossinthischapter.

Whenintroducinghistograms,theteachershoulddifferentiateclearlybetweenbargraphsandhistograms.

Thedrawingofhistogramswillalsorequireclearinstructions.Therearenogapsbetweenthebarsofthehistograms,andthescalechosenwillhavetobeaccuratelyrepresentedonthegraph.

Explainthatthe‘x’valueissometimesgivenintheformofgroups,andthatthevalueswillbewrittenonthegraphonthesidesofeachbarratherthanthemiddle(asisdoneinbargraphs.)

x f

50–6060–7070–80

573

A grouped data histogram

7654321

50 60 70 80

f

x'Averages'progressesto‘mean’atthislevel,asfrequencyhastobeindividuallymultipliedbythevalueof‘x’togetthe‘fx’product.

Example

2,2,2,3,3,5,6,6,6

Mean = ∑ ffx = 9

(2)(3) (3)(2) (5)(1) 6(3)+ + + = 359 = 3.88 or 3.9

Activity

Thischaptercouldbeusedasaprojecttopicwherethestudentscanbeencouragedtocollectdataandthenrepresentitonahistogram.Thedatacollectedcouldbe:

• numberofpeopleinvestinginfourorfivetypesofshares.

• battingstatisticsoffourorfivefamousbatsmen.

• ageinyears.

• heightsorweightsofstudentsinclass.

v1051 Information Handling

Summary of Key Facts• Rawdataisthefirstorinitialcollectionofdatathathasnotbeenorganisedortabulated.

• Classificationistheprocessofarrangingdataingroupsorclasses.

• Afrequencydistributiontablerepresentsasetofclassintervalsandtheircorrespondingfrequencies.

• Continuousfrequencydistributionisrepresentedbyhistograms.Therearenointervalsorgapsonthe‘x’axis.

• Themeasuresofcentraltendencyisthesinglevalueofthedatathatgivesinformationaboutthecentralvalueofthedata.

• Therearethreetypesofaverages:mean,median,andmode.

• Themeanistheaverageofgroupedorweighteddata.

• Themedianisthemiddlevalueofthedata.

• Themodeisthemostfrequentlyoccurringdatavalue.

?OOPS

!

Frequently Made MistakesStudentsgenerallymakemistakesindifferentiatingbetweenbargraphsandhistogramswherethefirsttypehasintervalsorgaps.Thisisarelativelysimplechapterandstudentstendtoenjoythedatapresentation.

Lesson Plan

Sample Lesson Plan

Topic

Informationhandling

Specific Learning ObjectiveModeandmedian


Key VocabularyMedian,Mode,Frequency

Method and Strategy

Thestudentsshouldunderstandtheconceptofthemean.

Theteachershouldthenproceedtoexplaintheothermeasuresofcentraltendency.

Beforeexplainingmodeandmedian,theteachershouldexplainthemostimportantruleofdatacollection:toorganisethegivendatainascendingorder.

Ifitemsofthedatahavetobegroupedortabulated,thefirstruleistoarrangethemfromthesmallesttothegreatestvalue.

Oncethisisdone,themodewillbethevaluethatoccursmostfrequentlyinagroupofdata.Therefore,the‘x’valuewiththehighestfrequencyisthemode.

iv106 1

Themedianisthemiddlevalue,butifthefrequencyiseven-numberedtwomiddlevaluesareselected.Ifthefrequencyisodd-numberedthenthereisonlyonemiddle‘x’value.Thisconceptcanbedemonstratedwithasmallactivityinclass.

Activity

Lineuptenstudentsaccordingtotheirheightsinascendingorder.Theteacherwillpointoutthatthemeasureoftheirheightsaretheir‘x’value.She/hewillthenremoveonestudentatatimefromtherightandthelefthandside.Intheend,twomiddleheightstudentswillbeleft.Theaverageoftheirheightswillbethemedian.

Similarly,theteacherwilladdontheeleventhstudentandtheprocessrepeated.Thestudentswillseethatonlyonestudentwillbeleftinthemiddle.Themiddlevalueisthemedian,withequalnumberofvaluesoneithersides.

Written AssignmentsQuestions14,15,and16ofExercise23canbedoneinclassbythestudentsintheirnotebooks.

EvaluationSincethischapterisallaboutpresentationandtabulation,thestudentscanbemarkedontheirclassworkandhomeworkassignments.Theyshouldbeassessedontheaccuracyoftheirpresentationanddrawings.

Atestcomprisingmultiple-choicequestions,trueandfalse,andconcept-basedquestionscanbegivenattheendofthechapter.

After completing this chapter, students should be able to:• defineprimaryandsecondarydata,

• usevariousmethodsofcollectingprimarydata,

• classifyandtabulatedata,

• findrangeandclassinterval,

• presentdatagraphically:bargraphsandhistograms,and

• findthemean,weightmean,median,andmode.

Information Handling

v1071

Ateacher'sjourneyinvolvesthreestagesExposition,Practice,andConsolidation.

Expositionisthesettingforthofcontent,andthequalityandextentoftheinformationrelayed.

Practiceinvolvesproblemsolving,reasoningandproof,communication,representations,andcorrection.

Assessmentisthefinalstageofconsolidationoftheprocessoflearning.Assessmentofteachingmeanstakingameasureofiteffectiveness. Formativeassessmentismeasurementforthepurposeofimprovingit. Summativeassessmentiswhatwenormallycallevaluation.

Anidealandfairevaluationinvolvesaplanthatiscomprehensive.Itcoversabroadspectrumofallaspectsofmathematics.Theassessmentpapersshouldtestallaspectsoftopicsthought.Thesecanbedemarcatedintocategories:basic,intermediate,andadvancedcontent.Theadvancedcontentshouldbeminimalasitteststhemostablestudentsonly.

Multiplechoicequestions,alsoknownasfixedchoiceorselectedresponseitems,requiredstudentstoidentifiedthecorrectanswerfromagivensetofpossibleoptions.

Structuredquestionsassessvariousaspectsofstudents'understanding:knowledgeofcontentandvocabulary,reasoningskills,andmathematicalproofs.

Allinalltheteaching'sassessmentofstudents'abilitymustbebasedonclassroomactivity,informalassessment,andfinalevaluationattheendofatopicand/ortheyear.

Assessment

iv108 1Assessment

Specimen PaperMathematics

Grade 8

Section ATime: 1 Hour Total Marks: 40

Q1. If A = {1, 2, 3, 4}, then the proper subset of A is

A. 0 B. {0 }

C. {2,3}

D. {1, 2, 3, 4}

Q2. Which of the following is a set of composite numbers.

A. {2, 4, 6}

B. {2, 3, 5}

C. {0, 1, 2}

D. {16, 25, 36}

Q3. (A ∩ B) ∪ C′

A B

C

A B A B

C

C D

A B A

C

B

C

Q4.A B

C

1

2

5 6

8 10

3

4

7

What is the value of A ∩ B ∩ C ?

A. {4, 5, 3, 6}

B. {4, 3}

C. {5, 4, 6}

D. {4}

Q5. Which of the following is correct?

A. > 1 43

1 53

B. ≥ 1 43

1 53

C. > 1 53

1 43

D. ≤ 1 53

1 33

Q6. We can write base 10 as

A. 1310

B. 1310

C. 1013

D. 1013

v1091 Assessment

Q7. 12 + 1

2

A. 102

B. 22

C. 1002

D. 210

Q8. A machine that allows people to withdraw cash from their account without going to the bank.

A. Debit card

B. Automated Teller Machine

C. Cheque

D. Automatic Cash Machine

Q9. A written instruction from the account holder to the bank for cash is a

A. cheque

B. pay order

C. demand draft

D. credit card

Q10. A written agreement by which a renter can use property on rent for a specific period is called

A. over draft.

B. running finance.

C. demand finance.

D. leasing.

Q11. If CP = Rs 100 and SP = Rs 90, then

which of the following statement is correct?

A. Profit of 10%

B. Loss of 10%

C. Loss of 90%

D. Profit of 90%

Q12. The polynomial 8x2y3 + 4x2y3 + xy2 + x2

has ‘n’ number of terms. Find the value of n.

A. n = 9

B. n = 4

C. n = 6

D. n = 5

Q13. In the polynomial 3x4 + 5x2 + 7x + 8 the

highest degree is

A. 1

B. 2

C. 3

D. 4

Q14. 2a + 8 = 7

A. – 1 2

B. 1 2

C. 2

D. – 2

Q15. (105)2 is equal to

A. (100)2 + 2(100) (25) + (5)2

B. (100)2 + 2(100) (5) + (5)2

C. (100)2 + 2(10) (25) + (5)2

D. (100)2 + 2(10) (5) + (5)2

Q16. Ali is 3 years older than Sara. In two years time his age will be: (take Ali's age as x and Sara's age as y)

A. x = y + 5 B. x – 2 = y C. x + 2 = y + 3D. x – y = 3

iv110 1Assessment

Q17. If 2x + y = 11 and x – y = 10, then x is equal to

A. – 7

B. 7

C. – 1 3

D. 1 3

Q18. In the simultaneous equations 4x + y = 7

and 5x + 2y = 12, the value of x is

A. 2 3

B. 1 3

C. 3 2

D. 0

Q19. If 4z = x and 4wz = v, then elimination of ‘z’ by substitution method gives

A. W = v x

B. W = x v

C. V = 4w

D. V = 16wz2

Q20. A

C B D

A. AB is parallel to CD.

B. AB is perpendicular CD.

C. AB is vertically opposite to CD.

D. AB and CD are not perpendicular lines.

Q21. Sum of the interior angles of a regular pentagon is

A. 90º

B. 108º

C. 540º

D. 180º

Q22.

ad

f cgh

ebA B

C D

If AB is parallel to CD, then which of the following statement is correct? A. ∠ a = ∠ e

B. ∠ d = ∠ b

C. ∠ b = ∠ g

D. ∠ h = ∠ f

Q23. Which of the following statement is correct?

A. Any enclosed shape is a polygon.

B. Circles and squares are polygons.

C. All polygons are shapes with more than or equal to 3 sides.

D. A triangle is not a polygon.

Q24. ABCD is a parallelogram. Which of the two angles are equal?

1

3

2

4A

DC

B

A. ∠1 and ∠2 B. ∠3 and ∠4 C. ∠1 and ∠3 D. ∠1 and ∠4

v1111 Assessment

Q25.

P Q

R

S

L

T

M

O

Which of the following is a chord of a circle?

A. OR

B. LM

C. ST

D. PQ

Q26. C

25x

A 20 B

The value of x in the above right-angled triangle is

A. 10 B. 15 C. √45 D. 16

Q27. The area of Δ PQR is

A. 25 cm2

B. 15 cm2

C. 30 cm2

D. 10 cm2

Q28. The surface area of a sphere with radius 7 cm is

A. 600 cm2

B. 616 cm2

C. 88 cm2

D. 636 cm2

Q29. A fundamental statement related to geometrical figure is

A. a theorem.

B. a postulate.

C. an arc.

D. a corollary.

Q30. Volume of a cone is equal to

A. πr (r + ℓ )

B. 1 3 πr 2h

C. 4 3 πr3

D. 4πr2

Q31.

10 cm 5 cm

3 cm

The volume of the given cone will be

A. 12π

B. 40π

C. 50π

D. 150π

Q32. 'Every even number is divisible by 2'.

The given statement represents:

A. a corollary.

B. an axiom.

C. a postulate.

D. a theorem.

R

P Q

10 cm5 cm

iv112 1Assessment

Q33. An axiom is the type of assumptions which is related to

A. numbers.

B. geometrical figures.

C. corollary.

D. angles..

Q34. Cos 30º equals to

A. 1 2

B. √31

C. √32

D. 2 1

Q35. Which of the following has value 1?

A. Sin 45º

B. Cos 45º

C. Tan 45º

D. Sec 45º

Q36. 2Sin 30º + √2Cos 45º

A. √22

B. 2

C. √21

D. 1

Q37. Sin (90º – θ) equals to

A. Sin θ

B. Cos θ

C. 1/tan θ

D. Tan θ

Q38. 19, 21, 20, 18, 23, 19, 20, 18, 19, 20, 19, 0

The total frequency of the following data is

A. 20

B. 19

C. 11

D. 12

Q39. Mode of 2, 4, 6, 8, 7, 8, 9, 10, 13 is

A. 8

B. 9

C. 10

D. 13

Q40. √2 is

A. a rational number.

B. a whole number.

C. an irrational number.

D. an odd number.

v1131 Assessment

Section B

Time: 2 Hours Total Marks: 60

1. 𝕌 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,} [6]

A = {1, 2, 3, 4, 6,}

B = { 2, 4, 5, 7,8}

Prove that (A∪B)′ = A′∩B′.

2. 𝕌 = {x | x ∊ w and | 10 ≤ x ≤ 30} [6]

A = {x| z ∊ w and | 12 ≤ x ≤ 25}

B = {x| z ∊ w and | 14 ≤ x ≤ 27}

Prove that (A∩B)′ = A′∪B′.

3. Find the value of

(i) 5123

[3]

(ii) ∙ ∙ 1 2

–3

[3]

4. Salman's salary is Rs 38000. If his relief is Rs 8000, then calculate his income tax at the rate of 5% per annum. [6]

5. (i) A fruit seller sold 2 dozen oranges and 3 dozen bananas on Monday. On Tuesday he sold 4 dozen oranges and 1 dozen bananas . How many oranges and bananas did he sell in two days? [4]

(ii) What is 3 7∙ ∙ 6

– 4 – –3

7∙ ∙0 4 ? [2]

iv114 1Assessment

6. Sara and Ali together have Rs 1500. If Sara spends twice as much as Ali, how much does Ali spend? [6]

7. Construct a right-angled triangle PQR, where ∠Q = 90º, QR = 4 cm and hypotenuse

PR = 5 cm. Find the measure of QP. Also write the steps of construction. [6]

8. (i) Write 33 = 27 in logarithmic form. [1]

(ii) Write log4 256 = 4 in exponential form. [1]

(iii) Find the value of x, if log8 x = 2. [2]

(iv) Find the value of a, if loga 729 = 3. [2]

9. A ship is 150 m away from a lighthouse. If the angle between the top of the lighthouse and base of the ship is 45º, find the height of the lighthouse. [6]

10. Draw a histogram of the following data. [6]

0 < x ≤ 10 7

10 < x ≤ 20 10

20 < x ≤ 30 5

30 < x ≤ 40 2

40 < x ≤ 50 8

v1151 Assessment

Marking SchemeMarking criteria for Section A: 1 mark for each correct answer.

1.C 2.D 3.A 4.D 5.A 6.A 7.A 8.B

9.A 10.D 11.B 12.B 13.D 14.A 15.B 16.A

17.B 18.A 19.A 20.B 21.C 22.C 23.C 24.D

25.D 26.B 27.A 28.B 29.B 30.B 31.A 32.A

33.A 34.C 35.C 36.B 37.B 38.D 39.A 40.C

Marking criteria for Section B:

Q.1 6 Marks Answer

• ForfindingA∪B

• Forfinding(A∪B)′

• ForfindingA′

• ForfindingB′

• ForfindingA′∩ B′

• Forcorrectproof

1mark

1mark

1mark

1mark

1mark

1mark

A∪B={1,2,3,4,5,6,7,8}

(A∪B)′={9,10}

A′ ={5,7,8,9,10}

B′ ={1,3,6,9,10}

A′∩ B′ ={9,10}

(A∪B)′ =A′∩ B′

Q.2 6 Marks Answer

• Forfindingcorrectvaluesof𝕌, AandB

• ForfindingA∩B

• Forfinding(A∩B)′

• ForfindingA′

• ForfindingB′

• ForfindingA′∪B′

• Proved

1mark

1mark

1mark

1mark

1mark

1mark

𝕌={10,11,12,.........,30}A={12,13,14,.........,25}

B={14,15,16,.........,27}

A∩B={14,15,16,.........,25}

(A∩B)′ ={10,11,12,13,26,27,28,29,30}

A′ ={10,11,26,27,28,29,30}

B′ ={10,11,12,13,28,29,30}

A′∪B′ ={10,11,12,13,26,27,28.29,30}

(A∩B)′ =A′∩ B′

Q.3 6 Marks Answer

(i)

(ii)

• Forcorrectprimefactorisation

• Forcorrectgrouping

• Foraccurateanswer

• Forapplyingcorrectlawofindices


1mark

1mark

1mark

2marks

1mark

512=2×2×2×2×2×2×2×2×2

2×2×2

8

43

64

Q.4 6 Marks Answer

• Forfindingtheamountonwhichtaxistobepaid

• Forfindingtaxat5%perannum


3marks

2marks

1mark

38,000–8000=30,000

5%of30,000

Rs1500

iv116 1Assessment

Q.5 6 Marks Answer

(i)

(ii)

• Forwritinginformationinformofamatrix

• Foraddition

• Foraccuracy

• Forcorrectsubtraction

• Foraccuracy

2marks

1mark

1mark

1mark

1mark

Oranges Bananas

Monday=[2×12 3×12]

Tuesday=[4×12 1×12]

Monday=[2436]andTuesday=[4812]

[24 36]+[48 12]=[24+48 36+12]

72Oranges 48Bananas

6–0 3+3–4–4 7–7∙ ∙

=

66–80∙ ∙

Q.6 6 Marks Answer

• Forcorrectstatement

• Forsolution

• Foraccuracy

2marks

2marks

2marks

2x + x=1500

Rs500

Q.7 6Marks Answer

• Foraccuratedrawing

• ForfindingQP

• Forwritingstepsofconstruction

3marks

1mark

2marks

QP=3cm

Q.8 6Marks Answer

(i)

(ii)

(iii)

(iv)

• Forcorrectlogarithmicform

• Forcorrectexponentialform

• Forcorrectexponentialform


• Forwritinginexponential


1mark

1mark

1mark

1mark

1mark

1mark

log327=3

44

82 = x∴ x = 64

a3=729

a3=93 ∴ a = 9

Q.9 6 Marks Answer

• Forusingcorrecttrigonometricratio

• Forapplyingcorrectvalues

• Foraccuracy

2marks

1mark

3marks

Tanθ = Perpendicular Base

Tan45º= Height 150

Height= Tan45º×150Height= 1×150=150m

Q.10 6 Marks Answer

• Forchoosingcorrectscaleonbothaxes.

• Forplottingvalues

• Foraccuratedrawingofthehistogram

1mark

2marks

3marks

P

Ship

Lighthouse

150m30º

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