CCEA GCE Specification in Mathematics

GCE

For first teaching from September 2018For first award of AS level in Summer 2019For first award of A level in Summer 2019Subject Code: 2210

CCEA GCE Specification in

Mathematics

Contents 1 Introduction 3 1.1 Aims 41.2 Keyfeatures 51.3 Priorattainment 51.4 Classificationcodesandsubjectcombinations

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2 Specification at a Glance

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3 Subject Content 7 3.1 OverarchingthemesinGCEMathematics 73.2 UnitAS1:PureMathematics 93.3 UnitAS2:AppliedMathematics 143.4 UnitA21:PureMathematics 183.5 UnitA22:AppliedMathematics

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4 Scheme of Assessment 25 4.1 Assessmentopportunities 254.2 Assessmentobjectives 254.3 Assessmentobjectiveweightings 264.4 SynopticassessmentatA2 264.5 Higherorderthinkingskills 274.6 Reportingandgrading

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5 Grade Descriptions

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6 Guidance on Assessment 32 6.1 UnitAS1:PureMathematics 326.2 UnitAS2:AppliedMathematics 326.3 UnitA21:PureMathematics 326.4 UnitA22:AppliedMathematics

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7 Links and Support 33 7.1 Support 337.2 Curriculumobjectives 337.3 Examinationentries 347.4 Equalityandinclusion 347.5 Contactdetails 35

Thisspecificationisavailableonlineatwww.ccea.org.uk

SubjectCodeQANASLevelQANALevel

2210603/1761/9603/1717/6

ACCEAPublication©2017

CCEAGCEMathematicsfromSeptember2018

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1 Introduction ThisspecificationsetsoutthecontentandassessmentdetailsforourAdvancedSubsidiary(AS)andAdvanced(Alevel)GCEcoursesinMathematics.FirstteachingisfromSeptember2018.Studentscantake:

• theAScourseasafinalqualification;or• theASunitsplustheA2unitsforafullGCEAlevelqualification.WeassesstheASunitsatastandardappropriateforstudentswhohavecompletedthefirstpartofthefullcourse.A2unitshaveanelementofsynopticassessment(toassessstudents’understandingofthesubjectasawhole),aswellasmoreemphasisonassessmentobjectivesthatreflecthigherorderthinkingskills.ThefullAdvancedGCEawardisbasedonstudents’marksfromtheAS(40percent)andtheA2(60percent).Theguidedlearninghoursforthisspecification,asforallGCEs,are:

• 180hoursfortheAdvancedSubsidiarylevelaward;and• 360hoursfortheAdvancedlevelaward.WewillmakethefirstASawardsforthespecificationin2019andthefirstAlevelawardsin2019.ThespecificationbuildsonthebroadobjectivesoftheNorthernIrelandCurriculum.Ifthereareanymajorchangestothisspecification,wewillnotifycentresinwriting.Theonlineversionofthespecificationwillalwaysbethemostuptodate;toviewanddownloadthispleasegotowww.ccea.org.uk


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1.1 Aims Thisspecificationaimstoencouragestudentsto:

• understandmathematicsandmathematicalprocessesinawaythatpromotesconfidence,fostersenjoymentandprovidesastrongfoundationforprogresstofurtherstudy;

• extendtheirrangeofmathematicalskillsandtechniques;• understandcoherenceandprogressioninmathematicsandhowdifferentareasofmathematicsareconnected;

• applymathematicsinotherfieldsofstudyandbeawareoftherelevanceofmathematicstotheworldofworkandtosituationsinsocietyingeneral;

• usetheirmathematicalknowledgetomakelogicalandreasoneddecisionsinsolvingproblemsbothwithinpuremathematicsandinavarietyofcontexts,andcommunicatethemathematicalrationaleforthesedecisionsclearly;

• reasonlogicallyandrecogniseincorrectreasoning;• generalisemathematically;• constructmathematicalproofs;• usetheirmathematicalskillsandtechniquestosolvechallengingproblemsthatrequirethemtodecideonthesolutionstrategy;

• recognisewhentheycanusemathematicstoanalyseandsolveaproblemincontext;

• representsituationsmathematicallyandunderstandtherelationshipbetweenproblemsincontextandmathematicalmodelsthattheymayapplytosolvethese;

• drawdiagramsandsketchgraphstohelpexploremathematicalsituationsandinterpretsolutions;

• makedeductionsandinferencesanddrawconclusionsbyusingmathematicalreasoning;

• interpretsolutionsandcommunicatetheirinterpretationeffectivelyinthecontextoftheproblem;

• readandcomprehendmathematicalarguments,includingjustificationsofmethodsandformulae,andcommunicatetheirunderstanding;

• readandcomprehendarticlesconcerningapplicationsofmathematicsandcommunicatetheirunderstanding;

• usetechnologysuchascalculatorsandcomputerseffectively,andrecognisewhensuchusemaybeinappropriate;and

• takeincreasingresponsibilityfortheirownlearningandtheevaluationoftheirownmathematicaldevelopment.


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1.2 Key features Thefollowingareimportantfeaturesofthisspecification.

• Itincludesfourexternallyassessedassessmentunits.• Itallowsstudentstodeveloptheirsubjectknowledge,understandingandskills.• AssessmentatA2includesmoredemandingquestiontypesandsynopticassessmentthatencouragesstudentstodeveloptheirunderstandingofthesubjectasawhole.

• Itgivesstudentsasoundbasisforprogressiontohighereducationandtoemployment.

• Arangeofsupportisavailable,includingspecimenassessmentmaterials.1.3 Prior attainment ThisspecificationassumesknowledgeofHigherTierGCSEMathematics.1.4 Classification codes and subject combinations Everyspecificationhasanationalclassificationcodethatindicatesitssubjectarea.Theclassificationcodeforthisqualificationis2210.Pleasenotethatifastudenttakestwoqualificationswiththesameclassificationcode,schoolsandcollegesthattheyapplytomaytaketheviewthattheyhaveachievedonlyoneofthetwoGCEs.ThesamemayoccurwithanytwoGCEqualificationsthathaveasignificantoverlapincontent,eveniftheclassificationcodesaredifferent.Becauseofthis,studentswhohaveanydoubtsabouttheirsubjectcombinationsshouldcheckwiththeuniversitiesandcollegesthattheywouldliketoattendbeforebeginningtheirstudies.


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2 Specification at a Glance ThetablebelowsummarisesthestructureoftheASandAlevelcourses:

Content

Assessment

Weightings

AS1:PureMathematics

Externalwrittenexamination1hour45minsStudentsanswerallquestions.

60%ofAS24%ofAlevel

AS2:AppliedMathematics


40%ofAS16%ofAlevel

A21:PureMathematics

Externalwrittenexamination2hours30minsStudentsanswerallquestions.

36%ofAlevel

A22:AppliedMathematics


24%ofAlevel


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3 Subject Content Wehavedividedthiscourseintofourunits:twounitsatASlevelandtwounitsatA2.Thissectionsetsoutthecontentandlearningoutcomesforeachunit.Theuseoftechnology,inparticularmathematicalandstatisticalgraphingtoolsandspreadsheets,mustpermeatetheteachingoftheunitsinthisspecification.Calculatorsusedmustinclude:

• aniterativefunction;and• theabilitytocomputesummarystatisticsandaccessprobabilitiesfromstandardstatisticaldistributions.

Studentsmustnothaveaccesstotechnologywithacomputeralgebrasystemfunctionduringexaminations.3.1 Overarching themes in GCE Mathematics ThisGCEMathematicsspecificationgivesstudentsopportunitiestodemonstratethefollowingknowledgeandskills.Theymustapplythese,alongwithassociatedmathematicalthinkingandunderstanding,acrossthewholecontentoftheASandA2unitssetoutbelow.ASandAlevelstudentsshouldbeableto:

• understandandusemathematicallanguageandsyntax,includingequals,identicallyequals,therefore,because,implies,isimpliedby,necessary,sufficient,∴,=,º, ¹,⇒,⇐and⇔;

• understandanduseVenndiagrams,languageandsymbolsassociatedwithsettheory,includingcomplement,Æ, Ç, È, Î, Ï and ε, andapplythesetosolutionsofinequalitiesandprobability;

• understandandusethestructureofmathematicalproof,proceedingfromgivenassumptionsthroughaseriesoflogicalstepstoaconclusion;

• usemethodsofproof,includingproofbydeductionandproofbyexhaustion;• usedisproofbycounterexample;• comprehendandcritiquemathematicalarguments,proofsandjustificationsofmethodsandformulae,includingthoserelatingtoapplicationsofmathematics;

• recognisetheunderlyingmathematicalstructureinasituationandsimplifyandabstractappropriatelytosolveproblems;

• constructextendedargumentstosolveproblemspresentedinanunstructuredform,includingproblemsincontext;

• interpretandcommunicatesolutionsinthecontextoftheoriginalproblem;• evaluate,includingbymakingreasonedestimates,theaccuracyorlimitationsofsolutions;

• understandtheconceptofaproblem-solvingcycle,includingspecifyingtheproblem,collectinginformation,processingandrepresentinginformationandinterpretingresults,whichmayidentifytheneedtorepeatthecycle;


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• understand,interpretandextractinformationfromdiagramsandconstructmathematicaldiagramstosolveproblems,includinginmechanics;

• translateasituationincontextintoamathematicalmodel,makingsimplifyingassumptions;

• useamathematicalmodelwithsuitableinputstoengagewithandexploresituations(foragivenmodeloramodelconstructedorselectedbythestudent);

• interprettheoutputsofamathematicalmodelinthecontextoftheoriginalsituation(foragivenmodeloramodelconstructedorselectedbythestudent);

• understandthatamathematicalmodelcanberefinedbyconsideringitsoutputsandsimplifyingassumptions;

• evaluatewhetheramathematicalmodelisappropriate;and• understandandusemodellingassumptions.

Alevelstudentsshouldalsobeableto:

• understandanduseproofbycontradiction;• constructandpresentmathematicalargumentsthroughappropriateuseofdiagrams,sketchinggraphs,logicaldeduction,precisestatementsinvolvingcorrectuseofsymbolsandconnectinglanguage,includingconstant,coefficient,expression,equation,function,identity,index,termandvariable;

• understandthatmanymathematicalproblemscannotbesolvedanalytically,butnumericalmethodspermitsolutiontoarequiredlevelofaccuracy;and

• evaluatetheaccuracyorlimitationsofsolutionsobtainedusingnumericalmethods.


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3.2 Unit AS 1: Pure Mathematics ThisunitcoversthepurecontentofASMathematics.ItiscompulsoryforbothASandAlevelMathematics.Theunitisassessedbya1hour45minuteexternalexamination,with6–10questionsworth100rawmarks.

Content

LearningOutcomes

Algebraandfunctions

Studentsshouldbeableto:

• demonstrateunderstandingofandusethelawsofindicesforallrationalexponents;

• useandmanipulatesurds,includingrationalisingthedenominator;

• workwithquadraticfunctionsandtheirgraphs;• demonstrateunderstandingofandusethediscriminantofaquadraticfunction,includingtheconditionforrealandrepeatedroots;

• completethesquareinaquadraticfunction;• solvequadraticequations,includingquadraticequationsinafunctionoftheunknown;

• solvesimultaneousequationsintwovariablesbyeliminationandbysubstitution,includingonelinearandonequadraticequation;

• solvesimultaneousequationsinthreevariables;• solvelinearandquadraticinequalitiesinasinglevariableandinterpretsuchinequalitiesgraphically,includinginequalitieswithbracketsandfractions;

• manipulatepolynomialsalgebraically,includingexpandingbracketsandcollectingliketerms,factorisationandsimplealgebraicdivision;

• usetheremainderandfactortheorems;and• sketchcurvesdefinedbysimpleequations,includingpolynomials.


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Content

LearningOutcomes

Algebraandfunctions(cont.)


• sketchcurvesdefinedbyequationsoftheform𝑦 = '(

and𝑦 = '(2(includingtheirverticalandhorizontal

asymptotes);• interpretthealgebraicsolutionofequationsgraphically;• useintersectionpointsofgraphstosolveequations;• demonstrateunderstandingoftheeffectofsimpletransformationsonthegraphof𝑦 = f(𝑥),includingsketchingassociatedgraphs:

𝑦 = 𝑎f(𝑥),𝑦 = f 𝑥 + 𝑎,𝑦 = f(𝑥 + 𝑎)and𝑦 = f(𝑎𝑥)

Co-ordinategeometryinthe𝒙, 𝒚 plane

• demonstrateunderstandingofandusetheequationofastraightline,includingtheforms𝑦 − 𝑦5 = 𝑚(𝑥 − 𝑥5)and𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0

• demonstrateunderstandingofhowtofindthemid-pointofalinesegment;

• usethegradientconditionsfortwostraightlinestobeparallelorperpendicular;

• usestraightlinemodelsinavarietyofcontexts;• demonstrateunderstandingofandusetheco-ordinategeometryofthecircle,includingusingtheequationofacircleintheforms:(𝑥 − 𝑎): + (𝑦 − 𝑏): = 𝑟:and𝑥: + 𝑦: + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0

• findthecentreandradiusofacirclebycompletingthesquare;

• usethestandardcircleproperties:angleinasemicircleisarightangle,perpendicularfromcentretoachordbisectsthechordandperpendicularityofradiusandtangent;and

• findtheequationofthetangenttoacirclethroughagivenpointonthecircumference.


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Content

LearningOutcomes

Sequencesandseries


• demonstrateunderstandingofandusethebinomialexpansionof(𝑎 + 𝑏𝑥)>forpositiveinteger𝑛

• demonstrateunderstandingofandusethenotations𝑛!and𝑛C𝑟

Trigonometry • demonstrateunderstandingofandusethedefinitionsofsine,cosineandtangentforallarguments;

• demonstrateunderstandingofandusethesineandcosinerules;

• calculatetheareaofatriangleintheform5

:𝑎𝑏 sin 𝐶

• demonstrateunderstandingofandusethesine,cosineandtangentfunctions,includingtheirgraphs,symmetriesandperiodicity;

• demonstrateunderstandingofandusetan 𝜃 = sin𝜃cos𝜃

• demonstrateunderstandingofanduse

sin:𝜃 + cos:𝜃 = 1• solvesimpletrigonometricequationsinagiveninterval,includingquadraticequationsinsin,cosandtanandequationsinvolvingmultiplesoftheunknownangle;

Exponentialsandlogarithms

• demonstrateunderstandingofandusethefunction𝑎(anditsgraph,where𝑎ispositive;

• demonstrateunderstandingofandusethefunction𝑒(anditsgraph;

• demonstrateunderstandingofandusethedefinitionoflog' 𝑥astheinverseof𝑎(,where𝑎ispositiveand𝑥 ≥ 0

• demonstrateunderstandingofandusethefunctionln 𝑥anditsgraph;and

• demonstrateunderstandingofanduseln 𝑥astheinversefunctionof𝑒(


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Content

LearningOutcomes

Exponentialsandlogarithms(cont.)


• demonstrateunderstanding,proveandusethelawsoflogarithms:

log' 𝑥 + log' 𝑦 = log' 𝑥𝑦log' 𝑥 − log' 𝑦 = log'

(P

𝑘 log' 𝑥 = log' 𝑥R (including,forexample𝑘 = −1and𝑘 = − 5

:)

• solveequationsoftheform𝑎( = 𝑏• solveinequalitiesinvolvingexponentialfunctions,forexample𝑎( < 𝑏

• demonstrateunderstandingofanduseexponentialgrowthanddecay;

• useexponentialgrowthanddecayinmodellingcontinuouscompoundinterest,populationgrowth,radioactivedecayanddrugconcentrationdecay;

Differentiation

• demonstrateunderstandingofandusethederivativeoff(𝑥)asafunctionforthegradientofthetangenttothegraphof𝑦 = f(𝑥)atageneralpoint(𝑥, 𝑦)

• demonstrateunderstandingofthegradientofthetangenttoacurveasalimit;

• interpretthegradientofatangentasarateofchange;• demonstrateunderstandingofandfindsecondderivatives;• demonstrateunderstandingofandusethesecondderivativeastherateofchangeofgradient;

• differentiate𝑥>,forrationalvaluesof𝑛,andrelatedconstantmultiples,sumsanddifferences;

• applydifferentiationtofindgradients,tangentsandnormals,maximaandminimaandstationarypoints;and

• identifyincreasinganddecreasingfunctions.


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Content

LearningOutcomes

Integration Studentsshouldbeableto:

• demonstrateunderstandingofanduseindefiniteintegrationasthereverseofdifferentiation;

• integrate𝑥>(excluding𝑛 = −1)andrelatedsums,differencesandconstantmultiples;

• evaluatedefiniteintegrals;• useadefiniteintegraltofindtheareadefinedbyacurveandeitheraxis;

Vectors • usevectorsintwodimensions(includingiandjunitvectors);

• calculatethemagnitudeanddirectionofavectorandconvertbetweencomponentformandmagnitude/directionform;

• performthealgebraicoperationsofvectoradditionandmultiplicationbyscalars,andunderstandtheirgeometricalinterpretations;

• demonstrateunderstandingofandusepositionvectors;and

• calculatethedistancebetweentwopointsrepresentedbypositionvectors.


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3.3 Unit AS 2: Applied Mathematics Thisunit,whichassumesknowledgeofUnitAS1,coverstheappliedcontentofASMathematicsandiscompulsoryforbothASandAlevelMathematics.Theunitaddressesaspectsofbothmechanics(50%oftheassessment)andstatistics(50%oftheassessment).Itassessesmodellingandtheapplicationofmathematics.Theunitisassessedbya1hour15minuteexternalexamination,with5–10questionsworth70rawmarks.Theexaminationhastwosections:SectionAassessesmechanicsandSectionBassessesstatistics.Studentsanswerallquestionsinbothsections.Thestatisticalcontentofthisunitshouldbetaughtthroughtheuseandinterrogationofalargedataset.Theexaminationtestsstudents’abilityto:

• interpretrealdatapresentedinsummaryorgraphicalform;and• usedatatoinvestigatequestionsarisinginrealcontexts.Studentsshouldbefamiliarwithmethodsofpresentingdata,includingfrequencytablesforungroupedandgroupeddata,boxplotsandstem-and-leafdiagrams.Theyshouldalsobefamiliarwithmean,modeandmedianassummarymeasuresoflocationofdata.Wewillnotsetquestionsthatdirectlyteststudents’abilitytoconstructsuchtablesanddiagramsandcalculatesuchmeasures,butstudentswillbeexpectedtointerpretanddrawinferencesfromthem.Section A: Mechanics

Content

LearningOutcomes

Quantitiesandunitsinmechanics


• demonstrateunderstandingofandusefundamentalquantitiesandunitsintheSIsystem:length,timeandmass;

• demonstrateunderstandingofandusederivedquantitiesandunits:velocity,acceleration,forceandweight;

Kinematics • demonstrateunderstandingofandusethelanguageofkinematics:position,displacement,distancetravelled,velocity,speedandacceleration;and

• demonstrateunderstandingof,useandinterpretgraphsinkinematicsformotioninastraightline:- displacementagainsttimeandinterpretationofgradient;and

- velocityagainsttimeandinterpretationofgradientandareaunderthegraph.


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Content

LearningOutcomes

Kinematics(cont.)


• demonstrateunderstandingofandusetheformulaeforconstantaccelerationformotioninastraightline;

• demonstrateunderstandingofandusetheconstantaccelerationformulaeintwodimensionsusingvectors;

ForcesandNewton’slaws

• demonstrateunderstandingofanduseNewton’sfirstlawandtheconceptofaforce;

• resolveforcesintwodimensions;• demonstrateunderstandingofanduseadditionofforcestofindtheresultantofasystemofforces;

• demonstrateunderstandingofanduseNewton’ssecondlaw,includingforcesgivenas2Dvectors;

• demonstrateunderstandingofandusethegravitationalacceleration,g,anditsvalueinSIunitstovaryingdegreesofaccuracy;

• demonstrateunderstandingofanduseweightandmotioninastraightlineundergravity;

• demonstrateunderstandingofanduseNewton’sthirdlaw;

• demonstrateunderstandingofanduseNewton’ssecondandthirdlawstosolveproblemsinvolvingconnectedparticles;

• solveproblemsinvolvingequilibriumofforcesonaparticle;

• demonstrateunderstandingofandusethe𝐹 ≤ 𝜇𝑅modeloffriction;

• demonstrateunderstandingofandusethecoefficientoffriction;

• solveproblemsinvolvingthemotionofabodyonaroughsurface;and

• solveproblemsinvolvinglimitingfrictionandstatics.


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Section B: Statistics

Content

LearningOutcomes

Statisticalsampling


• demonstrateunderstandingofandusethetermspopulationandsample;

• usesamplestomakeinformalinferencesaboutthepopulation;

• demonstrateunderstandingofandusesamplingtechniques,includingsimplerandomsamplingandstratifiedsampling;

• selectorcritiquesamplingtechniquesinthecontextofsolvingastatisticalproblem,includingunderstandingthatdifferentsamplescanleadtodifferentconclusionsaboutthepopulation;

Datapresentationandinterpretation

• interpretdiagramsforsingle-variabledata,includingunderstandingthatareainahistogramrepresentsfrequencyandconnectionstoprobabilitydistributions;

• interpretmeasuresofcentraltendencyandvariation,includingstandarddeviationandvariance;

• calculatestandarddeviationandvarianceofapopulationorsample,includingfromsummarystatistics;

• interpretscatterdiagramsandregressionlinesforbivariatedata,includingrecognitionofscatterdiagramsthatincludedistinctsectionsofthepopulation(excludingcalculationsinvolvingregressionlines);

• demonstrateunderstandingofinformalinterpretationofcorrelation;

• calculateandinterprettheproduct-momentcorrelationcoefficient;

• demonstrateunderstandingthatcorrelationdoesnotimplycausation;and

• recogniseandinterpretpossibleoutliersindatasetsandstatisticaldiagrams.


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Content

LearningOutcomes

Datapresentationandinterpretation(cont.)


• selectorcritiquedatapresentationtechniquesinthecontextofastatisticalproblem;

• cleandata,includingdealingwithmissingdata,errorsandoutliers;

Probability • demonstrateunderstandingofandusetheadditionandmultiplicationlaws;

• demonstrateunderstandingofandusethefollowingconcepts:- mutuallyexclusiveevents;- exhaustiveevents;and- statisticaldependenceandindependence;

• calculatecombinedprobabilitiesofuptothreeevents,usingtreediagrams,Venndiagramsandtwo-waytables;

Statisticaldistributions

• demonstrateunderstandingofandusethebinomialdistributionasanexampleofadiscreteprobabilitydistribution;

• calculateprobabilitiesusingthebinomialdistribution;and• linkbinomialprobabilitiestothebinomialexpansionandtreediagrams.


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3.4 Unit A2 1: Pure Mathematics ThisunitassumesknowledgeofUnitsAS1andAS2.ItcoversthepurecontentofA2MathematicsandiscompulsoryforAlevelMathematics.Theunitisassessedbya2hours30minuteexternalexamination,with7–12questions.Itisworth150rawmarks.

Content

LearningOutcomes

Algebraandfunctions


• simplifyrationalexpressions,includingbyfactorisingandcancelling,andalgebraicdivision;

• demonstrateunderstandingofandusethedefinitionofafunction;

• demonstrateunderstandingofandusethetermsdomainandrangeinthecontextoffunctions;

• demonstrateunderstandingofandusecompositefunctions;

• demonstrateunderstandingofanduseinversefunctionsandtheirgraphs;

• demonstrateunderstandingofandusethemodulusfunction(including 𝑥 − 𝑎 < 𝑏)

• demonstrateunderstandingoftheeffectofcombinationsofsimpletransformationsonthegraphof𝑦 = f(𝑥)asrepresentedby𝑦 = 𝑎f(𝑥),𝑦 = f 𝑥 + 𝑎,𝑦 = f(𝑥 + 𝑎)and𝑦 = f(𝑎𝑥)

• decomposerationalfunctionsintopartialfractions(denominatorsnotmorecomplicatedthansquaredlinearterms);

• usefunctionsinmodelling,includingconsiderationoflimitationsandrefinementsofthemodels;

Co-ordinategeometryinthe(𝒙, 𝒚)plane

• demonstrateunderstandingofandusetheparametricequationsofcurvesandconversionbetweenCartesianandparametricforms;and

• useparametricequationsinmodellinginavarietyofcontexts.


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Content

LearningOutcomes

Sequencesandseries


• workwithsequences,includingthosegivenbyaformulaforthe𝑛thtermandthosegeneratedbyasimplerelationoftheform𝑥>X5 = f 𝑥>

• demonstrateunderstandingofthebehaviourofsequences,includingconvergence,divergenceandoscillation;

• demonstrateunderstandingofandusesigmanotationforsumsofseries;

• demonstrateunderstandingofandworkwitharithmeticsequencesandseries,includingtheformulaefor𝑛thtermandthesumto𝑛terms;

• demonstrateunderstandingofandworkwithgeometricsequencesandseries,includingtheformulaeforthe𝑛thtermandthesumofafinitegeometricseries;

• provetheformulaforthesumofthefirst𝑛termsofanarithmeticseriesorageometricseries;

• findthesumtoinfinityofaconvergentgeometricseries,includingtheuseof 𝑟 < 1

• demonstrateunderstandingofandusetheexpansionof(𝑎 + 𝑏𝑥)>foranyrational𝑛,includingitsuseforapproximationandknowledgethattheexpansionisvalid

for 𝑏𝑥𝑎 < 1• usesequencesandseriesinmodelling;

Trigonometry • workwithradianmeasure,includinguseforarclengthandareaofsector;and

• demonstrateunderstandingofandusethedefinitionsofsecant,cosecantandcotangentandofarcsin,arccosandarctan,includingtheirrelationshipstosine,cosineandtangent,theirgraphsandtheirdomainsandranges.


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Content

LearningOutcomes

Trigonometry(cont.)


• demonstrateunderstandingofthegraphsofthesecant,cosecant,cotangent,arcsin,arccosandarctanfunctions,includingtheirrangesandappropriaterestricteddomains;

• demonstrateunderstandingofandusesec:𝜃 = 1 + tan:𝜃andcosec:𝜃 = 1 + cot:𝜃

• demonstrateunderstandingofandusethecompoundangleformulaeforsin(𝐴 ± 𝐵),cos(𝐴 ± 𝐵)andtan(𝐴 ± 𝐵)

• demonstrateunderstandingof,useandprovethedoubleangleformulae;

• demonstrateunderstandingofanduseexpressionsfor𝑎 cos 𝜃 + 𝑏 sin 𝜃intheequivalentformsof𝑟 cos(𝜃 ± 𝛼)or𝑟 sin(𝜃 ± 𝛼)

• constructproofsinvolvingtrigonometricfunctionsandidentities;

• usetrigonometricfunctionstosolveproblemsincontext;

Differentiation • differentiate𝑒R(,ln 𝑘𝑥,sin 𝑘𝑥,cos 𝑘𝑥,tan 𝑘𝑥andrelatedsums,differencesandconstantmultiples;

• differentiateusing:- theproductrule;- thequotientrule;and- thechainrule;

• differentiatecosec 𝑥,sec 𝑥andcot 𝑥• differentiatesimplefunctionsandrelationsdefinedimplicitlyorparametrically,includingfindingthesecondderivative;and

• constructsimpledifferentialequationsinpuremathematicsandincontext.


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Content

LearningOutcomes

Integration


• integrate𝑒R(,1𝑥, sin 𝑘𝑥,cos 𝑘𝑥andrelatedfunctions;• useadefiniteintegraltofindtheareabetweentwocurves;• demonstrateunderstandingofanduseintegrationasthelimitofasum;

• carryoutsimplecasesofintegrationbysubstitutionandintegrationbypartsandunderstandthesemethodsastheinverseprocessesofthechainandproductrulesrespectively;

• integrateusingpartialfractions;• evaluatetheanalyticalsolutionofsimplefirstorderdifferentialequationswithseparablevariables,includingfindingparticularsolutions;

• interpretthesolutionofadifferentialequationinthecontextofsolvingaproblem,includingidentifyinglimitationsofthesolution;

• evaluateavolumegeneratedbytherotationoftheareaunderasinglecurveaboutthe𝑥-axis;

Numericalmethods

• locaterootsoff 𝑥 = 0byconsideringchangesofsignoff(𝑥)inanintervalof𝑥inwhichf(𝑥)iscontinuous;

• solveequationsapproximatelyusingsimpleiterativemethods,forexampletheNewton–Raphsonmethod;

• demonstrateunderstandingofandusenumericalintegrationoffunctions(viatrapeziumrule),includingfindingtheapproximateareaunderacurve;and

• usenumericalmethodstosolveproblemsincontext.


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3.5 Unit A2 2: Applied Mathematics ThisunitassumesknowledgeofUnitsAS1,AS2andA21.ItcoverstheappliedcontentofA2MathematicsandiscompulsoryforAlevelMathematics.Theunitaddressesaspectsofbothmechanics(50percentoftheassessment)andstatistics(50percentoftheassessment).Itassessesmodellingandtheapplicationofmathematics.Theunitisassessedbya1hour30minuteexternalexamination,with6–10questionsworth100rawmarks.Theexaminationhastwosections:SectionAassessesmechanicsandSectionBassessesstatistics.Studentsanswerallquestionsinbothsections.Thestatisticalcontentofthisunitshouldbetaughtthroughtheuseandinterrogationofalargedataset.Theexaminationwillteststudents’abilityto:

• interpretrealdatapresentedinsummaryorgraphicalform;and• usedatatoinvestigatequestionsarisinginrealcontexts.Section A: Mechanics

Content

LearningOutcomes

Kinematics


• usecalculusinkinematicsformotioninastraightline:

𝑣 =d𝑠𝑑𝑡

𝑎 =d𝑣d𝑡 =

d:𝑠d𝑡:

𝑠 = 𝑣 d𝑡

𝑣 = 𝑎 d𝑡

• usecalculusinkinematicsintwodimensions:

𝐯 =d𝐫d𝑡

𝐚 =d𝐯d𝑡 =

d:𝐫d𝑡:

𝐫 = 𝐯 d𝑡

𝐯 = 𝐚 d𝑡

• modelmotionundergravityintwodimensionsusingvectors;and

• solveproblemsinvolvingprojectiles.


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Content

LearningOutcomes

Moments Studentsshouldbeableto:

• demonstrateunderstandingofandusemomentsinsimplestaticcontexts,includingrods,laddersandhingedbeams;

Impulseandmomentum

• demonstrateunderstandingofanduseimpulseandmomentum;and

• demonstrateunderstandingofandusetheprincipleofconservationoflinearmomentumtosolveproblemsinvolvingdirectcollisionsandexplosions.

Section B: Statistics

Content

LearningOutcomes

Probability Studentsshouldbeableto:

• demonstrateunderstandingofanduseconditionalprobability,includingtreediagrams,Venndiagramsandtwo-waytables;

• demonstrateunderstandingofandusetheconditional

probabilityformula:P 𝐴 𝐵 = P(𝐴∩𝐵)P(𝐵)

• modelwithprobability,includingcritiquingassumptionsmadeandthelikelyeffectofmorerealisticassumptions;

Statisticaldistributions

• demonstrateunderstandingofandusethenormaldistributionasanexampleofacontinuousprobabilitydistribution;

• findprobabilitiesusingthenormaldistribution;and• selectanappropriateprobabilitydistributionforacontext,withappropriatereasoning,includingrecognisingwhenabinomialornormalmodelmaynotbeappropriate.


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Content

LearningOutcomes

Statisticalhypothesistesting


• demonstrateunderstandingandusethelanguageofstatisticalhypothesistesting:- nullhypothesis;- alternativehypothesis;- significancelevel;- teststatistic;- 1-tailtest;- 2-tailtest;- criticalvalue;- criticalregion;- acceptanceregion;and- p-value;

• demonstrateunderstandingthatasampleisbeingusedtomakeaninferenceaboutthepopulationandappreciatethatthesignificancelevelistheprobabilityofincorrectlyrejectingthenullhypothesis;

• conductastatisticalhypothesistestfortheproportioninthebinomialdistributionandinterprettheresultsincontext;

• conductastatisticalhypothesistestforthemeanofanormaldistributionwithknown,givenorassumedvarianceandinterprettheresultsincontext;and

• interpretagivencorrelationcoefficientusingagiven p-valueorcriticalvalue.


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4 Scheme of Assessment 4.1 Assessment opportunities Eachunitisavailableforassessmentinsummereachyear.ItispossibletoresitindividualASandA2assessmentunitsonceandcountthebetterresultforeachunittowardsanASorAlevelqualification.Candidates’resultsforindividualassessmentunitscancounttowardsaqualificationuntilwewithdrawthespecification.4.2 Assessment objectives Therearethreeassessmentobjectivesforthisspecification.Candidatesmust:

AO1 useandapplystandardtechniques,by:• selectingandcorrectlycarryingoutroutineprocedures;and• accuratelyrecallingfacts,terminologyanddefinitions;

AO2 reason,interpretandcommunicatemathematically,by:

• constructingrigorousmathematicalarguments(includingproofs);• makingdeductionsandinferences;• assessingthevalidityofmathematicalarguments;• explainingtheirreasoning;and• usingmathematicallanguageandnotationcorrectly;

AO3 solveproblemswithinmathematicsandinothercontexts,by:

• translatingproblemsinmathematicalandnon-mathematicalcontextsintomathematicalprocesses;

• interpretingsolutionstoproblemsintheiroriginalcontextand,whereappropriate,evaluatetheiraccuracyandlimitations;

• translatingsituationsincontextintomathematicalmodels;• usingmathematicalmodels;and• evaluatingtheoutcomesofmodellingincontext,recognisethelimitationsofmodelsand,whereappropriate,explainhowtorefinethem.


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4.3 Assessment objective weightings ThetablebelowsetsouttheassessmentobjectiveweightingsforeachassessmentunitandtheoverallAlevelqualification:

AssessmentObjective

AssessmentUnitWeighting

AS1 AS2 A21 A22

AO1 50% 50% 50% 50%

AO2 25% 25% 25% 25%

AO3 25% 25% 25% 25%

Total 100% 100% 100% 100%

(Weightingshaveatoleranceof±3%)

4.4 Synoptic assessment at A2 TheA2assessmentunitsincludesomesynopticassessment,whichencouragescandidatestodeveloptheirunderstandingofthesubjectasawhole.InourGCEMathematics,synopticassessmentinvolves:

• buildingonmaterialfromtheASunits;and• bringingtogetherandmakingconnectionsbetweenareasofknowledge,understandingandskillsthattheyhaveexploredthroughoutthecourse.


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4.5 Higher order thinking skills TheA2assessmentunitsprovideopportunitiestodemonstratehigherorderthinkingskillsbyincorporating:

• moredemandingunstructuredquestions;and• questionsthatrequirecandidatestomakemoreconnectionsbetweensectionsofthespecification.

4.6 Reporting and grading Wereporttheresultsofindividualassessmentunitsonauniformmarkscalethatreflectstheassessmentweightingofeachunit.WeawardASqualificationsonafivegradescalefromAtoE,withAbeingthehighest.WeawardAlevelqualificationsonasixgradescalefromA*toE,withA*beingthehighest.Todeterminecandidates’grades,weaddtheuniformmarksobtainedinindividualassessmentunits.TobeawardedanA*,candidatesneedtoachieveagradeAontheirfullAlevelqualificationandatleast90percentofthemaximumuniformmarksavailablefortheA2units.IfcandidatesfailtoattainagradeE,wereporttheirresultsasunclassified(U).ThegradesweawardmatchthegradedescriptionsinSection5ofthisspecification.


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5 Grade Descriptions Gradedescriptionsareprovidedtogiveageneralindicationofthestandardsofachievementlikelytohavebeenshownbycandidatesawardedparticulargrades.Thedescriptionsmustbeinterpretedinrelationtothecontentinthespecification;theyarenotdesignedtodefinethatcontent.Thegradeawardeddependsinpracticeupontheextenttowhichthecandidatehasmettheassessmentobjectivesoverall.Shortcomingsinsomeaspectsofcandidates’performanceintheassessmentmaybebalancedbybetterperformancesinothers.ASGradeDescriptions

Grade

Description

ASAGrade

ForAO1,candidatescharacteristically:

• selectandaccuratelycarryoutalmostallroutineprocedurescorrectly;and

• accuratelyrecallalmostallfacts,terminologyanddefinitions.


• independentlyconstructrigorousmathematicalargumentsinalmostallrelevantcontexts;

• makevaliddeductionsandinferencesinalmostallrelevantcontexts;

• assess,critiqueandimprovethevalidityofamathematicalargumentinalmostallrelevantcontexts;

• constructextendedchainsofreasoningtoachieveagivenresult,findandcorrecterrorsandexplaintheirreasoning,evaluatingevidenceinalmostallrelevantcontexts;and

• usemathematicallanguageandnotationcorrectlyinalmostallrelevantcontexts.


• translateproblemsinmathematicalornon-mathematicalcontextsintomathematicalprocessesinalmostallrelevantcontexts;

• interpretsolutionstoproblemsintheiroriginalcontextand,whereappropriate,evaluatetheiraccuracyandlimitationsinalmostallrelevantcontexts;

• translatesituationsincontextintomathematicalmodelsinalmostallrelevantcontexts;

• usemathematicalmodelsinalmostallrelevantcontexts;and• evaluatetheoutcomesofmodellingincontext,recognisethelimitationsofmodelsand,whereappropriate,explainhowtorefinetheminalmostallrelevantcontexts.


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Grade

Description

ASEGrade


• selectandaccuratelycarryoutsomeroutineprocedurescorrectly;and

• accuratelyrecallsomefacts,terminologyanddefinitions.


• independentlyconstructrigorousmathematicalargumentsinsomerelevantcontexts;

• makevaliddeductionsandinferencesinsomerelevantcontexts;

• assess,critiqueandimprovethevalidityofamathematicalargumentinsomerelevantcontexts;

• constructextendedchainsofreasoningtoachieveagivenresult,findandcorrecterrorsandexplaintheirreasoning,evaluatingevidenceinsomerelevantcontexts;and

• usemathematicallanguageandnotationcorrectlyinsomerelevantcontexts.


• translateproblemsinmathematicalornon-mathematicalcontextsintomathematicalprocessesinsomerelevantcontexts;

• interpretsolutionstoproblemsintheiroriginalcontextand,whereappropriate,evaluatetheiraccuracyandlimitationsinsomerelevantcontexts;

• translatesituationsincontextintomathematicalmodelsinsomerelevantcontexts;

• usemathematicalmodelsinsomerelevantcontexts;and• evaluatetheoutcomesofmodellingincontext,recognisethelimitationsofmodelsand,whereappropriate,explainhowtorefinetheminsomerelevantcontexts.


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A2GradeDescriptions

Grade

Description

A2AGrade


• selectandaccuratelycarryoutalmostallroutineprocedurescorrectly;and

• accuratelyrecallalmostallfacts,terminologyanddefinitions.


• independentlyconstructrigorousmathematicalargumentsinalmostallrelevantcontexts;

• makevaliddeductionsandinferencesinalmostallrelevantcontexts;

• assess,critiqueandimprovethevalidityofamathematicalargumentinalmostallrelevantcontexts;

• constructextendedchainsofreasoningtoachieveagivenresult,findandcorrecterrorsandexplaintheirreasoning,evaluatingevidenceinalmostallrelevantcontexts;and

• usemathematicallanguageandnotationcorrectlyinalmostallrelevantcontexts.


• translateproblemsinmathematicalornon-mathematicalcontextsintomathematicalprocessesinalmostallrelevantcontexts;

• interpretsolutionstoproblemsintheiroriginalcontextand,whereappropriate,evaluatetheiraccuracyandlimitationsinalmostallrelevantcontexts;

• translatesituationsincontextintomathematicalmodelsinalmostallrelevantcontexts;

• usemathematicalmodelsinalmostallrelevantcontexts;and• evaluatetheoutcomesofmodellingincontext,recognisethelimitationsofmodelsand,whereappropriate,explainhowtorefinetheminalmostallrelevantcontexts.


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Grade

Description

A2EGrade


• selectandaccuratelycarryoutsomeroutineprocedurescorrectly;and

• accuratelyrecallsomefacts,terminologyanddefinitions.


• independentlyconstructrigorousmathematicalargumentsinsomerelevantcontexts;

• makevaliddeductionsandinferencesinsomerelevantcontexts;• assess,critiqueandimprovethevalidityofamathematicalargumentinsomerelevantcontexts;

• constructextendedchainsofreasoningtoachieveagivenresult,findandcorrecterrorsandexplaintheirreasoning,evaluatingevidenceinsomerelevantcontexts;and

• usemathematicallanguageandnotationcorrectlyinsomerelevantcontexts.


• translateproblemsinmathematicalornon-mathematicalcontextsintomathematicalprocessesinsomerelevantcontexts;

• interpretsolutionstoproblemsintheiroriginalcontextand,whereappropriate,evaluatetheiraccuracyandlimitationsinsomerelevantcontexts;

• translatesituationsincontextintomathematicalmodelsinsomerelevantcontexts;

• usemathematicalmodelsinsomerelevantcontexts;and• evaluatetheoutcomesofmodellingincontext,recognisethelimitationsofmodelsand,whereappropriate,explainhowtorefinetheminsomerelevantcontexts.


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6 Guidance on Assessment Therearefourexternalassessmentunitsinthisspecification,twoatASlevelandtwoatA2:

• UnitAS1:PureMathematics;• UnitAS2:AppliedMathematics;• UnitA21:PureMathematics;and• UnitA22:AppliedMathematics.6.1 Unit AS 1: Pure Mathematics Thisunitisassessedbya1hour45minuteexternalexamination,with6–10questionsworth100rawmarks.6.2 Unit AS 2: Applied Mathematics Thisunitisassessedbya1hour15minuteexternalexamination,with5–10questionsworth70rawmarks.Theexaminationhastwosections:SectionAassessesmechanicsandSectionBassessesstatistics.Candidatesanswerallquestionsinbothsections.Questionsonthestatisticsandmechanicscontentoftheunitareeachworth50percentoftheavailablerawmarks.6.3 Unit A2 1: Pure Mathematics Thisunitisassessedbya2hour30minuteexternalexamination,with7–12questionsworth150rawmarks.6.4 Unit A2 2: Applied Mathematics Thisunitisassessedbya1hour30minuteexternalexamination,with6–10questionsworth100rawmarks.Theexaminationhastwosections:SectionAassessesmechanicsandSectionBassessesstatistics.Candidatesanswerallquestionsinbothsections.Questionsonthestatisticsandmechanicscontentoftheunitareeachworth50percentoftheavailablerawmarks.


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7 Links and Support 7.1 Support Thefollowingresourcesareavailabletosupportthisspecification:

• ourMathematicsmicrositeatwww.ccea.org.uk• specimenassessmentmaterials;and• guidancenotesforteachers.Wealsointendtoprovide:

• pastpapersandmarkschemes;• ChiefExaminer’sreports;• planningframeworks;• supportdaysforteachers;• aresourcelist;and• exemplificationofstandards.7.2 Curriculum objectives ThisspecificationsupportscentrestobuildonthebroaderNorthernIrelandCurriculumobjectivestodeveloptheyoungperson:

• asanindividual;• asacontributortosociety;and• asacontributortotheeconomyandenvironment.ItcancontributetomeetingtherequirementsoftheNorthernIrelandEntitlementFrameworkatpost-16andtheprovisionofabroadandbalancedcurriculum.CurriculumProgressionfromKeyStage4ThisspecificationbuildsonlearningfromKeyStage4andgivesstudentsopportunitiestodeveloptheirsubjectknowledgeandunderstandingfurther.StudentswillalsohaveopportunitiestocontinuetodeveloptheCross-CurricularSkillsandtheThinkingSkillsandPersonalCapabilitiesshownbelow.Theextentofthisdevelopmentdependsontheteachingandlearningmethodologytheteacheruses.Cross-CurricularSkills• Communication:–TalkingandListening–Reading–Writing

• UsingMathematics• UsingICT


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ThinkingSkillsandPersonalCapabilities• ProblemSolving• WorkingwithOthers• Self-ManagementForfurtherguidanceontheskillsandcapabilitiesinthissubject,pleaserefertothesupportmaterialsonthesubjectmicrosite.7.3 Examination entries EntrycodesforthissubjectanddetailsonhowtomakeentriesareavailableonourQualificationsAdministrationHandbookmicrosite,whichyoucanaccessatwww.ccea.org.ukAlternatively,youcantelephoneourExaminationEntries,ResultsandCertificationteamusingthecontactdetailsprovided.7.4 Equality and inclusion Wehaveconsideredtherequirementsofequalitylegislationindevelopingthisspecificationanddesignedittobeasfreeaspossiblefromethnic,gender,religious,politicalandotherformsofbias.GCEqualificationsoftenrequiretheassessmentofabroadrangeofcompetences.Thisisbecausetheyaregeneralqualificationsthatpreparestudentsforawiderangeofoccupationsandhigherlevelcourses.Duringthedevelopmentprocess,anexternalequalitypanelreviewedthespecificationtoidentifyanypotentialbarrierstoequalityandinclusion.Whereappropriate,wehaveconsideredmeasurestosupportaccessandmitigatebarriers.Wecanmakereasonableadjustmentsforstudentswithdisabilitiestoreducebarrierstoaccessingassessments.Forthisreason,veryfewstudentswillhaveacompletebarriertoanypartoftheassessment.Itisimportanttonotethatwhereaccessarrangementsarepermitted,theymustnotbeusedinanywaythatunderminestheintegrityoftheassessment.YoucanfindinformationonreasonableadjustmentsintheJointCouncilforQualificationsdocumentAccessArrangementsandReasonableAdjustments,availableatwww.jcq.org.uk


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7.5 Contact details Ifyouhaveanyqueriesaboutthisspecification,pleasecontacttherelevantCCEAstaffmemberordepartment:

• SpecificationSupportOfficer:NualaTierney(telephone:(028)90261200,extension2292,email:[email protected])

• SubjectOfficer:JoeMcGurk(telephone:(028)90261200,extension2106,email:[email protected])

• ExaminationEntries,ResultsandCertification(telephone:(028)90261262,email:[email protected])

• ExaminerRecruitment(telephone:(028)90261243,email:[email protected])

• Distribution(telephone:(028)90261242,email:[email protected])

• SupportEventsAdministration(telephone:(028)90261401,email:[email protected])

• Moderation(telephone:(028)90261200,extension2236,email:[email protected])

• BusinessAssurance(ComplaintsandAppeals)(telephone:(028)90261244,email:[email protected]@ccea.org.uk).

© CCEA 2017

CCEA GCE Specification in Mathematics

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