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COUNTDOWNSecond Edition

Teaching Guide

1

7

viii1

Introduction iv

Curriculum 1

• StrandsandBenchmarks

• SyllabusMatchingGrid

Teaching and Learning 11

• GuidingPrinciples

• MathematicalPractices

• LessonPlanning

• FeaturesoftheTeachingGuide

• SampleLessonPlans

Assessment 84

• 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,andeffectsofoperationsinvarioussituations;

• computefluentlywithfractions,decimals,andpercentages,and

• manipulatedifferenttypesofsequencesandapplyoperationsonmatrices.

Benchmarks

Grades VI, VII, VIII

• Identifydifferenttypesofsetswithnotations

• Verifycommutative,associative,distributive,andDeMorgan’slawswithrespecttounionandintersectionofsetsandillustratethemthroughVenndiagrams

• Identifyandcompareintegers,rational,andirrationalnumbers

• Applybasicoperationsonintegersandrationalnumbersandverifycommutative,associative,anddistributiveproperties

• Arrangeabsolutevaluesofintegersinascendinganddescendingorder

• FindHCFandLCMoftwoormorenumbersusingdivisionandprimefactorization

• 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


• 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: Sets1.1 Set

i) Expressasetin • descriptiveform, • setbuilderform, • tabularform.

Chapter1

1.2 Operations on Sets i) Defineunion,intersectionanddifferenceoftwosets.ii) Find • unionoftwoormoresets, • intersectionoftwoormoresets, • differenceoftwosets.iii) Defineandidentifydisjointandoverlappingsets.iv) Defineauniversalsetandcomplementofaset.v) Verifydifferentpropertiesinvolvingunionofsets,intersectionofsets,

differenceofsetsandcomplementofaset,e.g.,A ∩ A′ = ø.

Chapter1

1.3 Venn Diagram i) RepresentsetsthroughVenndiagram.ii) Performoperationsofunion,intersection,differenceandcomplementon

twosetsAandBwhen • AissubsetofB, • BissubsetofA, • AandBaredisjointsets, • AandBareoverlappingsets, throughVenndiagram.

Chapter1

Unit 2: Rational Numbers2.1 Rational Numbers i) Definearationalnumberasanumberthatcanbeexpressedintheform

pq ,

wherepandqareintegersandq > 0.ii) Representrationalnumbersonnumberline.

Chapter2

2.2 Operations on Rational Numbers i) Addtwoormorerationalnumbers.ii) Subtractarationalnumberfromanother.iii) Findadditiveinverseofarationalnumber.iv) Multiplytwoormorerationalnumbers.v) Dividearationalnumberbyanon-zerorationalnumber.vi) Findmultiplicativeinverseofarationalnumber.vii) Findreciprocalofarationalnumber.

Curriculum

iv6 1

viii)Verifycommutativepropertyofrationalnumberswithrespecttoadditionandmultiplication.

ix) Verifyassociativepropertyofrationalnumberswithrespecttoadditionandmultiplication.

x) Verifydistributivepropertyofrationalnumberswithrespecttomultiplicationoveraddition/subtraction.

xi) Comparetworationalnumbers.xii) Arrangerationalnumbersinascendingordescendingorder.

Chapter2

Unit 3: Decimals3.1 Conversion of Decimals to Rational Numbers

Convertdecimalstorationalnumbers.

Chapter3

3.2 Terminating and Non- terminating Decimals

i) Defineterminatingdecimalsasdecimalshavingafinitenumberofdigitsafterthedecimalpoint.

ii) Definerecurringdecimalsasnon-terminatingdecimalsinwhichasingledigitorablockofdigitsrepeatsitselfinfinitenumberoftimesafterdecimalpoint

iii) Usethefollowingruletofindwhetheragivenrationalnumberisterminatingornot.

Rule:Ifthedenominatorofarationalnumberinstandardformhasnoprimefactorotherthan2,5or2and5,thenandonlythentherationalnumberisaterminatingdecimal.

iv) Expressagivenrationalnumberasadecimalandindicatewhetheritisterminatingorrecurring.

3.3 Approximate Value

Getanapproximatevalueofanumber,calledroundingoff,toadesirednumberofdecimalplaces.

Chapter3

Unit 4: Exponents4.1 Exponents/Indices

Identifybase,exponentandvalue.

Chapter5

4.2 Laws of Exponents/Indices

i) Userationalnumberstodeducelawsofexponents. • ProductLaw: whenbasesaresamebutexponentsaredifferent: am × an = am+n, whenbasesaredifferentbutexponentsaresame: an × bn=(ab)n, • QuotientLaw: whenbasesaresamebutexponentsaredifferent: am ÷ an = am–n, whenbasesaredifferentbutexponentsaresame: an ÷ bn = ∙ a

b ∙n,

Curriculum

v71

• Powerlaw:(am)n = amn. • Forzeroexponent:a0=1. • ForexponentasnegatIvemteger:a–m =

1am

ii) Demonstratetheconceptofpowerofintegerthatis(–a)”whennisevenoroddinteger.

iii) Applylawsofexponentstoevaluateexpressions.

Chapter5

Unit 5: Square Root of Positive Number5.1 Perfect Squares i) Defineaperfectsquare.ii) Testwhetheranumberisaperfectsquareornot.iii) Identifyandapplythefollowingpropertiesofperfectsquareofanumber. • Thesquareofanevennumberiseven. • Thesquareofanoddnumberisodd. • Thesquareofaproperfractionislessthanitself. • ThesquareofadecimallessthanIissmallerthanthedecimal.

Chapter4

5.2 Square Roots

i) Definesquarerootofanaturalnumberandrecogniseitsnotation.ii) Findsquareroot,bydivisionmethodandfactorizationmethod,of • naturalnumber, • fraction, • decimal, whichareperfectsquares.iii) Solvereal-lifeproblemsinvolvingsquareroots.

Chapter4

Unit 6: Direct and Inverse Variation

6.1 Continued Ratioi) Definecontinuedratioandrecalldirectandinverse,proportion.ii) Solvereal-lifeproblems(involvingdirectandinverseproportion)using

unitarymethodandproportionmethod.

Chapter6

6.2 Time, Work and Distance i) Solvereal-lifeproblemsrelatedtotimeandworkusingproportion.ii) Findrelation(i.e.speed)betweentimeanddistance.iii) Convertunitsofspeed(kilometreperhourintometrepersecondandvice

versa).iv) Solvevariationrelatedproblemsinvolvingtimeanddistance.

Chapter6

Unit 7 Financial Arithmetic7.1 Taxes

i) Explainpropertytaxandgeneralsalestax.ii) Solvetax-relatedproblems. Chapter7

Curriculum

iv8 1

7.2 Profit and Markup

i) Explainprofitandmarkup.ii) Findtherateofprofit/markupperannum.iii Solvereal-lifeproblemsinvolvingprofit!markup.

Chapter77.3 Zakat and U shr

i) Definezakatandushr.ii) Solveproblemsrelatedtozakatandushr.

Unit 8: Algebraic Expressions8.1 Algebraic Expressions i) Defineaconstantasasymbolhavingafixednumericalvalue.ii) Recallvariableasaquantitywhichcantakevariousnumericalvalues.iii) Recallliteralasanunknownnumberrepresentedbyanalphabet.iv) Recallalgebraicexpressionasacombinationofconstantsandvariables

connectedbythesignsoffundamentaloperations.v) Definepolynomialasanalgebraicexpressioninwhichthepowersof

variablesareallwholenumbers.vi) Identifyamonomial,abinomialandatrinomialasapolynomialhavingone

term,twotermsandthreetermsrespectively.

Chapter8

8.2 Operations with Polynomials i) Addtwoormorepolynomials.ii) Subtractapolynomialfromanotherpolynomial.iii) Findtheproductof

• monomialwithmonomial,

• monomialwithbinomial/trinomial,

• binomialswithbinomial/trinomial.

iv) Simplifyalgebraicexpressionsinvolvingaddition,subtractionandmultiplication.

Chapter8

8.3 Algebraic Identities

Recogniseandverifythealgebraicidentities:

• (x +a)(x+b)=x2+(a +b)x +ab,

• (a+b)2=(a +b)(a +b)=a2+2ab +b2,

• (a–b)2=(a –b)(a –b)=a2–2ab +b2,

• a2–b2=(a–b)(a+b).

Chapter9

8.4 Factorization of Algebraic Expressions

i) Factoriseanalgebraicexpression(usingalgebraicidentities).ii) Factoriseanalgebraicexpression(makinggroups). Chapter10

Curriculum

v91

Unit 9: Linear Equations9.1 Linear Equation

i) Definealinearequationinonevariable.

Chapter11

9.2 Solution of Linear Equation

i) Demonstratedifferenttechniquestosolvelinearequation.ii) Solvelinearequationsofthetype: • ax +b = c,

•ax + bcx + d =

mn

iii) Solvereal-lifeproblemsinvolvinglinearequations.

Unit 10 Fundamentals of Geometry

10.1 Properties of Angles

i) Defineadjacent,complementaryandsupplementaryangles.ii) Defineverticallyoppositeangles.iii) Calculateunknownanglesinvolvingadjacentangles,complementaryangles,

supplementaryanglesandverticallyoppositeangles.iv) Calculateunknownangleofatriangle.

Chapter12

10.2 Congruent and Similar i) Identifycongruentandsimilarfigures.ii) Recognisethesymbolofcongruency.iii) Applythepropertiesfortwofigurestobecongruentorsimilar.

Chapter1510.3 Congruent Triangles

Applyfollowingpropertiesforcongruencybetweentwotriangles. • SSS ≅ SSS,

• SAS ≅ SAS,

• ASA ≅ ASA,

• RHS ≅ RHS.

10.4 Circle

i) Describeacircleanditscentre,radius,diameter,chord,arc,majorandminorarcs,semicircleandsegmentofthecircle.

ii) Drawasemicircleanddemonstratetheproperty;theangleinasemicircleisarightangle.

iii) Drawasegmentofacircleanddemonstratetheproperty;theanglesinthesamesegmentofacircleareequal.

Chapter14

Unit 11: Practical Geometry

11.1 Line Segment

i) Dividealinesegmentintoagivennumberofequalsegments.ii) Dividealinesegmentinternallyinagivenratio.

*CoveredinBook8

Curriculum

iv10 1

11.2 Triangles i) Constructatrianglewhenperimeterandratioamongthelengthsofsides

aregiven.ii) Constructanequilateraltrianglewhen • baseisgiven,

• altitudeisgiven.

iii) Constructanisoscelestrianglewhen • baseandabaseanglearegiven,

• verticalangleandaltitudearegiven,

• altitudeandabaseanglearegiven.

Chapter13

11.3 Parallelogram

i) Constructaparallelogramwhen

• twoadjacentsidesandtheirincludedanglearegIven,

• twoadjacentsidesandadiagonalaregiven.

ii) Verifypracticallythatthesumof

• measuresofanglesofatriangleis180º

• measuresofanglesofaquadrilateralis360º

Chapter16

Unit 12: Circumference, Area and Volume

12.1 Circumference and Area of Circle

i) Expressπastheratiobetweenthecircumferenceandthediameterofacircle.

ii) Findthecircumferenceofacircleusingformula.iii) Findtheareaofacircularregionusingformula.

Chapter17

12.2 Surface Area and Volume of Cylinder

i) Findthesurfaceareaofacylinderusingformula.ii) Findthevolumeofacylindricalregionusingformula.iii) Solvereal-lifeproblemsinvolvingsurfaceareaandvolumeofacylinder.

Chapter18

Unit 13: Frequency Distribution

13.1 Frequency Distribution

i) Demonstratedatapresentation.ii) Definefrequencydistribution(i.e.frequency,lowerclasslimit,upperclass

limit,classinterval).

Chapter1913.2 Pie Graph Interpretanddrawpiegraph.

Curriculum

v111

Guiding Principles1. Studentsexploremathematicalideasinwaysthatmaintaintheirenjoymentofandcuriosity

aboutmathematics,helpthemdevelopdepthofunderstanding,andreflectreal-worldapplications.

2. Allstudentshaveaccesstohighqualitymathematicsprogrammes.3. Mathematicslearningisalifelongprocessthatbeginsandcontinuesinthehomeandextends

toschool,communitysettings,andprofessionallife.4. Mathematicsinstructionbothconnectswithotherdisciplinesandmovestowardintegrationof

mathematicaldomains.5. Workingtogetherinteamsandgroupsenhancesmathematicallearning,helpsstudents

communicateeffectively,anddevelopssocialandmathematicalskills.6. Mathematicsassessmentisamultifacetedtoolthatmonitorsstudentperformance,improves

instruction,enhanceslearning,andencouragesstudentself-reflection.

Principle 1

Studentsexploremathematicalideasinwaysthatmaintaintheirenjoymentofandcuriosityaboutmathematics,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

iv12 1

• 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 prekindergartenstudents'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.


v131

• 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.


iv14 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 Plan

Along-termlessonplanextendsovertheentireterm.GenerallyschoolshavecoordinatorstoplanthebigpictureintheformofCoreSyllabusandUnitStudies.Coresyllabiarethetopicstobecoveredduringaterm.Twothingswhichareveryimportantduringplanningarethe‘TimeFrame’andthe‘Prerequisites’ofthestudents.Anexperiencedcoordinatorwillknowthedepthofthetopicandtheabilityofthestudentstograspitintheassignedtimeframe.

Suggested Unit Study Format

Weeks Dates Months Days Remarks

Short-term Lesson Planning

Ashort-termplanisaday-to-daylessonplan,basedonthesub-topicschosenfromthelong-termplan.


v151

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.


iv16 1

Lesson Plan

Sample Lesson Plan Planningyourworkandthenimplementingyourplanarethebuildingblocksofteaching.

Teachersadoptdifferentteachingmethods/approachestoatopic.

Asamplelessonplanisprovidedineverychapterasapreliminarystructurethatcanbefollowed.Atopicisselectedandalessonplanwrittenunderthefollowingheadings:

Topic

Thisisthemaintopic/sub-topic.

Specific Learning Objectives

Thisidentifiesthespecificlearningobjective/softhesub-topicbeingtaughtinthatparticularlesson.

Suggested Duration

Suggesteddurationisthenumberofperiodsrequiredtocoverthetopic.Generally,classdynamicsvaryfromyeartoyear,soflexibilityisimportant.

Theteachershoulddrawhis/herownparameters,butcanadjusttheteachingtimedependingonthereceptivityoftheclasstothattopic.Notethatintroductiontoanewtopictakeslonger,butfamiliartopicstendtotakelesstime.

Key vocabulary

Listofmathematicalwordsandtermsrelatedtothetopicthatmayneedtobepre-taught.

Method and Strategy

Thissuggestshowyoucoulddemonstrate,discuss,andexplainatopic.

Theintroductiontothetopiccanbedonethroughstarteractivitiesandrecapofpreviousknowledgewhichcanbelinkedtothecurrenttopic.

Resources (Optional)

Thissectionincludeseverydayobjectsandmodels,exercisesgiveninthechapter,worksheets,assignments,andprojects.

Written Assignments

Finally,writtenassignmentscanbegivenforpractice.Itshouldbenotedthatclassworkshouldcomprisesumsofalllevelsofdifficulty,andoncetheteacherissurethatstudentsarecapableofindependentwork,homeworkshouldbehandedout.Forcontinuity,alternatesumsfromtheexercisesmaybedoneasclassworkandhomework.

Supplementary Work (Optional): Aprojectorassignmentcouldbegiven.Itcouldinvolvegroupworkorindividualresearchtocomplementandbuildonwhatstudentshavealreadylearntinclass.

Thestudentswilldotheworkathomeandmaypresenttheirfindingsinclass.

Evaluation

Attheendofeachsub-topic,practiceexercisesshouldbedone.Forfurtherpractice,thestudentscanbegivenapracticeworksheetoracomprehensivemarkedassessment.


v17

Operations on Sets

1

11

Specific Learning Objectives• Operationsofsetsinvolvingunions,intersections,complements,subsets,anddifferenceofsets

• Venndiagrams

• Commutativeandassociativepropertiesofsets

Suggested Time Frame4to5periods

Prior Knowledge and RevisionStudentshavebeenintroducedtotheconceptsofsetsandtheirdescriptiveandtabularnotations.Aquickreviewquizcanbeconductedwheretheteacherwritesthevarioustypesofsetsfromthelistbelowandencouragesthestudentstocomeupwiththecorrectterminology.

– Emptyset

– Finiteandinfinitesets

– Disjointandoverlappingsets

– Equivalentandequalsets

– Universalset

– Subsetandsuperset

Real-life Application and ActivitiesThesymbolstableonpage8needstobehighlightedinthelesson.Theteachercandisplaythesignsonchartpaperonthesoftboard.Atthebeginningofeverylesson,thesymbolscouldberevisedorallybylookingatthechartpresentationforaminute.

Thiswillensurethatthestudentsarewell-versedinthesymbols.Theyneedtobeabletoreadthesymbolsasalanguageofmathematics.Theycanfeelconfidentwhenateacherwritesasetnotationontheboardandthestudentsareabletoexplainitinwords.

iv18 1Operations on Sets

Example

Symbols used in sets

Symbol Meaning Example

1. = isequalto {1,2,5}={5,1,2} (Setshavingexactly thesamemembers areequalsets.)

2. ≠ isnotequalto {1,2,5}≠{1,2,4}

3. ∈ isamemberof 3∈{1,2,3}

4. ∉ isnotamemberof 3∉{0,1,2}

5. Φ istheemptyset {} orthenullset

Starter activity

TheteachercanmakeatableofafictionalsetofstudentswhogotA’sinmathematicsandEnglish.Whiledoingso,thestudentswillobservethatsomegotA’sinbothsubjects.

SetofstudentswhogotanAinmathematics:

{Ali,Maya,Myra,Sara,Ahmed,Ameera}

SetofstudentswhogotanAinEnglish:

{Fatima,Maheen,Zain,Ali,Maya,Myra}

TheteacherwillhighlightthefactthatthreestudentsgotanAinbothsubjects.

Mathematics English

SaraAhmedAmeera

FatimaMaheenZain

AliMayaMyra

Activity

Aninterestingtenminuteactivitycanbeconductedbybringingtwohoolahoopstoclass.PlacethemonthefloorandlabelEnglishandMathematicswithflashcardsplacedonthefloorjustoutsidethehoops.Makesurethehoopsoverlap.

Callout'Begin'andthestudentsshouldscrambletotaketheirrespectivepositions.Changethelabellingofthehoopsforanothersetofsubjectsandstartagain.

Thisactivitywillnotonlyexplaintheconceptofsets,butwillalsobehelpfulintheunderstandingofVenndiagramsandtheirintersectionsandunions.

v191 Operations on Sets

Summary of Key Facts• Unionisthecombinationoftwosets,wherethecommonelementsarewrittenonce.Itisdenotedby‘U’whichiseasytorememberasunionbeginswithU.

• Intersectionisthecommonelementsonlyoftwoormoresetsandthisisobviousfromthetermitself.Itisdenotedby'∩'.

• IfasetAisasubsetofthegivenuniversalset,thenthesetofelementsnotinAiscalleditscomplementset.

Example Universalset:{1,2,3,4,5,6,….,10}

SetA :{1,2,3,4}

SetB :{5,6,7,8,9,10}

SetA’ :{5,6,7,8,9,10}

Anintersectingconcepttoaddonwouldbethattheintersectionofacomplementanditssetwillalwaysbeanullset.

Similarlytheunionofacomplementanditssetwillbetheuniversalset.

• Thedifferenceoftwosets,setAandSetB,wouldbetheelementsofSetAthatarenotinSetB.

• Setscanbeoverlapping,disjoint,orcanbeplacedasasubsetofeachother.

Venndiagramsofallthreetypes.

Subsets

B⊂ A

X

X X

B

AA

Disjoint

B

X

XX

A

Overlapping

B

X

XX

✓

✓

Changingtheplacesofthesetsduringuniondoesnotaltertheoperation.Thisisthecommutativepropertyoftheunionofsets.ThatisA∪ C=C∪ A

Similarly,thecommutativepropertyofintersectionofsetsidentifiesthesameconceptintheoperationofintersection.Theorderdoesnotaffecttheresult.ThatisA∩ B=B∩ A

• Theassociativepropertyofunionandintersectionofsetshighlightsthefactthatchangingtheorderofagroupofsetsdoesnotaltertheresult. Thatis(A∪ B)∪ C=A∪ (B∪ C)and(A∩ B)∩ C=A∩(B∩ C)

?OOPS

!

Frequently Made MistakesStudentsgenerallygetconfusedwiththesymbols.Ifthesuggestionsstatedearlierareimplementedtheysurelywillnotfinditdifficulttodecodethelanguageofsets.

iv20 1Operations on Sets

Lesson Plan

Sample Lesson Plan

Topic

Associativeproperty


Understandingtheassociativepropertyofunionofsets.

Suggested Duration

1period

Key Vocabulary

Union,Venndiagram,Associativeproperty

Method and strategy

Activity

Consideranexample:

Universalset:{1,2,3,4,…..,10}

SetA:{2,4,6,8}

SetB:{2,3,5,7}

SetC:{1,2,3,4,5}

A∪(B∪C)=(A∪B)∪C

Bothorderswillproducethefollowinganswer

{1,2,3,4,5,6,7,8}.

Theteachershoulddosimilarexamplesontheboardandprovetheassociativepropertyoftheunionofsets.ThiscanalsoberepresentedbyaVenndiagrambyshadingtheentireunion.

Written Assignment

Questions11,12,15,and16ofExercise1canbedoneasclasswork.Fivesimilarsumscanbegivenforhomework.

EvaluationAmarkedassignmentcanbedoneinclassfortheentireExercise1asthestudentsprogressduringthecourseoftheweek.

Thischapterismorepresentation-basedandthesymbolsareofutmostimportance.Marksshouldbeawardedforthecorrectuseofsymbols.

After completing this chapter, students should be able to:• findtheunion,intersection,anddifferencesetforgivensets,• drawVenndiagramsforunionsets,intersectionsets,andsubsets,and• provethecommutativeandassociativepropertiesofsets.

v211

2 Rational Numbers

Specific Learning Objectives• Introductionofrationalnumbers

• Operationsonrationalnumbers

• Additiveandmultiplicativeidentity

• Expressingarationalnumberinstandardform

• Comparingrationalnumberswithunlikedenominators


Prior Knowledge and RevisionStudentsarealreadyawareofnaturalandwholenumbersastaughtinearlierclasses.Theyhavebeenintroducedtothenumberlineandunderstandthelawsofaddition,subtraction,multiplication,anddivisionofintegers.

Itwouldbeadvisabletorevisetherulesusinganumberlinedrawnontheboard.

(+)+(+) [Addandwriteapositive(+)signintheanswer.]

(–)+(–) [Addandwriteanegative(–)signintheanswer.]

(+)+(–) [Subtractandwritethesignofthelargernumberintheanswer.]

(+)×(+) [Multiplyandwriteapositive(+)signintheanswer.]

(–)×(–) [Multiplyandwriteapositive(+)signintheanswer.]

(+)×(–) [Multiplyandwriteanegative(–)signintheanswer.]

(+)÷(+) [Divideandwriteapositive(+)signintheanswer.]

(–)÷(–) [Divideandwriteapositive(+)signintheanswer.]

(+)÷(–) [Divideandwriteanegative(–)signintheanswer.]

iv22 1Rational Numbers

Real-life Application and ActivitiesThefollowingactivitycanbedoneontheboardasafungame.

Dividethestudentsintogroupsofthree.

Writeasum.

Example:

14 ÷ ∙ 12 – 34 + 14 ∙Askonegrouptoattemptthesumlefttoright.AskthenextgrouptofollowtheorderofoperationofBODMAS.Seewhogetsthehighervalueandpointoutthatorderofoperationmattersastheyendupwithtwodifferentanswers.

Thisactivitywillnotonlymakethestudentspractisetogether,butwillalsomakethemappreciatethesignificanceofBODMAS.Sincetheywillbeworkingingroupstheycanhelpeachotherbypointingoutanymistakesandgivingtherightclueifanyoneisunabletograsptheconcept.

Summary of Key Facts• Rationalnumbersarenumberspresentedonthenumberline.Theyincludefractionsand

integers.Rationalnumbersarenumbersthatcanbeexpressedintheformpq ,whereq ≠0.

• Irrationalnumberscannotbeexpressedintheformofafraction.Forexample√2 ,√7 .

• Whenthereciprocalofarationalnumberismultipliedwithitsnumber,theresultis1.

• Thesumofarationalnumberanditsadditiveinverseis0.

• Thecommutativepropertyofrationalnumbersinmultiplicationstatesthattheproductoftworationalnumberswillremainthesameregardlessoftheorder.

• Theassociativepropertyofthreerationalnumbersinmultiplicationstatesthattheproductremainsthesameregardlessoftheorderofoperation.

• Distributivepropertywithrespecttomultiplicationoveradditionstatesthatwhenmultiplyingandaddingthreerationalnumbers,theresultisthesameirrespectiveoftheorderofoperation.

• Whenarationalnumberisdividedbyanon-zerorationalnumber,thequotientisarationalnumber.

• Thestandardformofarationalnumberhasapositivedenominator.

?OOPS

!

Frequently Made MistakesStudentsgenerallygetconfusedwiththeterminology(orvocabulary)ofrationalandirrationalnumbersandtheirreciprocals.Itisimportantthattheearlierterminology(orvocabulary)andconceptsofnatural,wholenumbersandintegersarethoroughlyrevisedbeforetheconceptofrationalnumbersisintroduced.Thisisimportantasthischapterformsthebasisofalgebra.Thestudentsshouldrecognisethesignificanceoftheorderofoperationsandtherulesofthesigns.

v231 Rational Numbers

Lesson Plan

Sample Lesson Plan

Topic

Comparingrationalnumbers

Specific Learning Objective

Comparingrationalnumbers

Suggested Duration

1period

Key Vocabulary

Rationalnumber,Unlikedenominators

Method and Strategy

Studentsshouldunderstandthatinordertocomparerationalnumbers,therationalnumbersshouldhavecommondenominators.Askthemtorewritethegivenrationalnumberswithpositivedenominatorstogetcommondenominators.Toobtaincommondenominatorstherationalnumbersaremultipliedbythecommonfactor.Oncethisisdone,thenumeratorwhichissmallerisplacedfirstandthentheinequalitysigniswritten.

Example Write<,>or=inthebox

34 57

2128 > 2028

∴ 34 > 57

Written Assignment

Questions 3,7,8,and9ofExercise2bcanbedoneinclass.Thesumsthatarenotcompletedcanbegivenforhomework.

EvaluationAnassessmentcanbeplannedalongthelinesofExercise2b.Sincethischapteristechnical,studentscanalsobegivena'Fillintheblankstest'.Theblankscanbebasedonthedefinitions,rules,andpropertiestaughtinthischapter.

After completing this chapter, students should be able to:

• identifyrationalnumbersandplotthemonanumberline,

• add,subtract,multiply,anddividerationalnumbers,

• compareandorderrationalnumbersinascendinganddescendingorder,and

• findthereciprocalofarationalnumber.

iv24 1

3 Decimal Numbers

Specific Learning Objectives• Terminatingandnon-terminatingdecimals

• Conversionoffractionsandpercentagesintodecimals

• Conversionofdecimalsintofractionsandpercentages

• Conversionofdecimalsintorationalnumbers

• Approximationandroundingoffofdecimalnumbers.


Prior Knowledge and RevisionStudentsareawareofdecimalsandtheidentificationoftheplacevalueofdecimals.Teacherscanrevisetheplacevalueofdecimalsbymentioningtenths,hundredths,andthousandthsasthefirst,second,andthirddecimalplaces.Operationsindecimalsshouldalsoberevisedasstudentscandostorysumscontainingallthefouroperationsoffindingthesum,difference,product,anddivisionofdecimals.

Revisionofplacevalueofdecimalscanbecalled‘pinningthedecimalpoint’,alongthelinesofpinningthedonkey'stailgame.Itisashortfive-minuteactivitywherethestudentscanscrambleintogroupsandtheteacherdividestheboardintoasmanycolumnsasthenumberofgroups.Theteacherwrites5sumsineachcolumnandeachgroupsendsavolunteer.Itcanbecomearowdygameasthestudentsareallowedtohelptheirvolunteers.Thegroupthatfinishesallfivesumsfirstandcorrectlygainsapoint.

Thesumsontheboardcanbeasfollows:

1) 7643,3hundredths,sothestudentwillplacethedecimalpointafter6.

76.43

2) 807945,5thousandths,thedecimalpointwillbeplacedafter7.

807.945

v251 Decimal Numbers

Real-life Application and ActivitiesDecimalsareassociatedwithmoney.Theteachercanaskfornewspaperclippingswherethegrowthrateornationalreservesofthecountryarementionedindecimals.Similarly,moneyconversionshavedecimalpoints.

ExampleDollarscanbeconvertedtorupeesaccordingtotheconversionrate.

Alistofcurrenciesandtheirconversionscouldbesharedinclassbytheteacher.

Whenexplainingterminatingandnon-terminatingorrecurringdecimals,theliteralmeaningsofthewordscouldbeexplained.Terminatingmeanstoend;thereforeterminatingdecimalshavedecimalplacesthatarefixedandcomplete.Recurringdecimalshavedecimalplacesthatkeeponrepeatingindefinitely.Thesedecimalplacescangoontoinfinityandtherecurrencecansometimesbeingroupsorsequencesofnumbers.

Summary of Key Facts• Fractionscanbeconvertedtodecimalsbythelongdivisionmethodwhereadecimalpointisintroducedandazeroisadded.Thealternatemethodistoconverttoanequivalentfractionwithadenominatorwithmultiplesoften.Whendoingso,thefactoridentifiedtoconvertthedenominatortoamultipleoftenisalsomultipliedwiththenumeratortocreateequivalence.Thenumberofdecimalplacesdependsonthenumberofzeroesinthedenominator.

• Conversionofpercentagesintodecimalsisquitesimpleasthepercentageitselfisadenominatorof100,whichiseasilyconvertedtoafractionbycreatingtwodecimalplaces.

• Whenconvertingadecimalintoapercentage,thedecimalisfirstconvertedintoafractionwithadenominatorwithmultiplesof10andthenmultipliedby100.

• Whenconvertingdecimalsintorationalnumbers,thefractionhastobereducedorsimplified.Generally,thefractionwithadenominatorofmultiplesofteniseasilyreducedorsimplified.

• Terminatingdecimalshaveafinitenumberofdigitsafterthedecimalpoint.

• Whenaninfinitenumberofdigitsoccur,afterthedecimalpointtheyarecalled non-terminating,orrecurring,decimals.

• Approximatevalueorroundingoffhasasetofsimplerules.Theplacevaluethatneedstoberoundedoffiscircled.Ifthenumberafterthecircleddigitis5orgreaterthan5,thenthevalueofthecircleddigitisincreasedby1.Ifthenumberafterthecircleddigitislessthan5,thenthevalueremainsthesame.

?OOPS

!

Frequently Made MistakesWhendoinglongdivision,studentssometimesgetconfusedaboutwhentointroducethedecimalpointandhencethezerotothedividend.Alotofpracticequestionsinlongdivisiontocreatedecimalsneedstobedoneinclass,ontheboardinitiallyandtheninstudents'notebooks.

iv26 1Decimal Numbers

Lesson Plan

Sample Lesson Plan

Topic

Roundingoffandapproximation


Roundingoffofdecimals

Suggested Duration

1period

Key Vocabulary

Roundingoff,Approximation,Decimalplaces,Decimalpoint

Method and Strategy

Anumberthatneedstoberoundedoffhastobecircled.Ifthenumbertoitsrightis5ormore,thenthecircleddigitisincreasedby1andtherestisdeleted.

ExampleRoundoff28.89tothenearestwholenumber.28.89 ↑

Thisroundingoffcanalsobeexplainedbyusinganumberline.Wheneveradecimalhastoberoundedofftothenearestwholenumber,anumberlinecomesinhandy.

Anumberlinecanbemadeonthefloorinthecorneroftheclassroomandkeptduringthedurationofthischapter.Colouredelectricaltapecanbetapedtothegroundtoformthenumberlineandthenumberscanbeplacedasflashcardsthatcanbereplacedaccordingtothedemandsofthesumsornumbersets.Thegradingsorthedashesonthenumberlinecanalsobemade semi-permanentbymakingthemarkingswithadifferentcolouredelectricaltape.

Written Assignments

Questions12,13,and14ofExercise3canbedoneinclass.Similarsumscanbegivenforhomework.

EvaluationAcomprehensivetestcanbeconductedwherelearningofallconceptstaughtcanbeassessed.Storysumsinvolvingroundingoffandconversionscanbeasked.Thiswilldevelopcriticalthinkingskills.


• convertfractionsandpercentagestodecimalnumbers,andviceversa,

• identifyterminatingandnon-terminatingdecimalnumbers,and

• roundoffdecimalnumberstothenearestwholenumberandtherequireddecimalplace.

v271

4Squares and Square Roots

Specific Learning Objectives• Introductionoftheconceptofperfectsquaresandsquareroots

• Derivationofthesquarerootofaperfectsquare

• Derivationofpositivesquarerootbyprimefactorization

• Derivationofpositivesquarerootbydivision


Prior Knowledge and RevisionStudentsareawareofsquarenumbersandtheareaofasquare.Thecorrelationofthefactthatasquarenumberandareaofasquarearethesameisimportant.Theteachercanholdaquickoneminutequizwherehe/shecallsoutanumberandthestudentsmultiplyitbyitselfandstateitssquare.

Agameofsnapcanalsobeplayed.

Makeasetof20flashcardswiththesquaresof1to10writtenoneachcardtwice.Cardsareshuffledanddistributedbetweentwostudents;whenoneofthemcallsoutsnapashe/shegetsthesamesquarenumber,he/sheneedstocalloutthenumberitisasquareof.Thiscanbeplayedbyallstudentsinturnandcanbeafiveminutefuntimewiththerestcheering.

Themultiplicationruleoftwonegativenumberscanberecalledanditcanbepointedoutthatwhentwonegativenumbersaremultiplied,wegetapositivesquare.

Example13 × 13 = 169

(–13) × (–13) = 169

Thefactthatallconceptsconvergeandbuilduptoformnewconceptshastoberecognisedbytheteacher.Thisisimperativeinordertocreateanetworkofmathematicalconceptsamongststudents.

iv28 1Squares and Square Roots

Real-life Application and ActivitiesAlthoughthisisanentirelycomputation-basedchapter,youcancreateagamesothatstudentscanlearnthestepsoftheprimefactorizationanddivisionmethodsfaster.

Youwillrequireawhiteboard,differentcolouredboardmarkers,astopwatch,andflashcards.

Thestudentsselectasquarenumberfromtheflashcards.Theyalsopickouttheoptionofprimefactorizationordivisionmethod.Thestudentgoestotheboard,solvesthesum,andhis/herfinishingtimeinthecaseofacorrectansweriswritteninacolumnatthesideoftheboard.Allstudentstaketurnstillthewholeclasshashadaturn.Thisactivitywillinvolvethewholeclasswhileeachsumisbeingsolved.Thiswillresultinalotofpracticeasstudentswillallfolloweachsumdoneontheboardandwillbeencouragedtopointoutanymistakes.Thiswillalsoquickentheirmathematicalcomputation.

Summary of Key Facts• Aperfectsquarecanbeexpressedastheproductoftwointegersofequalvalue.

• Theareaofasquareisasquarenumber.

• Whendeterminingthepositivesquarerootofnumbersbytheprimefactorizationmethod,thenumberisdividedbyitsprimefactorstillitcomestoa1.Theprimefactorsaregroupedinpairsandonefactorfromeachpairistakenandtheproductiscalculated.Thisisthesquarerootofthesquarenumber.

• Whendeterminingthepositivesquarerootbythedivisionmethod,thenumberispairedbyputtingbarsoneachsetstartingfromtheunit.Foreachset,thelargestnumberwhosesquareiscontainedineachpair,isusedasadividend.Thewholeprocessofdivisionisthusfollowedbysubtractingthesquarefromthepairandthecyclecontinuedwiththeseconddividend.Fortheseconddivisor,theoriginalquotientisaddedtoitself.Findoutthenewdivisorandplaceitnexttothesumfoundearlier.Multiplythenewdivisorwiththeentiresetandplacetheproductunderthedividendandsubtract.Repeatthisprocesstillthedividendbecomeszero.

• Wheneverasquarerootisfound,alwayscheckbymultiplyingitbyitselftoseeifitcomestothegivensquarenumber.

?OOPS

!

Frequently Made MistakesThisisanentirelymathematicalconceptwithstepstobelearnt.Studentssometimesmakemistakesiftheydon’trememberthesteps.Thereforetheyshouldfocusonlearningthesteps.Thestepsofthedivisionandprimefactorizationmethods,alongwithaworkedexample,canbedisplayedonchartpaperforthestudents'perusalduringthecourseoftheweek.

v291 Squares and Square Roots

Lesson Plan

Sample Lesson PlanTopicSquaresandsquareroots

Specific Learning ObjectivesFindingthesmallestwholenumbertobemultipliedtomakeanumberasquarenumber.

Suggested Duration1period.

Key Vocabulary

Squarenumbers,Primefactorization,Exponents,Power

Method and Strategy

Inordertofindthenumberwhichwillmakethenumberacompletesquare,thestudentsfirsthavetorevisetheprimefactorizationmethod.Theexponentialformrepresentationisimportant.

Example 120 = 23× 3 × 5

2,3,and5needtobeintroducedtomakecompletepairs.

Hence30isthesmallestnumbertobemultipliedto120tomakeitaperfectsquare.

Thisisadifficultandconceptualtopic.Alotofsumsshouldbedoneontheboardandthen,oncetheteacherfeelsthatthestudentscanworkindependently,theycandosumsintheirexercisenotebooks.

Written Assignment

SumslikeQ#12ofExercise4bcanbedoneinclassandtherestcanbegivenforhomework.

Findthesmallestwholenumberbywhichthefollowingnumberscanbemultipliedtomakethemaperfectsquare.

1) 120

2) 325

3) 66

4) 35

5) 260

6) 180

7) 95

8) 21

9) 45

10) 500

EvaluationAsmentionedearlier,thisisanextremelyconceptualchapterwithstepsofmathematicalcomputation.Anoralquizonthestepsofthedivisionmethodcanbegiventwiceorthriceatthebeginningofeachlesson.Awrittenquizcanalsobegivenforthestepsofdivisionmethod.Acomprehensivetestincludingwordproblemsshouldbegivenattheendofthechapter.Assessmentoflearningduringthecourseofthetopicisimportantandcanbeimplementedintheformoffiveminutequizzes.


• statethesquarerootsofthefirst12perfectsquares,

• performtestsforperfectsquares,

• findthesquareofanumber,and

• derivethesquarerootofapositiveintegerbyprimefactorizationanddivision.

iv30 1

5 Exponents

Specific Learning Objectives• Conceptofexponentialnotation

• Expressingrationalnumbersinexponentialnotation

• Lawsgoverningthepowersorexponentsinmultiplicationanddivision


Prior Knowledge and RevisionTheteacherexplainstherepetitivemultiplicationofanumberbyitselfandlinksittosquaresandcubesofanumber.Theterm'powers'isalreadyknownbythestudents.Theyalsoknowthefollowingterms:variable,power,constant,product,andquotient.

Studentshavebeendoingprimefactorizationandexpressingnumbersintheirindexform.

Real-life Application and ActivitiesTheconceptofexponentialnotationcanbereintroducedwithafunactivity.

Youneed2or3decksofcards.

Removetheaces,kings,queens,jacks,andjokers.

Dividethestudentsintogroupsoffour.Askonestudentfromeachgrouptodealthecards.Thepackscanbeshuffledanddividedequallyamongthegroups.

Theythendivideandorganisetheircardsinexponentialorder.

Forexample,ifthestudenthas3fiveshewillwriteitas53.

Next,askthestudentstofindtheproductoftheirexponentiallist.

v311 Exponents

Example53 = 5 × 5 × 5 = 125,andsoon.

Nowaskthemtoaddalltheproducts.

Thegroupwiththehighestscoreisthewinner.

Youcantimethemusingastopwatchandyouwillhaveanotherwinnerintermsoftimings.

Thisactivitynotonlydevelopstheirmentalskillsandabilitytoorganisedata,butalsopractisestheterminologiesofpowers/exponents/index.

Summary of Key Facts• 2tothepoweroffivemeansthat2isrepetitivelymultipliedwithitselffivetimes,where2isthebaseand5isthepowerorexponent.

• Whenanegativebaseisraisedtoanevenpower,theproductisalwayspositive.

• Similarly,thenegativebaseofanoddnumberpowerstaysnegative.

• Reciprocalsofrationalnumberswithpowersremainthesame,onlythebaseisswappedfromnumeratortothedenominatororviceversa.

Example

• The6LawsofExponentsareasfollows:

LawI: xm × xn = xm+n

LawII: xn × yn=(x × y)n (wherey isalsoanon-zerorationalnumber)

LawIII: xm ÷ xn = xm –n

LawIV: xn ÷ yn=(xy )

n (wherey isalsoanon-zerorationalnumber)

LawV: (xm)n = xmn

LawVI: x0=1 (ifx isanynon-zerorationalnumber)

?OOPS

!

Frequently Made MistakesStudentsinvariablydonotapplythelawswhiledoingtheoperationofmultiplicationanddivisionandtendtoresorttocancellingindivisionandcountingthebasesinmultiplication.

Also,thereciprocallawshavetobeexplainedmeticulously.Studentstendtoputthenegativepowerofthebasetotheexponents.Theytendtogetconfusedwhendealingwithnegativebases.

iv32 1Exponents

Lesson Plan

Sample Lesson PlanTopic

Lawsofexponents

Specific Learning ObjectivesTwolawsofexponentswillbeintroduced.

xm × xn = xm+n

xm ÷ xn = xm–n

Suggested Duration

1period

Key Vocabulary

Product,Quotient,Power,Exponent,Base,Variable

Method and Strategy

Whenexplainingthetwolaws,theteachershouldmanuallybreakupthebasesandshowthatwhenmultiplyingweactuallyendupaddingthepowers.

Similarly,theteachershouldshowdivisionontheboard,wherethebasesarecancelled,whichisactuallythedifferenceofthepowersofthetwobases.

Asmallactivitycanbedonewithsweets.

Collecttensweetsofthesametypeandarrangetheminaline.Tellthestudentsthatthesweetsaretothepoweroften.

Ifthesweetsareputintwosets:fourtogetherandsixtogetherwhichbecomestenwhenadded.

Exponentially,

S4 × S6 = S10

Similarly,indivision,thetwosetsofsweetscanbeputonafractionalbaranddivided.

Thisactivitywillhelpthestudentsunderstandthelawseasily.

Written Assignments

Mixedsumsofmultiplicationanddivisioncanbegiventobedoneinstudents'notebooks.

Askthestudentstohighlightthelawsonaseparatepageoftheirnotebooksforreference.

EvaluationExercise5andtherevisionexerciseonpage71canbegivenasanassessment.Sumsshouldbemixedandchosenfromtheseexercisesinsuchawaythatthestudentsareassessedontheirrecallandassociationskillsrelatedtothelaws.


• identifythebaseandpowerofanumber,

• expressnumbersinexponentialform,and

• applytheproduct,quotient,andpowerlawsofexponentstosolvequestions.

v331

Direct and Inverse Variation

Specific Learning Objectives• Directandinversevariations

• Continuedratio

• Unitaryandproportionmethod

• Ratioproblemssolvingthreequantities


Prior Knowledge and RevisionThischapterisacontinuationofthetopicofratios.Theteachershouldconductarecallsessioninwhichrulesofratioarerevised.Thefactstoberevisedare:

• Ratiosarealwaysexpressedintheirsimplestform.

• Quantitiesareinthesameunits.

• Ratiosareplacedinthefollowingordernew:old

Examples a. 4:8

1:2

b. 400g : 1000kg

400g : 1,000,000

4 : 10,000

1g : 2500kg

Real-life Application and ActivitiesReal-lifeexamplesonpages73and74canbediscussedinclassandabrainstormingsessioncanbeconducted.

iv34 1Direct and Inverse Variation

Example

• Costandnumberofapples

• Speedofthecarandtime

• Amountoffoodandthedaysitwilllast

• Numberofpipesfillingupatankandthetimetaken

Withthehelpoftheseexamples,studentsshouldbeencouragedtoexploreparity.Ifonequantityincreases,theotheralsoincreases.Sometimesifonequantityincreases,theotherquantityorvaluedecreases.

Thedifferencebetweendirectandinverseproportionshouldbeexplainedthroughreal-lifeexamplesandapplications.Onlywhenthestudentsareabletodistinguishbetweenthetwo,shouldtheteacherproceed.

Summary of Key Facts• Directvariationoccurswhenbothquantitiesincreaseordecreasesimultaneously.Themethodstosolvethemaretheunitaryandproportionalmethods.

• Intheproportionmethod,thedataissetandcrossmultiplicationoccurs.

• Ininversevariation,onequantityincreasesandtheotherdecreases.

• Ininversevariation,oncethedataisset,horizontalmultiplicationtakesplace.

• Continuedratioisanexpressionofthreeratios.Twosetsofratiosarecombinedintoonelinearratio.Thecommonquantitybecomesthelowestcommonmultipleandtheratiosarecombined.

Examplered:green

2:3

green :blue

5 : 7

2 :3 (multiplyby5)

5 :7 (multiplyby3)

Hence,

10:15and15:21

Therefore,

10:15:21

• Dividinganamountintoproportionalpartsisgenerallydoneintheformofawordproblem.Eachproportionismadeintoafraction.Eachfractionalvalueisthenmultipliedtogetitsproportionalpart.

?OOPS

!

Frequently Made MistakesThisisaneasychapterandthestudentsenjoyitaslongastheycandifferentiatebetweendirectandinverseproportion.

v351 Direct and Inverse Variation

Lesson Plan

Sample Lesson Plan

Topic

Directandinversevariation


Directandinversevariation

Suggested Duration

1period

Key Vocabulary

Directvariation,Inversevariation,Proportion

Method and Strategy

Real-lifeexamplesofinverseproportionsshouldbegiveninclass.Theteachershouldhighlightthefactthatifonequantityincreases,theotherdecreases.

Averylogicaldeductionisthatifthespeedisgreater,thecarwilltakelesstimetofinishajourney.

Activity

Asimpleactivitycanbedoneinclass,inwhichtwoidenticaltoycarsarebroughtintothelessonandarepushedwithdifferentforcestotravelagivendistance.Thestudentswillrecordthetimesonthestopwatchandseethatmoreforceresultsinlesstime,andviceversa.

Oncethestudentshavedecidedtheproportion,whetherdirectorinverse,theteachershouldthenexplainthemethod.Whentheoperationisinverse,horizontalmultiplicationisdone.

Written Assignment

Questions4to13ofExercise6shouldbegiventogethersothatstudentscandistinguishbetweendirectandinversevariationandthencarryoutthemathematicalcomputation.

EvaluationAquizshouldbegivenaftereachconceptsothattheteachercanassesswhethertomoveontothenextconceptorreinforceearlierlearning.Quizzesareassessmentoflearningwhichareverybeneficial.

AcomprehensiveassessmentcanbegivenalongthelinesofExercise6andstudents,canbeevaluated.


• identifyproblemsinvolvingdirectvariation,inversevariation,andcontinuedratio,

• applytheunitaryandproportionmethodstosolveratioproblems,and

• solveproblemsinvolvingcontinuedratio.

iv36 1

Financial Arithmetic7

Specific Learning ObjectivesTheconceptsof:

• discount

• profitandlosspercent

• taxation,propertytax,andgeneralsalestax

• simpleinterest

• zakatandushr

Suggested Time FrameAtleast10periods

Prior Knowledge and RevisionStudentsstudiedpercentagesinGrade6.Financialtransactionsaretheapplicationofpercentagesinreal-lifescenarios.Theteachershouldfirsthavearevisionworksheetpreparedforthelessoninwhichtheconceptsofpercentagesarerevised.Theteachershouldnotembarkonthisnewchapteruntilthestudentshavethoroughlyrevisedandrevisitedtheconcepts.

Real-life Application and ActivitiesProfitandlossisareal-lifeapplication.Studentscanbeshownnewspaperclippingsofsaleadvertisementsandtaughthowtocalculatethediscountfromthemarkedprice.

Studentscanbeencouragedtocreatetheirownbusinessplanandpresentitinclass.

Activity

Aclassoutingcanbeorganisedtoamanufacturingunite.g.ashoefactory.

Themanagercouldbeaskedtopresentasimplebreak-downoftheproductioncoststothestudents.

v371 Financial Arithmetic

Overheadcosts = Rsx

Materialcosts = Rsy

Labour = Rs w

Totalcosts = Rs z

Saleprice = Rs v

Profit = v – z

Profit% = v – z × 100 z

Itshouldbehighlightedthattheabovevaluesareforoneshoeorperunit.Itshouldalsobepointedoutthatthesalepricehastobehigherthanthecostpricetomakeaprofit.

Activity

Theteachercanaskparentstotaketheirchildtoadiscountstore.Theycanwriteanessayontheirexperienceandfindings.

ExampleItem1

Markedprice: RS400

Discount: 30%

= 30×400100

Discount =Rs120

Salepriceafterdiscount =400 – 120 = Rs280

Theycanmakealistofvariousitemsinthisformat,butitisimportantthattheyvisitasaleshopandhaveahands-onexperience.

Activity

Studentsshouldbeaskedtoprepareamockreportofassetsandsavings,e.g.goldjewellery,savings,wheretheycalculatetheirvalueandthenworkoutthezakatontheassets.

Alltheactivitiescouldberecordedonchartpaperanddisplayedinclassforalltoview.Inthiswaymathematicscanbemadeinterestingandrelevant.

Summary of Key Facts• Areductionmadeonthemarkedpriceiscalleddiscount.

• Profitisincurredwhensellingprice>costprice.

• Lossisincurredwhenthecostprice>sellingprice.

• Overheadsareincludedwhencalculatingcostprice.

• Taxpaidasapercentageofthevalueofpropertytothegovernmentiscalledpropertytax.

• Thetaxpaidasapercentageofthesellingpricebythebuyertotheselleriscalledgeneralsalestax.

• Interestisincurredwhenaprincipalamountiseitherinvestedorborrowedatacertainpercentageofinterestrateoveraperiodoftime.

iv38 1Financial Arithmetic

• Theamountisthemoneypaidafteraprincipalamountisinvestedanditaccruesasimpleinterest.Hencetheamountisthesumoftheprincipalamountandthesimpleinterestaccruedoveraspecifiedperiodoftime.

• ZakatisanobligatorytaxpaidattheendofeachyearbyeveryMuslimatarateof2.5%ofthetotalvalueofhis/hersavingsandassets.

• UshristhetaxpaidbyaMuslimonhis/heragriculturalassets,leviedattherateof10%onagriculturaloutputifirrigatedbynaturalsources,and5%ifirrigatedbyartificialmeans.

?OOPS

!

Frequently Made MistakesStudentsusuallythinkthatifadiscountisgiven,therewillbealoss.Thisisnottrueasthediscountisgivenonthemarkedpricetowhichsomepercentageofprofithasalreadybeenadded.

Lesson Plan


Simpleinterest


Calculatingsimpleinterest

Suggested Duration

1period

Key Vocabulary Words

Principal,Rate,Perannum,Interest,Amount

Method and Strategy

Formulaforcalculatingsimpleinterest:

Simpleinterest= Principal×Rate×Time 100

where,

Pistheprincipalamount,

Ristherateperannumoryear,and

Tisthetimeinyears.

Amount=principal+simpleinterest.

Iftheinterestisgivenandalternatevaluesareasked,theformulahastobemanipulated.

If SI = PRT100

ThenT = SI× 100PR

R =SI× 100PT

v391 Financial Arithmetic

Written Assignments

SelectedsumsfromExercise7ccanbedoneinclassandtherestforhomework.

Worksheetswithsumswiththenamesofrealbanksandstudents'namescanbemadeandhandedoutasassignments.

Example

Erum’sfather,MrAyubinvestsRs100,000intheNationalBankofPakistanforayearataninterestrateof4%.Calculatetheamounthewouldreceivefromthebankattheendoftheyear.Alsoiftaxisdeductedbythegovernmentattherateof1%ofhistotalinterest,calculatethetaxpaidbyMrAyubandhisnettotaltaking.

Evaluation

Thischapterisverycomprehensiveandcritical.Beforeatestoffinancialarithmeticistaken,anassessmentoflearningshouldbecarriedoutaftercompletionofeverytopic(attheend).


• findthecostprice,sellingprice,profitorlosspercentonatransaction,

• calculatesimplediscountwhengiventhepriceanddiscountrates,

• computepropertytaxandgeneralsalestax,

• calculatesimpleinterestwhengiventheprincipal,time,andinterestrate,and

• calculatezakatandushrbyapplyingthecorrectrates.

iv40 1

8Algebraic Expressions

Specific Learning Objectives• Terminologyofalgebraicexpressions,comprisingvariables,coefficients,andconstants

• Orderofalgebraicterms

• Thefouroperationsofaddition,subtraction,multiplication,anddivisionofalgebraicterms


Prior Knowledge and RevisionThestudentshaveabasicknowledgeofalgebra.Theyshouldtorecallthatitisabranchofmathematicswherequantitiesareexpressedinlettersandvariables,andunknownvaluesarefiguredoutbyformingalgebraicexpressionsandequations.

Theteachercanhavearecapsessioninclassbywritingsumsontheboardandelicitingresponsesfromthestudentswhohavetosaythealgebraicexpressionsoutloud.

Example 1) Saima'sageafter5years = s + 7

2) Thecostofxapplesifonecosts50cents = 50x

3) Theweightofafulltruckwithbricksiswandtheweightofemptytruckisz: theweightofthebricks = w – z.

Alsoexplaintothestudentsthatthepowernotationstudiedearlierislinkedwithalgebra.

Example2tothepowerof2isthevalueof4.

∴ 22 = 4

y × y × y = y3

w × w × w × w × w = w5 (wtothepowerorexponentof5,wherewisthebasevariable)

v411 Algebraic Expressions

Real-life Application and ActivitiesAlgebrashouldbeconsideredasalanguageofmathematicswheretheunknownvalueisdenotedbyaletter.Therulesofoperationsthatweretaughtinclass6canberevisedagainbyplayingagamewithflashcards.

Eachchildisgivenasetofflashcardswithsignsofplusandminuswrittenseparatelyoneachcard.

Theteachercallsouttheoperationofaddition,subtraction,multiplication,anddivision.

Thechildpicksuptwoflashcards,lookatthesignsandfiguresoutthesignoftheresultoftheproduct,sum,ordifference.

Thestudentcanbetimedortheteachercangive30secondsandtheyhavetowriteasmanyoperationsandanswersaspossibleintheirnotebooks.

Thisencouragesthemtobequickwiththeirlawsandoperations.

Summary of Key Facts• Avariableisanunknownnumberdenotedbyaletter.

• Anumberplacedbeforethevariableisacoefficient.

• Anumberwithafixednumericalvalueisaconstant.

• Analgebraicexpressionconsistsoftermsconnectedbyeitheroftheoperations.

• Apolynomialcomprisesmorethanoneterm.

• Whenarrangingpolynomialterms,descendingorderisfollowed.Thepowerofthevariabledecidesthevalueoftheterm:thelargerthepower,thelargertheterm.

• Insubtraction,thetermtobesubtractedhasallitssignsswitched.

• Inmultiplication,powerswiththesamebaseareadded.

• Indivision,samebasepowersaresubtracted.

• Whenmultiplyingonealgebraicexpressionbyanotherexpression,eachtermofthefirstexpressionismultipliedbyeachtermofthesecondexpression.

Thesignrulesofadditionandsubtractionandtherulesofmultiplicationanddivisionshouldberevisedbeforesimplifyingpolynomials.

?OOPS

!

Frequently Made MistakesStudents'mostcommonmistakeisplacingtheincorrectsignwhileapplyingthefouroperations.Teachersneedtobeextremelycarefulwhenexplainingthesignconcept.

iv42 1Algebraic Expressions

Lesson Plan

Sample Lesson Plan

Topic

Divisionofpolynomials


Divisionofpolynomials

Suggested Duration

1period

Key vocabularyPolynomial,Coefficient,Power,Exponent

Method and Strategy

Studentsshouldbeintroducedtothisoperationbyrelatingittoregulardivision.Theonlydifferenceisthatalltermsaredividedorcancelledbythedivisortermbutonlythepowersofthesamebasevariablearesubtracted.

Lotsofpracticeworksheetsonthisoperationshouldbegivenandtherulesshouldberevised.

(+) and (+) = +

(+) and (–) = –

(–) and (–) = +

Written Assignments

Questions8,9,and10ofExercise8bcanbedoneinclassinthestudents'notebooks.Theycancompletetheexerciseathome.

Evaluation

ThischapterisrelativelyeasyandacomprehensiveassessmentbyusingsumsfromExercise8aand8bcanbegiven.

Boardquizzescanalsobedoneforalltheoperations.


• identifyandsolvealgebraicexpressions,

• arrangepolynomialsinincreasinganddecreasingorder,and

• add,subtract,multiply,anddividegivenalgebraicexpressions.

v431

Algebraic Indentities

Specific Learning Objectives• Introductionofthesumanddifferenceofaperfectsquare

• Thealgebraicidentityofthedifferenceoftwosquares

• Thenumericalapplicationoftheseidentitiestosolvequestions


Prior Knowledge and RevisionStudentshavenotstudiedalgebraicidentitiesbefore.Thistopicisanextensionoftheirunderstandingoftherulesandlawsgoverningalgebraicpolynomials.

Real-life Application and ActivitiesTheidentitiescanbeexplainedgeometricallythroughtheareasofrectanglesandsquares.

P QL

M

a2 ab

ab b2

a b

ba

N O

S P R

A BP

QR

T

S

C D

a

a

b(a –b)

b(a–b)(a–b)2

(a –b)

b2

iv44 1Algebraic Identities

A

N MP

D

B

C

La

b (a –b)

a

a +b

N P L

BCD

a b

a–b

Theteachershouldmakecutoutsofthesediagramsonchartpaperandthenexplainthembyprovingtheidentitiesgeometrically.Thesidesofthesquaresandrectanglesshouldbedenotedbyvariables‘a’and‘b’.

Summary of Key Facts• Thesumofthesquareisthesumofthesquaresof‘a’and‘b’andtwicetheproductofboth.

(a+b)2= a2+2ab+b2

• Thedifferenceofthesquareisthesumofthesquaresof‘a’and‘b’andtwicetheproductofbothwithaminussign.

(a–b)2= a2–2ab+b2

• Theproductofthesumanddifferenceoftwovariablesisthedifferenceofthetwosquares.

(a+b)(a–b) = a2–b2

Whensolvingnumericalvaluesofatermusingidentitiesthetermisfirstbrokendownintooneofthethreeidentitiesandthenexpandedintoanumericalgebraicidentitytocalculatethevalues.

?OOPS

!

Frequently Made MistakesWhensolvingthefirstandthesecondidentity,thestudentsgetconfusedwiththe2abexpression.Itistobehighlightedthatthisproducthasaminussigninthedifferenceofthewholesquareidentity.Thefactthatbsquaredcanneverhaveaminussignshouldalsobepointedout.

(–b)2=(–b)x(–b)=+b2

v451 Algebraic Identities

Lesson Plan


Algebraicidentities

Specific learning Objectives

Solvingarithmeticexpressionsusingthefirstalgebraicidentity

Suggested Duration

1period

Key Vocabulary

Thesumofaperfectsquare,Numericalvalue,Algebraicidentity

Method and Strategy

Whenexplainingthefactthatarithmeticexpressionsdonothavetobeevaluatedmanuallywitharithmeticoperations,itshouldbepointedoutthatalgebraicidentityisaquickerway.

Example105 × 105

Thestudentscanbetoldthattheproductcanbefoundarithmeticallybutthiswilltakelonger.Alsohighlightthefactthatthequestionmentionstheuseofalgebraicidentitytoevaluate.

105 × 105 = 1052

Usingthefirstalgebraicidentity:

a2 + 2ab + b2

= (100+5)2

= (100)2 + 2(100)(5) + (5)2

= 10000 + 1000 + 25

= 11025

Written Assignments

SumssimilartoQ#7ofExercise9canbegivenasaclassworkassignmentonceithasbeenexplained.

Example 1) 206 × 206

2) 505 × 505

3) 101 × 101

4) 702 × 702

5) 4001 × 4001

iv46 1Algebraic Identities

Evaluation

Assessmentoflearningwillplayakeyroleduringthischapter.Short,fiveminutequizzesshouldbegivenaftereachidentityandconcepttaught.Thiswillinformtheteacherwhethertoproceedornot.ItisimportantfortheteachertounderstandthatalgebraisarelativelyeasyandenjoyablebranchofMathematicsbuttheidentitiesareacriticalstepinthisarea.Theconceptofidentitiesandtheirapplicationisveryimportant.Thestudentsshouldbegivenenoughmini-teststoensurethattheyarewellversedineachidentitybeforeacomprehensiveassessmentincludingallconceptsisgiven.

RevisionExerciseonpages142and143canbeusedascomprehensivetestforthischapter.


• derivethealgebraicidentitiesrelatedtothesquareofasumandthedifferenceoftwotermsandtheproductofasumanddifferenceoftwoterms(a+b)(a–b),and

• solvequestionsbyapplyingthecorrectalgebraicidentities.

v471

10Factorization of Algebraic Expressions

Specific Learning Objectives• Introductionoffactorizationofalgebraicexpressions

• Introductionoffactorizationofexpressionsusingalgebraicidentities

• Introductionoffactorizationbymakinggroups


Prior Knowledge and RevisionStudentshavejuststudiedalgebraicidentitiesandthischapterisacontinuationofitsapplication.Itisalsoanextensionandbuild-upoftheconceptsofidentities.Theteachershouldnothaveanyissueswiththischapterasitisaprogressionoftheearliertopic.

Real-life Application and ActivitiesTheteachershouldbeawareofthecomplexitiesofthischapterandshoulddisplaythegeometriccutoutsoftheidentitiesonthesoftboardduringtheweek.Thethreeidentitiesshouldalsobewrittenonchartpaperanddisplayedonthesoftboard.Thisislatentlearningwherethestudentsareencouragedtouseitforreferencewhiledoingsumsandinthiswaytheidentitiesbecomeembeddedintheirminds.

Summary of Key Facts• Apolynomialwithmultipletermshasacommonfactorintermsofanumberoravariable.

• Anexpressionwhichcanbewrittenasthedifferenceoftwosquarescanbefactorisedastheirsumanddifference.Whiledoingthat,studentsfirstlookforacommonfactororterm.

• Whentherearemultipletermsandthefactorsarenotobvious,theyaregenerallyrearrangedandregroupedsothatacommonfactorisobvious.

iv48 1Factorization of Algebraic Expressions

?OOPS

!

Frequently Made MistakesThefactthatsometimesthesumsneedtobefactorisedmultipletimesisabitchallengingforstudents.Itisimportantthattheteacherhighlightsthatcheckingwhetheracommonfactorcanbefoundshouldbedonebeforeproceedingwiththefactorizationusingidentities.

Lesson Plan


Factorizationofalgebraicexpressions


Multiplefactorization

Suggested Duration

1period

Key Vocabulary

Factorization,Differenceoftwosquares,Commonfactor

Method and Strategy

Thefirstandforemostruletoteachistocheckforcommonnumbersasfactorsandthencommonvariables.Thecommonvariablesandnumbershavetobethesmallestforthemtobeafactorforallterms.

Example

4xy + 6x2y + 10xyz Thecommonfactorsthatarevariablesarexandy.2isthecommonnumberfactorforall.Hence2xyisthecommonterm.2xy(2+3x+5z)

Furthermore,formultiplefactorization,thedifferenceoftwosquareswillbeappliedafterthecommonfactorization.

Example

27x³ – 3x3x(9x² – 1)3x(3x + 1)(3x – 1)3xisthecommonfactor.

Written Assignment

SelectedsumsfromExercise10aand10bcanbedoneinclassoncetheexampleprovidedabovehasbeendoneontheboard.

Aworksheetcanbeprovidedasahomeworkassignment.

v491 Factorization of Algebraic Expressions

Example

Answers

1) 90x2y2 –10 1) 10(3xy +1)(3xy – 1)

2) 2x2 + 10x + 50 2) 2(x+5)2.

3) 49 – 4a2 3) (7+2a)(7 – 2a)

4) 81a4 –1 4) (9a2+1)(3a + 1)(3a – 1)

5) (x–4)2–25 5) (x+1)(x–9)

EvaluationAtthebeginningofeachlessonthereshouldbeatwo-minuterecaptestoftheconceptstaughtinthepreviouslesson.Peercheckingcanbedoneandthesumssolvedontheboard.Conductingthisactivitythroughoutthischapterwillnottakenotmorethanfiveminutesineverylessonandwillensurethattherearenogapsintheunderstandingandapplicationoftheconcepts.


• factorisealgebraicexpressionsbyapplyingalgebraicidentities,and

• factorisealgebraicexpressionsbymakinggroups.

iv50 1

11 Simple Equations

Specific Learning Objectives• Theconceptofalgebraicequations

• Axiomsofsolvingalgebraicequations

• Procedureforsolvinganalgebraicequation

• Forming,orconstructing,analgebraicequation


Prior Knowledge and RevisionStudentshavebeenformingalgebraicexpressionsoutofsentencesandstatements.Thealgebraicequationhasbeenintroducedinpreviouslessons.Theconceptoftransposingshouldberevised.Thekeypropertiesshouldalsobedoneontheboard.

Thefactthat'LHS = RHS'shouldberevised.

Analgebraicequationisacombinationoftermsthatareconjoinedbyan'equalto'signstatingthatthevariablesandnumbersoneachsidesareequal.Thisaperfectwayoffindingthevalueofanunknownvariable.Theformationofsuchmathematicalstatementsisfundamentaltoproblemsolvinginarithmeticandgeometry.Ithastobehighlightedthatalgebraisnotanisolatedbranchofmathematics,butthestructuralbaseofmathematics.

Real-life Application and ActivitiesTherearedifferenttypesofequations:linearequations,equationswithbrackets,andequationswithdenominators.Eachtypeofequationshouldbetaughtseparatelyinadifferentlesson.AmixedexercisesuchasExercise11acanthenbegiven.

Wordproblemscanbeconvertedtoreal-lifesituationsbysubstitutingthenamesofthestudentsinthequestions.Similarly,real-lifewordproblemscanbewrittenontheboardandcanberoleplayed.

v511 Simple Equations

Example

IfAlihastwobrothersinreal-life,theteachercanmakeawordproblemknowingthattheyallliketoplaycricket.

Question

IfAliscored53runsandhisbrotherAmirscored23runsandthecollectivescoreofallthreebrotherswas212,howmanyrunsdidUmar,thethirdbrother,score?

Solution

53 + 23 + x = 212

x + 76 = 212 x = 212 – 76

x = 136

Umarscored136runsinthecricketmatch.

Summary of Key Facts• A(+)plussigntermtransposestotheothersideasaminussign.

• Acoefficientinthedenominatorwillgoontheothersideasanumerator.

• AllvariabletermsarecollectedontheLHSandtheconstantsaremovedtotheRHS.

• Simplifyequationsbyopeningbrackets.Multiplythetermoutsidethebracketbyallthetermsinside.Thisiscalledexpansioninalgebra.

• Iftherearefractionalterms,thentheLCMshouldbefoundtosimplifytheterms.TheLCMwhichisinthedenominatoristhentransposedontheothersideoftheequation.

• Tosolvewordproblems,thefirststepistoconverteachphraseintoamathematicalexpressionwheretheunknownvalueissubstitutedwithavariable.

• Theprocessofsolvingtheequationisthesameastofindthevalueoftheunknownvariable.

?OOPS

!

Frequently Made MistakesStudentswhoareweakatalgebraicruleswillhavedifficultyintransposingandsolvingtheequation.Ifthatisthecase,theteachercanrevisitthenumberlineconceptandexplainhowtherulesofalgebraicsignsarederived.

Lesson Plan


Algebraicequations


Solvingequationswithfractionalterms

Suggested Duration

1period

iv52 1Simple Equations

Key Vocabulary

Denominators,LCM,Transpose

Method and Strategy

Example

Solve3x+116

+ 2x–37

= x+38+3x–1

14.

Solution:

3x+116

+ 2x–37

= x+38+3x–1

14

3x+116

–x+38

= 3–114

– 2x–37

3x+1–2(x+3)16

= 3x–1–2(2x–3)14

3x+1–2x–616

= 3x–1–4x+614

x–516

= –x+514

14(x–5) = 16(5–x)

14x–70 = 80–16x30x = 150

x = 5

Thisisthemostcomplexformofsolvinganequationatthislevel.Theconceptoftransposingcomesrightattheend.InitiallytheLCMisfoundandthenumeratorsaremultipliedbythenumberfoundbythedivisionoftheLCMandthedenominator.Whendoingso,thestudentshavetobeverycarefuloftheminussignasallsignswillchangewhenmultipliedbyanegativenumber.Thedenominatorsarethencross-multipliedoncetheLHSandRHSbothhavesingleterms,keepinginmindtherulesoftransposing,andtheequationisthensolved.

Thesetypesofsumsshouldbedoneontheboardandstudentscantaketurnstosolvethem.Alotofpracticeworksheetsshouldbehandedoutforhomeworkandclassworkassignments.

Written Assignment

Alleven-numberedsumsofExercise11acanbedoneinclassandtheodd-numberedsumscanbegivenforhomework.Thiswillensurethatsumsofalllevelsofdifficultyaredoneinclassandthenparallelsumsaregivenforhomework.

EvaluationThisisquiteacomprehensivechapterandtwoassessmentscanbegiven.Onecaninvolvealltypesofequationsandtheothercanbecompletelybasedonwordproblemswherethestudentswillbeexpectedtoformtheequationandthensolveit.


• solvealgebraicequationsbyapplyingrelevantaxiomsandidentities.

v531

12 Lines and Angles

Specific Learning Objectives• Conceptsofperpendicularandparallellines

• Constructingperpendicularandparallellines

• Propertyofatransversalanditsconstruction


Prior Knowledge and RevisionStudentsarewellawareofthegeometrystrandofmathematics.Itisimportanttorecallthecorrectuseofthegeometricinstruments.

• Aprotractorisusedtoconstructangles.

• Apairofcompassesisusedtoconstructlinesegments.

Recognitionanddefinitionsoflines,rays,andlinesegmentsshouldalsobequicklyrevisitedinclass.Whilerevisingthethreetypes,thedifferencebetweenthethreewithregardtotheendpointshouldbeemphasised.

Real-life Application and Activities

Activity

Afunwaytoexplainanglesformedbythetransversalonparallellineswouldbewithanactivity.

Everystudentwillneed:

A4paper,threestraws,apairofscissors,agluestick,andaprotractor.

AskeachstudenttopastetwostrawsontotheA4paperparalleltoeachotherandthethirdasatransversal.

iv54 1Lines and Angles

Labeltheanglesformedas:a,b,c,d,e,f,g,andh.

Thencompareandmeasurethem.

a b

d c

e f

g h

Pairsaddingupto180°:a+b,b+c,c+d,d+a,e+f,f+h,g+e,andg+h

Corresponding∠s:bandf,candh,aande,dandg(Theseareequalpairs.)

Alternate∠s:cande,dandf(Theseareequaltoeachother.)

Interior∠s:candf,dande(Thesearenotequalbutaddupto180°.)

Summary of Key Facts• Perpendicularlinesaretwolinesthatintersecteachotheratarightangle.

• Twolinesthathaveaconstantdistancebetweenthemandthereforenevermeetareparallellines.

• Whenparallellinesarecutbyatransversal,alternate,corresponding,andinterioranglesareformed.

CorrespondinganglesformanF. AlternateanglesformaZ. InterioranglesformaU.

• Adjacentanglesshareacommonvertex.

• Adjacentanglesthataddupto90ºarecalledcomplementaryangles.

• Adjacentanglesthataddupto180ºarecalledsupplementaryangles.

• Twononadjacentanglesformedbytwointersectinglinesareequalandarecalledverticallyoppositeangles.

?OOPS

!

Frequently Made MistakesStudentssometimesconfuseinteriorangleswithcorrespondingandalternateangles.Interioranglesaddupto180ºbutalternateandcorrespondinganglesareequaltoeachother.

v551 Lines and Angles

Lesson Plan


Linesandangles

Specific learning objectives

Constructionofperpendicularlines

Suggested duration

1period

Key vocabulary

Perpendicular

Method and Strategy

Thisactivitycanbemadefunbydoingitonchartpaper.

Thestepsofconstructioncanbewrittenontheboard.Twostudentsarechosenandeachisgivenasheetofchartpaper.

Theyaretimedforfiveminutesafterthesumshavebeenwrittenontheboard.Thepairthatmanagestofinishthemostconstructionsaccuratelyarethewinners.

Steps of constructions:

• LetABbethegivenline.TakeapointNonthelineABandapointPatacertaindistancefromtheline.

• Placethesetsquareinsuchawaythatitisatarightangletothebaseline.

• SlidetherulerandthesetsquareinsuchawaythatthepencilmarksthepointsPandNofthemeasureddistance.

• Withapencil,drawalinealongthesideofthesetsquare.PNisperpendiculartoAB.

Written Assignments

Oncetheactivityisdone,afewconstructionsumscanbedoneinstudents'notebooks.Thestepsofconstructioncanbewrittenasbulletpointsintheirnotebooksforquickreferralatanylatertime.

Evaluation

Thischapterisafunchapterandthestudentscanbemarkedontheirgroupactivitiesandassign-ments.


• constructperpendicularandparallellines,

• describethepropertiesofperpendicularandparallellines,

• identifyadjacent,complementary,andsupplementaryangles,and

• findtheunknownangleinatrianglewhentwoanglesaregiven.

iv56 1

13Geometrical Constructions

Specific Learning Objectives• Constructingperpendicularlinebisectors

• Constructinganangleequaltoagivenangle

• Constructinganangletwicethesizeofagivenangle

• Constructinganglesofmeasurement30º,90º,120º,135º,150º,and165º

• Constructingtriangleswiththreesides

• Constructingtriangleswithtwosidesandtheincludedanglegiven

• Constructingtriangleswithtwogivenanglesandthelengthoppositetotheangles

• Constructingaright-angledtrianglegiventhehypotenuseandoneside

• Constructingequilateralandisoscelestriangles


Prior Knowledge and RevisionThischapterinvolvestheuseofgeometricinstruments.Priortobeginningthischapter,thestudentsshouldrevisethekeywords,bisection,equilateral,andisoscelestriangles.Thepropertiesoftypesofanglesandtrianglesarealsoimportant.

Boardgeometricinstrumentsshouldbeusedtoteachthestudentsthecorrecthandlingoftheinstruments.

Real-life Application and ActivitiesTheteachercanstimulatetheinterestofthestudentsbyinformingthemthatwecanrole-playbyactingasarchitectsandcanplandesignsusingthegeometricinstruments.

Henceforth,allworkishands-onandastheteacherexplainsthestepsofconstruction,thestudentsshouldwritethemintheirnotebooksandconstructfiguresaccordingly.

v571 Geometrical Constructions

Summary of Key Facts• Toconstructaperpendicularbisectoroflinesegments,thecompassesshouldmeasuremorethanhalfthelengthoftheline.Drawtwoarcsaboveandbelowthelinesegmentthatcuteachother,andjointhemtogetabisector.

• Toconstructanglebisectors,thecompassesshouldmeasurelessthanhalfthelengthofthelines.Drawtwoarcsbetweenthearmsoftheanglethatintersecteachother.Drawalinethroughthepointofintersectionandthevertexoftheangle.

• Toconstructa60ºangle,drawthebaselineandwithasuitableradius,drawanarcthatcutsthebaseline.Withthesameradius,drawanotherarcwhichcutsthepreviousarc.Extendthelinefromtheendpointtothepointofintersectionofthearcs.

• Anangleof30ºisconstructedbybisectingtheconstructed60ºangle.

• Toconstructa90ºangle,followthestepsofconstructiongiveninthebook.

• Anangleof45ºisconstructedbybisectingtheconstructed90ºangle.

• Toconstructa120ºangle,drawtwo60ºanglesside-by-side.

• Toconstructanangleof150º,drawa90ºangleanda60ºangleside-by-side.

• Toconstructa165ºangle,drawa150ºangleandanadjacent30ºanglewhichisbisectedtogeta15ºangle.Hence150º+15º=165º.

• Atrianglecanbeconstructedinmanywaysdependingontheconditionsgiven.

Wecanconstructtrianglesif:

_ allthreesidesaregiven.

– twosidesandanincludedanglearegiven,

– onesideandtwobaseanglesaregiven,

– thelengthsofonesideandthehypotenuseofaright-angledtrianglearegiven,

– theperimeterandratiobetweenthelengthsofsidesaregiven,

– thealtitudeofanequilateraltriangleisgiven,

– theverticalangleandthealtitudeofanisoscelestrianglearegiven,

– thebaseangleandthealtitudeofanisoscelestrianglearegiven,

?OOPS

!

Frequently Made MistakesStudentsshouldbemadeawarethatthecorrectuseofgeometricinstrumentsforexamplehowtoholdandplacetheinstruments,isimportanttoproduceaccuratedrawings.

Lesson Plan


Geometricconstructions


Constructingtriangleswiththeratioofthesidesandperimetergiven

iv58 1 Geometrical Constructions

Suggested Duration

1period

Key vocabulary

Ratios,Perimeter,Compasses

Method and Strategy

Toconstructatrianglewithagivensetofratios,firstrevisetheconceptofproportionalratios.

Example: TriangleABChassidesintheratioof1:4:5.

Ifperimeteris30cm,thenthesideswillbecalculatedas:

AB = 110 × 30 = 3cm

BC = 410

× 30 = 12cm

AC = 510

× 30 = 15cm

DrawAC,thelongestside,asthebaseline.WithAandCascentresdrawtwoarcswithradius3cmand12cmrespectively,cuttingeachotheratB.

JoinBtoAandC.

Itshouldbepointedoutthatsometimesmathematicalcomputationsaredonebeforeproceedingwithconstructionofthetriangle.Similarly,toconstructatrianglewithgivenaltitudeorverticalangle,mathematicalworkingwillberequired.

Written Assignment

Practicesumsshouldbegiveninclassandtheteachershouldapproacheachchildindividuallytohelpthemusethegeometricinstrumentsalongwithhelpingwiththemathematicalconcepts.

Evaluation

Markedassignmentsshouldbegiveninclassandhomeworkperiodicallybeforetakingacomprehensiveassessmentofthischapter.

Thetestshouldhaveachoiceofoptionsofdifferentcasesofconstructionsandatleast5sumsofconstructionsshouldbegivenforthedurationofaone-periodtest.


• bisectagivenlinesegmentusingarulerandapairofcompasses,

• constructalineperpendiculartoagivenline,

• drawanglesofarequiredsizewithaprotractor,

• constructanglesmeasuring60°,90°,120°,135°,150°,and165°,

• bisect60°,90°,120°,135°,150°,and165°angles,and

• constructatrianglewhendifferentmeasurementsofsidesandanglesaregiven.

v591

14 Circles

Specific Learning Objectives• Thepropertiesofcircles

• Constructingcircles,semicircles,andsegments

Suggested Time Frame2to3periods.

Prior Knowledge and RevisionPartsofacirclehavebeentaughtearlier;however,thedifferencebetweenachord,diameter,and,radiusshouldbeexplainedwiththehelpofdiagrams.

Theradiusisthedistancefromthecentretothecircumferenceofthecircle,whereasthediameteristhemeasureofthecircleacross,passingthroughthecentre.

Theradiustouchesthecircumferenceofacircleatonepoint,whilethediametertouchesitattwopoints.

Theradiusanddiameterareconstantvalues.

Achordtouchesthecircleattwopointsbutdoesnotpassthroughthecentre.

Asemicircleishalfacircle,subtended(meetingattwopoints)byadiameter.Aquadrantisaquarterofacirclesubtendedbytworadii.

Thecircumferenceofacircleisitsperimeterandthecircularmeasureofitsboundary.

chord

radius

diameter

semicircle

iv60 1Circles

Real-life Application and ActivitiesConstructionofcirclesandsemicirclesisrelativelyeasyasonlytheuseofcompassesisrequiredandthestudentsneedtogetthevalueoftheradiusonthecompassesandthecircleorsemicirclecanbedrawn.

Thetheoremsofcirclesareextremelycriticalandthesecanonlybeexplainedifdonepractically.

A C

B D

O

A C

B D

O

L MO

A

P

QB

L MO

A

A B

O

A BO●

Activity

Youwillneedchartpaper,drawingpins,andthread.

Cutoutabigcirclewiththewidthofthechartpaperasthediameter.

Putthedrawingpinsattheendpointofthediameter.Putathreadaroundthedrawingpinstomakealoop.Pullthethreadandpinitoppositethediameteronthesemicircle.

Measuretheangleformedonthecircumferencewithaprotractororasetsquare:itwillbe90º.

Similarly,onthesamechartpaperloopathreadaroundthetwodrawingpinsandpinitonthecircleattwopointsonthecircumference,thistimetocreateachordandnotadiameter.Measurethedistanceofthechordfromthecentreandusethesamemeasurementtotieanotherchordontheothersideofthecentreofthecircle.Measurethelengthofthethreadsformingthechord:theywillbeequal.

Bythismethodalltheoremscanbeproved.Thestudentswritethetheoremonthechartpaper.Helpstudentstoproveallthetheoremspractically.

v611 Circles

Summary of Key Facts

Elements of a circle

Acircleistracedonaplanebyapointmovinginsuchawaythatitsdistancefromanotherfixedpointontheplaneisalwaysconstant.

Semicircle

D

r

r

D

B

A

C

r

r

segment

Centre:

Thecentreofacircleisthefixedpointontheplanefromwhichthedistanceofthemovingpointisalwaysconstant.

Circumference:

Thecircumferenceofacircleistheboundaryorperimeterofthecircle.

Radius:

Thedistancebetweenthecentreandapointonthecircumferenceiscalledtheradius.Itisdenotedby‘r’.

Sector:

Theareabetweentworadiiiscalledasector.

Minor sector:

Iftheanglebetweenthetworadiiislessthan180°,thenthesectorisaminorsector.

Major sector:

Iftheanglebetweenthetworadiiisgreaterthan180°,thenthesectorisamajorsector.

Diameter:

Thediameteristhedistancebetweentwopointsonacircumferencealongastraightlinethatpassesthroughthecentre.Thediameterisdenotedby‘D’.Itisequaltotwicetheradiusi.e.D=2r.

Arc:

Anypartofthecircumferenceorperimeterofacircleisknownasanarcofthecircle.ThecurvedsegmentABisanarcofthecircle.

iv62 1Circles

Chord:

Alinesegmentthatjoinstheendpointsofanarciscalledachord.ThelinesegmentABisachordofthecircle.Thediametercanbedefinedasachordthatpassesthroughthecentreofthecircle.Thediameteristhelongestchordofacircle.

Segment:

Asegmentofacircleistheareaenclosedbetweenanarcandthecorrespondingchord.

Major arc:

Whenacircleisdividedintotwopartsbyachord,thearcthatformsthelargerpartiscalledthemajorarc.

Minor arc:

Whenacircleisdividedintotwopartsbyachord,thearcthatformsthesmallerpartiscalledtheminorarc.

Major segment:

Whenacircleisdividedintotwopartsbyachord,thelargersegmentformediscalledthemajorsegment.

Minor segment:

Whenacircleisdividedintotwopartsbyachord,thesmallersegmentformediscalledtheminorsegment.

Semicircle:

Asemicircleisonehalfofacircleformedwhenacircleisdividedbythediameter.

• Concentriccirclesarecircleswithacommoncentrebutdifferentradii.

• Equalchordsareequidistantfromthecentre,andviceversa.

• Aperpendicularlinedrawnfromthecentretothechordbisectsthechord,andviceversa.

• Equalchordssubtendequalanglesatthecentreandviceversa.

• Theanglesubtendedbythediameterofacircleatthecircumferenceofthecircleisarightangle.

?OOPS

!

Frequently Made MistakesStudentsevenatahighergrademixuptheconceptsofchordsanddiameters.Thiscausesfurtherconfusionlateronwhileworkingonthetheorems.

v631 Circles

Lesson Plan

Sample Lesson Plan

Topic

Circles


Thepropertiesofthecircle:

• equalchordsareequidistantfromthecentre,

• theperpendicularlinefromthecentrebisectsthechord,and

• equalchordssubtendequalanglesatthecircumference.

Suggested Duration

1period

Key Vocabulary

Chords,Equidistant,Subtend,PerpendicularandBisect.

Method and Strategy

Theactivitystatedearliercanbeshowntothestudentstorevisethetheorems.However,forthetheoremstobemoreeffectiveinapplication,alotofpracticesumsshouldbedone.

Written Assignments

Exercise14bshouldbedoneinclassontheboardandthengivenforhomework.

EvaluationAcomprehensiveassessmentonthischaptershouldbegiven.Itshouldbepointedouttostudentsthatinaccuracyinconstructionwillresultinthelossofmarks.


• describeacircleintermsofitselements,

• constructcircles,semicircles,andsegmentsusingapairofcompasses,and

• demonstratethepropertiesofacircle.

iv64 1

15 Congruence and Similarity

Specific Learning Objectives• Propertiesofcongruenttriangles

• Propertiesofsimilartriangles

• Applyingpropertiesofsimilarityandcongruence

Suggested Time Frame6to8periods.

Prior Knowledge and RevisionStudentsareawareoftrianglesandotherpolygons;inthischapteranewconceptofcongruenceandsimilarityisintroduced.

Theteachershouldbrainstormwiththestudentsandpromptandexplainthemeaningsofcongruenceandsimilarity.

Congruence:exactsame(equal)size,angles,facesetc.

Similarity:sameshapebutdifferentsizes

Thestudentsmaycomeupwithgeometricinstrumentsofthesamebrandthatareexactlythesameandlinkitwithcongruence,andofdifferentbrandsthataresimilarinrelationtosizes.

Real-life Application and ActivitiesTheteachershouldexplaincongruencewithreal-lifeexamples.Variousexamplesareapartmentsinbuildingcomplexes,potsandpansoftheexactsamesize,andLegoblocks.Duetothefactthattheyareexactlyequalinallaspectsofmeasurement,theytendtolooklikeclones.Thisisahelpfulanalogytocreateandmakealistofreal-lifecongruence.

SimilarityisbestexplainedwiththeexampleoftheRussiandollsthatfitoneinsidetheother.Thoughtheyareofdifferentsizes,theyfitinsideeachotherasthecurves/anglesarethesame.Theteachercanbringinclayplantpotsthataresimilarandofdifferentsizes,andshowthatthemeasurementsoflengthareproportionalbuttheangularaspectremainsexactlythesame.

v651 Congruence and Similarity

Similarityisrelatedtoenlargementandmagnificationbyascalefactor.Anotherreal-lifeexamplecouldbeenlargedpicturesonacomputerprintingeachpictureinvarioussizes.

Studentsshouldbeencouragedtomakeatable/chartpresentationofsimilarandcongruentobjectsinreal-life.

Summary of Key Facts• Congruencyoftrianglesisdefinedbyfourproperties:SSS,SAS,ASAandRHS.

• Similarityiswhenallcorrespondinganglesareequalandcorrespondingsidesareinproportionalratios.

?OOPS

!

Frequently Made MistakesThedifferencebetweensimilarandcongruentfiguresmustbeexplainedclearlyandstudentsmustlearntheproperties.Theytendtojumbleuptheproofofsimilaritywiththatofcongruence.

Lesson Plan

Sample Lesson Plan

Topic

Congruence


ThecasesofRHSandSASandtheirapplication

Suggested Duration

1period

Key Vocabulary

Hypotenuse,Adjacentandincludedangle

Method and Strategy

StudentshavealreadybeenintroducedtothecasesofSSSandSAS.Itshouldbemadecleartothemthatwhenaright-angledtrianglehasasideandhypotenusecongruent,thecasebecomesRHS.However,toprovecongruencywiththeincludedangleshouldbebetweenthecongruentsides.Invariably,atrianglewithtwosidesandoneanglewhichisnotinbetweenthetwocongruentsidesmakesthecasenullandvoid.Similarly,inthecaseofRHS,tworight-angledtrianglesneednotbecongruentiftwoanglesandaside,orarightanglewithitsarmscongruentisgiven.Then,thecaseswouldbecomeASAandSASrespectively.

Example:

A B

C

D E

F

Included∠Aand∠Darenotgiven,thereforetrianglesarenotcongruent.

iv66 1Congruence and Similarity Similarity and Congruence

A DB E

C F

✗ NotRHS

✓ SAS

Written Assignment

PracticesumsfromExercise15inthechaptercanbedoneinclassandtherestcanbegivenforhomework.

Thefollowingssumcanbegiveninclassasaquiz.

1. Statethecaseofcongruencyifcongruent.

Answers

(a)

C D

BA

Yesthetrianglesarecongruent.

Property:AAS

(b)

A B

C

P Q

R


Property:SAS

(c)

A D

CB


Property:AASorSSS

v671 Congruence and SimilaritySimilarity and Congruence

(d)

60° 60°


Property:SAS

(e)

A B

C

P Q

R

Thetrianglesarenotcongruent.

Nopropertyissatisfied.

EvaluationAcomprehensiveassessmentshouldbegivenattheendofthetopicbutinbetweenshortquizzesontheboardcouldbeconductedtowardstheendofeachlessontocheckstudents'understanding.Thischapterintroducesanentirelynewconceptandstage-by-stageassessmentisnecessary.


• identifycongruentfigures,

• identifysimilarfigures,

• establishcongruenceandsimilaritybetweengeometricfigures,and

• testforthecongruencyoftwotrianglesusingtheSSS,ASA,andRHSpropertiesofcongruencyoftriangles.

iv68 1

1 Quadrilaterals

Specific Learning Objectives• Propertiesofquadrilaterals:parallelograms,rhombuses,rectangles,andsquares

• Constructingaparallelogram


Prior Knowledge and RevisionTheconceptofshapesmadeby3ormorelinesegmentshasalreadybeenintroducedtothestudents.Abrainstormingsessiononidentificationofvariousshapescanbedoneontheboard.Aftertheidentificationofvariousquadrilaterals,theteachershouldpromptthestudentstoidentifythepropertiesofeachshape.

ExampleArhombushasallsidesequalbutitisnotasquare.Why?(Theanglesarenotrightangles.)

Aparallelogramhaslengthandbreadthbutitisnotarectangle.Why?(Therearenorightanglesatthevertices.)

Akiteisanunusualquadrilateralwithequalsidesadjacenttoeachother(twosmalladjacentsidesequalandtwolongeradjacentsidesequal).

Atrapeziumisdifferentfromaparallelogram.Givetwopropertiessupportingthestatement(onlyonesetofparallellinesandtheparallellinesarenotequalinlength).

v691 Quadrilaterals

IsoscelesTrapezium

A D

B C

A D

B C

Rhombus

A D

B C

ParallelogramTrapezium

A D

B C

A D

B C

Square

A D

B C

Rectangle

A

C

B DKite

Real-life Application and ActivitiesToreinforceknowledgeoftheproperties,thestudentscanbedividedintogroupsandeachgroupcanbeassignedaquadrilateral.Thegroupthenmakesacut-outoftheshapeassignedfromchartpaperandgiveaminute-longpresentationonthepropertiesoftheassignedquadrilateral.


Quadrilaterals

Kite

Parallelogram

RhombusRectangle

Square

Trapezium

Isoscelestrapezium

NonParallelograms

iv70 1Quadrilaterals

• Aparallelogramisaquadrilateralinwhichtheoppositesidesareparallel.

• Oppositeanglesofaparallelogramarecongruent.

• Arhombusisaparallelograminwhichallfoursidesareequal.

• Anadditionalpropertyofarhombusnotfoundinaparallelogramisthatitsdiagonalsbisectatrightangles.

• Arectangleisaparallelograminwhichalltheanglesarerightangles.

• Asquareisarectanglewithfourequalsides.

• Atrapeziumisaquadrilateralwithonlytwosidesparallel.

• Anisosceles trapeziumisatrapeziuminwhichthenon-parallelsidesareequal.

• Akiteisaquadrilateralinwhichthetwopairsofadjacentsidesareequal(ingeneral,theoppositesidesarenotparallelorequal).

• Thetreediagramaboveexplainstherelationshipbetweenthedifferentpolygons.

• Thesumoftheanglesinaquadrilateralis360º.

Quadrilateral

parallelograms

rectangles

rhombus

Rhombus Parallelograms

allsidesareequal

oppositesidesareequal

diagonals bisectat(right∠s)

adjacentanglesare

supplementary,therearetwosetsofparallel

lines

Rectangles Parallelograms

90°∠satthevertices

anglesarenotat90ºadjacent∠sare

supplementary,equallengthsandequalbreadths

?OOPS

!

Frequently Made MistakesStudentsfindtherelationshipbetweentheshapesofquadrilateralsabitchallenging.Iftheshapesaretaughtinawaywheretheoverlappingpropertiesarefirstpointedoutandthentheadditionalpropertieswhichdifferentiateoneshapefromtheotherareexplaineditwillhelpthestudentsimmensely.Venndiagramsofsimilaritiesanddifferenceswillalsohelp.

v711 Quadrilaterals

Lesson Plan


Constructingaparallelogram


Constructingaparallelogram

Suggested Duration

1period

Key Vocabulary

Adjacent,Includedangle

Method and StrategyWhendoingconstructions,theinstrumentshavetobeinsoundcondition.Oftentheprotractorreadingsarenotclearorhavebeenerased.Similarlythecompassesisloosewhichresultsininaccuratemeasurements.

Case:whentwoadjacentsidesandincludedanglearegiven

Therearetwowaysofapproachingthiscase.Iftheincludedangleisgivenandweknowthatadjacentanglesaresupplementary,thesecondanglecanbecalculatedandbothanglescanthenbedrawnonthebaselinesegment.Arcsofmeasurementequaltothebreadtharethenmadeonthearmsoftheanglesandtheparallelogramiscompleted.

Thesecondapproachismentionedinthetextbookonpage201.Theneedforcalculatingthesecondangledoesnotariseasthearcmeasurementsofthelengthandbreadthgivesusthethirdvertex,andsubsequentlythefourthvertex.

Written Assignments5sumsofconstructioncanbegiventothestudenttodoinclassandthenasetofanother5sumscanbegivenforhomework.

Theworkedexamplesinthetextbookcanalsobedoneintheirnotebooks.

EvaluationExercise16isacomprehensiveexerciseontheconceptsofthischapterwithmultiplechoicequestions.Anassessmentalongthelinesofthisexercisecanbegiven.

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

• identifythepropertiesofparallelograms,rectangles,squares,andrhombuses,and

• constructparallelogramsandrhombuses.

iv72 1

17Perimeter and Area of Geometric Figures

Specific Learning Objectives• Calculatingtheperimeterofaparallelogram,rectangle,square,triangleandtrapezium

• Calculatingtheareaofaparallelogram,rectangle,square,triangle,trapezium,rhombus

• Introducingtessellations

• Calculatingtheperimeterandareaofacircle


Prior Knowledge and RevisionThischapterisacontinuationofconceptstaughtintheearliergrades.Thestudentsareawareoftheconceptsofareaandperimeter,sonoformalintroductionisneeded.

Revisionofshapesandcalculatingtheareaandperimeterofcompositeshapesmadeupofsquaresandrectanglescanbedone.

Activity

Theteachercanmaketherevisionfunbybringingcut-outsonchartpapersanddividingtheclassintogroupsandaskingthemtocalculatetheareaandperimeterofthecut-outs.Thesecut-outscanbeputonthefloorandthegroupscanworkonthefloor.Thisactivityshouldnottakemorethanfiveminutes.Thecalculationscanbedoneintheirexercisenotebooks.

Shapesareeverywhere;architectureinvolvesspatialgeometry,andtheconstructionofahouseinvolvescalculationofmaterialsrequired,areas,etc.Evensomethingasrelativelysimpleasmakingawoodentableorcupboardrequiresknowledgeoftheconceptstaughtinthischapter.Ifthereisanin-househandymanorcarpenterinschool,hecanbeinvitedtothelessontoexplainthedimensionsandmaterialrequirementsofmakingadesk.

Studentsshouldnotbegiventheformulaeasmathematicalcomputationsalone.Theyneedtounderstandthederivationtoappreciatethereal-lifeapplication.

v731 Perimeter and Area of Geometrical Figures

Activity

Thingsyouwillneed:

2sheetsofcolouredchartpaper,thickmarkers,ruler,andapairofscissors

Askthestudentstodrawaparallelogramofthesamesizeasthechartpaper.Youcanhelpthemdrawtheparallellines.Helpthestudentstousetheprotractortodrawtheinteriorangles.Labeltheshape.

Theywillseetheshapecanbedividedintotwocongruenttriangles.Theconceptofcongruencycanbeappliedhere.

Nowshowthederivationontheboardandexplainthatthetrianglesaretwocongruentshapessothe½oftheformulaiscancelledandweendupwithb × h.

Real-life Application and ActivitiesThevalueofpi(π)caneasilybeexplainedwithaninterestinghands-onactivity.

Activity

Youwillneed:a1metrelengthofyarn,(anythickthreadwillalsodo),amarker,differenteverydayobjectsthatarecirculare.g.aCD,circularplate,circularsharpener,a30cmruler,andplaydoughoranyadhesive.

Fastentheyarnaroundthecircularobjectwiththeplaydoughsoitstaysinposition.

Measuretheyarnandrecordthelengthincentimetre.

Nowplacetheyarnacross,makingsureitpassesthroughthecentre.

Measureandrecordthelength.

Askthestudentstocalculatethevalue:around/across.

Theyshouldcometoavaluecloseto3.142.

Theyshouldrepeatthisprocesswithtwomorecircularobjectsofdifferentsizes.

Theteachershouldthenpointouttheconstantvalueofπthatitis3.142forallcircles.

Oncetheformulaforthecircumferenceisintroducedhe/shewillrelatearoundtothecircumferenceandacrosstothediameter.


L

L

L

L

B B

L

L

A

C

B C D

B

A

H

iv74 1Perimeter and Area of Geometrical Figures

• Thedistancearounda2Dshapeistheperimeter.

Perimeterofarectangle = 2(l+b)

Perimeterofasquare = 4l

Perimeterofaparallelogram = 2(l+b)

Perimeterofatriangle = a+b+c

Perimeterofatrapezium = a+b+c+d

Perimeter(circumference)ofacircle=2⊼ror⊼d

• Theareaofa2Dshapeisthenumberofsquareunitsthatthefigurecovers.

Areaofarectangle = l × b

Areaofasquare = l × l = l2

Areaofaparallelogram = b × h

Itshouldbepointedoutthattheheightoftheparallelogramiscriticalincalculatingtheareaoftheshape.

Areaofatriangle =12 (b × h)

Areaofatrapezium = 12 (a+b)× h (aandborparellalsides)

Areaofarhombus= 12 (d1 × d2) (d1andd2orthetwodiagonal)

Areaofacircle=⊼r3

• Theconceptofaltitudeusedinparallelograms,triangles,andtrapeziumsshouldbemadeclear.Itistheperpendiculardistancetothebaseoftheshape.Incidentally,itisalsotheshortestdistancebetweenthetwosides.

• Tofindtheareasthatareborderstheconceptsofexternalareaandinternalareashouldbemadeclear.Oncethesehavebeenfound,theareasaresubtractedtogettheareaoftheborders.

• Tessellationsarerepetitivepatternsofthesameshape.Totessellateistorepeatapatterninsuchawaythatnogapsintheareaarecreated.

?OOPS

!

Frequently Made MistakesStudentsgenerallygetconfusedwiththeidentificationofthealtitudeandperpendicularlines.Theearlierchapteronthiscanberevisedtoobtaincorrectvalueswhicharesubstitutedintheformulae.

v751 Perimeter and Area of Geometrical Figures

Lesson Plan


Tesselation


Theconceptoftessellation

Suggested Duration

1period

Key Vocabulary

Tessellationortessellate,Polygon,Pattern

Method and Strategy

Real-lifeexamplesoftessellationscanbegiven.

ExampleHoneycombinabeehiveismadeupofhexagons.

Tilesonthefloorarerepetitivepatternsofthesameshape.

Activity

Ahands-onactivitycanbedoneinclass.

Eachstudentrequiressheetsofcolouredpaper,marker,ruler,gluestickandapairofscissors.

Asetofflashcardscanbemadewiththenameofadifferentshapewrittenoneachflashcard.Forexample,pentagon,parallelogram,equilateraltriangle,square,rectangleetc.

Eachstudentisaskedtopickaflashcardandthenaskedtomake10or12cut-outsoftheshape.Theyarethengluedsidebysideinhis/hernotebook,tocheckwhethertheshapestessellateornot.

Thisactivitywillbefunandtheconceptoftessellationwillbeveryclear.

Written Assignment

Anassignmentcanbegivenwherethestudentsareaskedtodrawtessellationsoffiveshapes.

iv76 1

Example

Tessellationsof:

Atessellationformedbyequilateraltriangles

Atessellationformedbysquares

Atessellationformedbyregularhexagons

A

BC

Atessellationformedbysquaresandregularoctagons

AtessellationformedbyparallelogramsAtessellationformedbyisoscelestrapeziums

Evaluation

AcomprehensivetestalongthelinesofExercises17aand17bcanbegivenonthecompletionofthechapter.


• calculatetheperimeterofaparallelogram,rectangle,square,andtrianglebyapplyingtherelevantformulae,

• calculatetheareaofaparallelogram,rectangle,square,triangle,andtrapeziumbyapplyingtherelevantformulae,

• calculatethecircumferenceofacirclewhenthediameterorradiusisgiven,

• calculatetheareaofacirclebyapplyingtheformula,and

• identifytessellatedpatternsintheenvironment.

Perimeter and Area of Geometrical Figures

v771

18Volume and Surface Area

Specific Learning Objectives• Calculatingthesurfaceareaofacube,cuboidandcylinder

• Calculatingthevolumeofacube,cuboid,andcylinder

Prior Knowledge and RevisionStudentscalculatedthesurfaceareaandvolumeofcubesandcuboidsinthepreviousgrade.

Abriefrevisionatthebeginningofthelessoncanbedonewherethefacesandedgesofacubeandacuboidareidentifiedandthesurfaceareaandvolumeformulaearerevised.

Activity

Netdiagramsofacubeandcuboidcanbephotocopiedandhandedtothestudents.Theycancutthemoutandtapetheedgestocreatea3Dshapeoutofa2Dcut-out.Thiswillhighlightthefactthata2Dshapecanbeconvertedintoa3Dshapethatwillhaveavolume.

Net diagram of a cubeSinceallthesidesareequal,thefacesareallequalinareaanddimensions.

Net diagram of a cuboidSincethedimensionsaredifferent,2faceseachhavethesamedimensionsandsamearea.

iv78 1Volume and Surface Area

Real-life Application and Activities

Activity

Therelationshipbetweenvolumeandbaseareaofcubes,cuboids,andcylinderscanbeexplained.Thebasicformulaforvolumeis:

Volume=basearea×height

Theshadedregionineachdiagrambelowisthebase.

Thereforethevolumeofeachshapecannowbeeasilycalculated.

i) Volumeofacuboid=basearea×height

V=(l × b)× h (baseisarectangle,thereforethebasearea=l × b)

ii) Volumeofacube=basearea×height

V=(l × l)× l = l3 (baseisasquare,thereforethebasearea=l × l) iii) Volumeofacylinder=basearea×height

V=(πr2)h (baseisacircle,thereforethebasearea=πr2)

l

ll

basearea

h

bbasearea

l

h

r

basearea

Summary of Key Facts• 1cubicmetre =1000litres

• 1litre =1000cm3

• Volumeofacube=l3

• Totalsurfaceareaofacube=6l2

• Volumeofacuboid=length×breadth×height

• Totalsurfaceareaofacuboid=2(l × b)+2(b × h)+2(l × h)

?OOPS

!

Frequently Made MistakesTheidentificationofdimensionswhenapplyingtheformulaisveryimportant.Thestudentstendtoputinthevalueofthediameterinsteadoftheradius.Similarmistakesalsooccurinthecaseofcubesandcuboids.

Thehands-onactivityofthenetdiagramwillensurethattheconceptsofthedimensionsandtheirshapesareclear.

v791 Volume and Surface Area

Lesson Plan


Volumeandsurfaceareaofacylinder


Calculatingthesurfaceareaofacylinder

Suggested Duration

1period

Key Vocabulary

Radius,Circumference,Height,Curvedsurfacearea,Totalsurfacearea

Method and Strategy

Activity

Themosteffectivewayofteachingtheformulaofthesurfaceareaofacylinderistotakeapieceofpaperandshowtheclassthelengthandbreadthofthepaper.

Highlightthefactthattherectangularpaperisactuallythecurvedsurfaceareaofthecylinderasitfoldstoformacylinder.

Puttwocirclecut-outsonthetopandbottomofthepapercylinder.Makesurethatthecircumferenceofthetwocirclesisequaltothelengthoftherectangle;onlythentheywillbeplacedperfectly.

Thisfactcanbepointedouttothestudents.

Curvedsurfaceareaofacylinder=2πr × h

Totalsurfaceareaofaclosedcylinder=2πrh+2πr2

r

r

r

h h

iv80 1Volume and Surface Area

Written Assignment

Questions6and7fromExercise18bcanbedoneinnotebooks.Theyshouldnotetheformulaedownwithmarkersintheirnotebooksbeforetheyproceedtodothesums.

5sumsoffindingtotalandcurvedsurfaceareasofcylinderscanbegivenforhomework.

EvaluationThisisanextremelyconceptualtopic.Quizzestofindtheareaorvolumeofanyoneshapeshouldbegivenatthebeginningofeachlesson.Thiswaytheconceptswillbefurtherenhancedasthechapterprogresses.Acomprehensiveassessmentcoveringallconceptsshouldbegivenoncethestudentsareconfident.Therevisionexerciseonpages236to237canbeusedtoassessmensuration.


• calculatethevolumeofacuboid,cube,andrightcircularcylinderbyapplyingtheformulae,

• calculatethesurfaceareaofacuboid,cube,andrightcircularcylinderbyapplyingtheformulae.

v811

19Information Handling

Specific Learning Objectives• Theimportanceofdatapresentation

• Differencebetweenungroupedandgroupeddata

• Conceptsofclassintervalandfrequency,rangeofdata,lowerandupperclasslimit

• Constructingafrequencydistributiontable

• Readingbarchartsandpiecharts

• Constructingbarcharts


Prior Knowledge and RevisionStudentsareawareoflinegraphsandbargraphs.TheteachercangiveaPowerPointpresentationandshowcolourfulslidesofvariousbargraphsandlinegraphs.Alistofquestionscanbereadoutandthestudentscananswerbylookingattheslides.Thisactivitywillnotonlyhelpthemrevisetheconcepts,butwillalsoaddvarietytomathematicslessons..

Real-life Application and ActivitiesStudentsareawareofsimpledistributionofdatawherefrequencyisnotmentioned.Thefactsthatthedataisnowgroupedandthequantityiswithinarangehavetobeexplainedclearly.Thestepsofconvertingrawdataintogroupeddataandthatofconstructingabargraph,havetobeexplainedclearlyandhighlightedasasoftboardpresentation.

Studentsshouldbeencouragedtodrawonchartpaperrepresentationsofabargraph.Theycanworkingroups.Thiswillenhancetheirunderstandingastheywillbenefitfrompeercooperation.

iv82 1Information Handling

Summary of Key Facts• Datainitsrawformiscalledungroupeddata.

• Tallymarksconsistofverticallineswiththefifthlinedrawndiagonallythroughthe4verticallines.Thisgivesabundleoffive.

• Afrequencydistributiontableconsistsofclassintervalsandtheircorrespondingfrequencies.

• Thedifferencebetweenthegreatestandsmallestdatavaluesiscalledtherangeofthedata.

• Apiechartrepresentsthedistributionintheformofsectorsofacircle.

?OOPS

!

Frequently Made MistakesStudentsoftenmakemistakesaddingfrequency.Inthecaseofapiechart,iftheanglesofthesectoraretobecalculated,thentheyshouldcheckthattheirsumis360º,asanglesatapointaddupto360º.Similarlycareshouldbetakenwhilecalculatingpercentagesastheirtotalshouldbe100%.

Lesson Plan


Piecharts


Tocalculatethevalueoftheanglesofapiechart

Suggested Duration

1period

Key Vocabulary

Sectors,Circles,Frequencydistribution,Piecharts

Method and Strategy

Apiechartrepresentsinformationinacircle.Eachdistributionisrepresentedbyasector.Allthesectorstogetherformonecompletecircle.Theangleofeachsectoriscalculatedarithmeticallyanditshouldbepointedoutthatsinceanglesatapointaddupto360º,theanglesofallsectorsshouldalsoaddupto360º.

Anydistributioncanbepresentedintheformofabargraphorapiechart.

v831 Information Handling

Example

Thenumberofstudentsindifferentclassesofaschoolwholiketoplayhockeyaregivenbelow.Drawapiecharttorepresentthesame.

GradeVI: 436

× 360º = 40°

GradeVII: 236

×360º = 20°

GradeVIII: 1036

× 360º = 100°

GradeIX: 1536

× 360º = 150°

GradeX: 536

× 360º = 50°

Tocheck:40°+20°+100°+150°+50°=360°

VI

IX

X

VIII

VII

20°

100°

150°

40°

50°

Itshouldbeexplainedthatinordertocalculatethevaluesoftheanglesofapiechart,wetakethefrequencyofthesubjectoverthetotaltocreateafractionandthenmultiplyby360ºasitisgoingtobeafractionofafullcircle.

Angleofasector = frequencytotalfrequency

×360º

Written Assignment

Questions6to10ofExercise19canallbeusedtoconvertthedatagivenintopiechartdata.Theteachershouldexplainanddoacoupleontheboardandaskfortheresttobedoneinclassworknotebooks.

Onlywhenthestudentsareclearaboutthecalculationsof‘how’and‘why’shouldtheteacheraskthestudentstobringprotractorstothenextlessonandproceedtoteachtheconstructionofpiecharts.

Evaluation

Thisisapresentation-basedchapter.Markscanbeawardedonassignmentsandclassworkinvolvingbar graphsandpiecharts.


• explaintheimportanceofpresentingdataclearlyandaccurately,

• groupdataintheformofafrequencydistributiontable,

• determineclasssizebyusingdatarangeandnumberofclasses,

• readbarchartsandpiecharts,and

• constructbarchartsforgivendata.

iv84 1

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,requirestudentstoidentifythecorrectanswerfromagivensetofpossibleoptions.

Structuredquestionsassessvariousaspectsofstudents'understanding:knowledgeofcontentandvocabulary,reasoningskills,andmathematicalproofs.

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

Assessment

v851

Specimen Paper

Mathematics

Grade 7

Section ATime: 1 hour Total marks: 40

1. AsetcontainingthecommonelementsofAandBisformedby

A. addingset

B. unionofsets

C. intersectionofset

D. differenceoftwosets

5. Whichofthefollowingisanon-terminatingdecimal?

A. 36

B. 18

C. 25

D. 1117

2. IfA={1,2,3,.....,50}and B={1,3,5,.....49},whatisA∪B?

A. SetA

B. SetB

C. A–B

D. B–A

6. 254 issameas

A. 625

B. 0.625

C. 6.25

D. 62.5

3. Whichpropertyisrepresentedby?

∙ ab ×

cd ∙ ×

gh = a

b × ∙ cd ×

gh ∙

A. commutativepropertyofaddition

B. associativepropertywithrespecttomultiplication

C. distributivewithrespecttomultiplication

D. BODMASrule

7. Whatmustbeaddedto1999.061tomake2000?

A. 0.0939

B. 0.939

C. 1.061

D. 1000

4. Whichnumbershouldbesubtracted

from– 94 toget–316?

A. 616

B. –3916

C. –3316

D. 3916

8. Thesquarerootof10×10×10×10is

A. 100

B. 1000

C. 10000

D. 10

Assessment

iv86 1

9. If36boysstandmakingequalnumberofrowsandcolumns,howmanyboysareineachrow?

A. 6

B. 18

C. 9

D. 4

15. Adecreaseinsellingpriceisa

A. loss

B. tax

C. profit

D. %discount

10. 0.49isthesquareof

A. 7

B. 0.07

C. 0.7

D. 4.9

16. ThecostpriceofanobjectisRs4500andthesellingpriceisRs4050.Calculatetheprofitorlosspercent.

A. 45%gain

B. 70%loss

C. 10%loss

D. 10%gain

11. 64125

issameas

A. 2553

B. (22)353

C. 4352

D. 24125

17. TheinterestgainedonRs25,000attherateof10%for3yearsis

A. Rs7500

B. Rs32,500

C. Rs750

D. Rs750,00

12. IfAlimakes50basketsin5days.Howmanybasketwillhemakein3days?

A. 250

B. 15

C. 30

D. 150

18. ApersonpaysRs36,000asincometaxattherateof12%peryear.Whatishisincome?

A. Rs72,000

B. Rs4,320

C. Rs300,000

D. Rs30,000

13. Ifa:b=4:7andb:c=7:9whatisa:b:c?

A. 4:14:9

B. 4:7:9

C. 4:1:9

D. 1:7:9

19. P+ P×R×T100

=

A. Simpleinterest

B. Rate

C. Amount

D. Incometax

14. Acartravels110kmin2hours.Howfarwillittravelin3hours?

A. 220km

B. 165km

C. 330km

D. 73.2km

20. Simplify9x(–x)2

A. –9x3

B. 9x3

C. 9x2

D. 9x

Assessment

v871

21. Threesidesofatriangleare(x+5)m,(x+8)mand(x–3)m.Whatwillbeitsperimeter?

A. (2x+10)m

B. (3x+10)m

C. (3x+16)m

D. (10x)m

26. Thevalueofa2+2a2–a=2,if

A. a=–1

B. a=1

C. a=0

D. a=2

22. Howmuchshouldbeaddedtox+9tomakeit3x–4?

A. –2x+13

B. 4x+13

C. 4x+5

D. 2x–13

27. If50x=(45)2–(35)2thenthevalueofxis

A. 2

B. 8

C. 400

D. 16

23. 9x2–25y2expressedasproductoftwotermswillbe

A. (3x–5y)(3x+5y)

B. (3x)2(5y)2

C. (9x–25y)(9x+25y)

D. (9x)(25y)

28. Whichofthefollowingsetofanglesisbothsupplementaryandverticallyopposite?

A. 45º,35º

B. 100º,80º

C. 90º,90º

D. 35º,75º

24. Iftheareaofasquareis 16x2–56xy+49y2,thenthelengthofeachsidesis

A. 4x+7y

B. 16x–49y

C. 4x–7y

D. 8x–7y

29. Whichofthefollowingangleisanobtuseangle?

A. 29 of90º

B. 29 of180º

C. 56 of180º

D. 710of90º

25. Factorise81–9p2. A. (9–3p)(9–3p)

B. 9(3–p)(3+p)

C. 3(3–p)(3+p)

D. (9–3p)

30. Thesumofinterioranglesofatriangleis

A. 90º

B. 180º

C. 180º

D. 120º

31. Whichofthefollowingisanisoscelestriangle?

A. mAB=5cm mBC=3cm mAC=3cm

B. mAB=7cm mBC=6cm mAC=5cm

C. mAB=8cm mBC=8cm mAC=8cm

D. mAB=5cm mBC=4cm mAC=7cm

36. Thesumofinterioranglesofaquadrilateralis

A. 180º

B. 270º

C. 630º

D. 360º

Assessment

iv88 1

32. IfAandBaretwoconcentriccirclesasshowninthefigurebelow,then

A

B

A. diameterofcircleA>diameterofcircleB

B. diameterofcircleA<diameterofcircleB

C. diameterofcircleAistwicethediameterofcircleB

D. diameterofcirclesAandBareequal

37. Thevalueofx inthegivenisoscelestrapeziumis

A. 50º

B. 90º

C. 130º

D. 120º

x x

50º 50º

33. Theboundaryofacircleiscalled

A. majorsector

B. circumference

C. majorarc

D. minorarc

38. Iftheradiusofacircleis14cm,thenitsareais

A. 44cm2

B. 616cm2

C. 154cm2

D. 160cm2

34. Twofiguresaresimilar,if

A. theyhavesameshapes

B. theyareofsamesize

C. theyhavesameangleswithdifferentsides

D. theyhavesameangleswithsidesinsameratio

39. Theradiusofacylinderis7cmanditsheightis10cm.Findthevolumeofthecylinder.

∙takeπ = 227 ∙ A. 1540cm2

B. 154cm2

C. 15.40cm2

D. 1.540cm2

35. Twotrianglesarecongruentiftheyhave

A. sameangles

B. sameanglesandsamesides

C. samesides

D. equalanglesandunequalsides

40. Thedifferencebetweenthegreatestandsmallestvalueinadataisthe

A. upperlimit

B. frequency

C. rangeofthedata

D. lowerlimit

Assessment

v891

Section B

Time: 2 hours Total marks: 60

Q.1 (i) Simplify∙– 75 × 3

14 ∙+∙ 23 × –310 ∙–∙– 87 × –21

4 ∙ [5]

(ii) If𝕌={0,1,2,3,4,5,6}, A={1,3,5},

B={0,1,2,3},

C={2,4,6},

find

(a) A∪ C (b) B

U

C (c) B′ [3]

(iii) Findthepositivesquarerootof181476.

[4] [Total marks 12]

Q.2 (i) Express64216aspowerofrationalnumbers. [2]

(ii) Express∙∙– 57 ∙–2∙–1withnegativeexponents. [2]

(iii) 260studentsinahostelhavefoodfor25days.If10studentsleavethehostel, howlongwillthefoodlastfortheremainingstudents? [5]

(iv) Asim’spropertyisworthRs4,000,000.Iftherateofpropertytaxis5%peryear, howmuchpropertytaxwillhepayin2years? [3]

[Total marks 12]

Q.3 Simplify:

(i) 4(x –5)+3(x–8)–(x–10) [3]

(ii) 49p2+112pq+64q2 [3]

(iii) Resolveintofactors25x2–81y2 [3]

(iv) when8issubtractedfrom5timesanumber,theresultis37.Findthenumber.

[3]

[Total marks 12]

Assessment

iv90 1

Q.4 (i) Constructthefollowingisoscelestriangle.

mAB=5cm m∠A=55° [5]

(ii) ABandCDaretwochordsofacirclewithcentreO.

IfmAB=3cm,m∠ AOB=m∠COD,findmCD. [3]

OA

C

D

B

3cm

(iii) Inthegivenparallelogramfind [4]

(a) mDC

(b) m∠ADB (c) m∠DCB (d) m∠ABD

[Total marks 12]

Q.5 (i) Thediameterofacircularhallis14m.Findthecostofflooringthehallat [4]

therateofRs400persquaremetre. ∙Takeπ = 227

∙ (ii) Findthevolumeandsurfaceareaofasolidcylinderwhoseradiusis21cm [5]

andheightin45cm.

(iii) Inaasurvey500peoplewereaskedthenameoftheirfavouritecountry.

Thepiechartshowsthenameofthecountry.

Turkey

Japan

Rome72º

180º

(a) FindtheangleofsectorofTurkey. [1]

(b) HowmanypeoplelikeJapan? [1]

(c) WhatpercentageofpeoplelikeRome? [1]

[Total marks 12]

30º70º

A

CD

B3cm

Assessment

v911

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

Answers

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

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

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

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

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

Marking criteria for Section B

Q.1 12 Marks Answer

(i) • Simplificationwithinbrackets [3] • SimplificationwithLCM [1] • Accuracy [1]

(ii) • Correctconceptof∪,∩andcomplementofa setandaccurateanswer [1]

• Correct [1] • Accurateanswer [1]

(iii)• Correctprocedureofprimefactorization andpairing [2]

• Correctuseofradicalsign [1] • Correctanswer [1]

5marks

3marks

4marks

–6 12

(a) {1,2,3,4,5,6,}(b) {2}(c) {4,5,6}

426

Q.2 12Marks Answer

(i) • Forfindingcorrectpowers [2]

(ii) • Accurateanswer [2]

(iii)• Identifyingtheproportion(inverse) [1] • Correctequation [3] • Correctanswers [1]

(iv)• Method/formula [1] • Calculation [1] • Accuracyinanswer [1]

2marks

2marks

5marks

3marks

( 46)3

(– 75)–2

26days

Rs400,000

Assessment

iv92 1

Q.3 12 Marks Answer

(i) • Openingofthebrackets [1] • Collectingtermsandsimplification [1] • Correctanswer [1]

(ii) • Identifythewholesquareformula [1] • Accurateapplication [1] • Correctanswer [1]

(iii)• Identifyingtheidentity [1] • Correctapplication [1] • Accuracyinanswer [1]

(iv)• Formingcorrectequation [1] • Solutionofequation [1] • CorrectAnswer [1]

3marks

3marks

3marks

3marks

6x–34

(7p+8q) (7p+8q)

(5x–9y)(5x+9y)

x=9

Q.4 12 marks Answer

(i) • Knowthatinanisoscelestriangletwosides andtwoanglesareequal [1]

• Construction [2]

• Accuratemeasurement [2]

(ii) • Useof3rdpropertyofcircle [1]

• Applicationandreasoning [2]

(iii)• Correctreasoningandcorrectanswer ineachcase [4]

5marks

3marks

4marks

3cm

(a) 3cm

(b) 70º

(c) 30º

(d) 80º

Q.5 12 marks Answer

(i) • Forfindingareaofcircle(usingcorrectformula)[2]

• Calculationofcost [1]

• Accurateanswer [1]

(ii) • Correctapplicationofformulaofvolume [1]

• Accurateanswer [1]

• Correctapplicationofformulaofsurfacearea [1]

• Correctcalculationandaccuracy [2]

(iii)• Correctusageofconceptsineachcase [3]

4marks

5marks

3marks

Rs61600

V=62370cm3

SA=8712cm3

(a) 108º

(b) 250

(c) 20%

Assessment

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