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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
Thestudentswillbeableto:
• analysenumberpatternsandinterpretmathematicalsituationsbymanipulatingalgebraicexpressionsandrelations;
• modelandsolvecontextualisedproblems;and
• interpretfunctions,calculaterateofchangeoffunctions,integrateanalyticallyandnumerically,determineorthogonaltrajectoriesofafamilyofcurves,andsolvenon-linearequationsnumerically.
Benchmarks
Grades VI, VII, VIII
• Identifyalgebraicexpressionsandbasicalgebraicformulae
• Applythefourbasicoperationsonpolynomials
• Manipulatealgebraicexpressionsusingformulae
• Formulatelinearequationsinoneandtwovariables
• Solvesimultaneouslinearequationsusingdifferenttechniques
Strand 3: Measurement and Geometry
Thestudentswillbeableto:
• identifymeasurableattributesofobjects,andconstructanglesandtwodimensionalfigures;
• analysecharacteristicsandpropertiesandgeometricshapesanddevelopargumentsabouttheirgeometricrelationships;and
• recognisetrigonometricidentities,analyseconicsections,drawandinterpretgraphsoffunctions.
Benchmarks
Grades VI, VII, VIII
• Drawandsubdividealinesegmentandanangle
• Constructatriangle(givenSSS,SAS,ASA,RHS),parallelogram,andsegmentsofacircle
• Applypropertiesoflines,angles,andtrianglestodevelopargumentsabouttheirgeometricrelationships
• Applyappropriateformulastocalculateperimeterandareaofquadrilateral,triangular,andcircularregions
• Determinesurfaceareaandvolumeofacube,cuboid,sphere,cylinder,andcone
• Findtrigonometricratiosofacuteanglesandusethemtosolveproblemsbasedon right-angledtriangles
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Strand 4: Handling Information
Thestudentswillbeabletocollect,organise,analyse,display,andinterpretdata.
Benchmarks
Grades VI, VII, VIII
• Read,display,andinterpretbarandpiegraphs
• Collectandorganisedata,constructfrequencytablesandhistogramstodisplaydata
• Findmeasuresofcentraltendency(mean,medianandmode)
Strand 5: Reasoning and Logical Thinking
Thestudentswillbeableto:
• usepatterns,knownfacts,properties,andrelationshipstoanalysemathematicalsituations;
• examinereal-lifesituationsbyidentifyingmathematicallyvalidargumentsanddrawingconclusionstoenhancetheirmathematicalthinking.
Benchmarks
Grades VI, VII, VIII
• Finddifferentwaysofapproachingaproblemtodeveloplogicalthinkingandexplaintheirreasoning
• Solveproblemsusingmathematicalrelationshipsandpresentresultsinanorganisedway
• Constructandcommunicateconvincingargumentsforgeometricsituations
Curriculum
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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
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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
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• 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
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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
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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
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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
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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.
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• 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.
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• 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.
Teaching and Learning
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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.
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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.
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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.
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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
Specific Learning Objectives
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
Suggested Time Frame5to6periods
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.
Suggested Time Frame5to6periods
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
Specific Learning Objectives
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.
After completing this chapter, students should be able to:
• convertfractionsandpercentagestodecimalnumbers,andviceversa,
• identifyterminatingandnon-terminatingdecimalnumbers,and
• roundoffdecimalnumberstothenearestwholenumberandtherequireddecimalplace.
v271
4Squares and Square Roots
Specific Learning Objectives• Introductionoftheconceptofperfectsquaresandsquareroots
• Derivationofthesquarerootofaperfectsquare
• Derivationofpositivesquarerootbyprimefactorization
• Derivationofpositivesquarerootbydivision
Suggested Time Frame4to5periods
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.
After completing this chapter, students should be able to:
• statethesquarerootsofthefirst12perfectsquares,
• performtestsforperfectsquares,
• findthesquareofanumber,and
• derivethesquarerootofapositiveintegerbyprimefactorizationanddivision.
iv30 1
5 Exponents
Specific Learning Objectives• Conceptofexponentialnotation
• Expressingrationalnumbersinexponentialnotation
• Lawsgoverningthepowersorexponentsinmultiplicationanddivision
Suggested Time Frame4to5periods
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.
After completing this chapter, students should be able to:
• identifythebaseandpowerofanumber,
• expressnumbersinexponentialform,and
• applytheproduct,quotient,andpowerlawsofexponentstosolvequestions.
v331
Direct and Inverse Variation
Specific Learning Objectives• Directandinversevariations
• Continuedratio
• Unitaryandproportionmethod
• Ratioproblemssolvingthreequantities
Suggested Time Frame4to5periods
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
Specific Learning Objective
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.
After completing this chapter, students should be able to:
• 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
Sample Lesson PlanTopic
Simpleinterest
Specific Learning Objective
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).
After completing this chapter, students should be able to:
• 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
Suggested Time Frame6to8periods
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
Specific Learning Objectives
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.
After completing this chapter, students should be able to:
• identifyandsolvealgebraicexpressions,
• arrangepolynomialsinincreasinganddecreasingorder,and
• add,subtract,multiply,anddividegivenalgebraicexpressions.
v431
Algebraic Indentities
Specific Learning Objectives• Introductionofthesumanddifferenceofaperfectsquare
• Thealgebraicidentityofthedifferenceoftwosquares
• Thenumericalapplicationoftheseidentitiestosolvequestions
Suggested Time Frame6to7periods
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
Sample Lesson PlanTopic
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.
After completing this chapter, students should be able to:
• derivethealgebraicidentitiesrelatedtothesquareofasumandthedifferenceoftwotermsandtheproductofasumanddifferenceoftwoterms(a+b)(a–b),and
• solvequestionsbyapplyingthecorrectalgebraicidentities.
v471
10Factorization of Algebraic Expressions
Specific Learning Objectives• Introductionoffactorizationofalgebraicexpressions
• Introductionoffactorizationofexpressionsusingalgebraicidentities
• Introductionoffactorizationbymakinggroups
Suggested Time Frame4to5periods
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
Sample Lesson PlanTopic
Factorizationofalgebraicexpressions
Specific Learning Objective
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.
After completing this chapter, students should be able to:
• factorisealgebraicexpressionsbyapplyingalgebraicidentities,and
• factorisealgebraicexpressionsbymakinggroups.
iv50 1
11 Simple Equations
Specific Learning Objectives• Theconceptofalgebraicequations
• Axiomsofsolvingalgebraicequations
• Procedureforsolvinganalgebraicequation
• Forming,orconstructing,analgebraicequation
Suggested Time Frame6to7periods
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
Sample Lesson PlanTopic
Algebraicequations
Specific Learning Objectives
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.
After completing this chapter, students should be able to:
• solvealgebraicequationsbyapplyingrelevantaxiomsandidentities.
v531
12 Lines and Angles
Specific Learning Objectives• Conceptsofperpendicularandparallellines
• Constructingperpendicularandparallellines
• Propertyofatransversalanditsconstruction
Suggested Time Frame2to3periods
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
Sample Lesson PlanTopic
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.
After completing this chapter, students should be able to:
• 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
Suggested Time Frame6to8periods
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
Sample Lesson PlanTopic
Geometricconstructions
Specific Learning Objectives
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.
After completing this chapter, students should be able to:
• 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
Specific Learning Objective
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.
After completing this chapter, students should be able to:
• 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
Specific Learning Objectives
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
Yesthetrianglesarecongruent.
Property:SAS
(c)
A D
CB
Yesthetrianglesarecongruent.
Property:AASorSSS
v671 Congruence and SimilaritySimilarity and Congruence
(d)
60° 60°
Yesthetrianglesarecongruent.
Property:SAS
(e)
A B
C
P Q
R
Thetrianglesarenotcongruent.
Nopropertyissatisfied.
EvaluationAcomprehensiveassessmentshouldbegivenattheendofthetopicbutinbetweenshortquizzesontheboardcouldbeconductedtowardstheendofeachlessontocheckstudents'understanding.Thischapterintroducesanentirelynewconceptandstage-by-stageassessmentisnecessary.
After completing this chapter, students should be able to:
• identifycongruentfigures,
• identifysimilarfigures,
• establishcongruenceandsimilaritybetweengeometricfigures,and
• testforthecongruencyoftwotrianglesusingtheSSS,ASA,andRHSpropertiesofcongruencyoftriangles.
iv68 1
1 Quadrilaterals
Specific Learning Objectives• Propertiesofquadrilaterals:parallelograms,rhombuses,rectangles,andsquares
• Constructingaparallelogram
Suggested Time Frame5to6periods
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.
Summary of Key Facts
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
Sample Lesson PlanTopic
Constructingaparallelogram
Specific Learning Objectives
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
Suggested Time Frame6to8periods
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.
Summary of Key Facts
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
Sample Lesson PlanTopic
Tesselation
Specific Learning Objectives
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.
After completing this chapter, students should be able to:
• 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
Sample Lesson PlanTopic
Volumeandsurfaceareaofacylinder
Specific Learning Objectives
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.
After completing this chapter, students should be able to:
• calculatethevolumeofacuboid,cube,andrightcircularcylinderbyapplyingtheformulae,
• calculatethesurfaceareaofacuboid,cube,andrightcircularcylinderbyapplyingtheformulae.
v811
19Information Handling
Specific Learning Objectives• Theimportanceofdatapresentation
• Differencebetweenungroupedandgroupeddata
• Conceptsofclassintervalandfrequency,rangeofdata,lowerandupperclasslimit
• Constructingafrequencydistributiontable
• Readingbarchartsandpiecharts
• Constructingbarcharts
Suggested Time Frame4to5periods
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
Sample Lesson PlanTopic
Piecharts
Specific Learning Objectives
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.
After completing this chapter, students should be able to:
• 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