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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
5
2 Specification at a Glance
6
3 Subject Content 7 3.1 OverarchingthemesinGCEMathematics 73.2 UnitAS1:PureMathematics 93.3 UnitAS2:AppliedMathematics 143.4 UnitA21:PureMathematics 183.5 UnitA22:AppliedMathematics
22
4 Scheme of Assessment 25 4.1 Assessmentopportunities 254.2 Assessmentobjectives 254.3 Assessmentobjectiveweightings 264.4 SynopticassessmentatA2 264.5 Higherorderthinkingskills 274.6 Reportingandgrading
27
5 Grade Descriptions
28
6 Guidance on Assessment 32 6.1 UnitAS1:PureMathematics 326.2 UnitAS2:AppliedMathematics 326.3 UnitA21:PureMathematics 326.4 UnitA22:AppliedMathematics
32
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
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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
Externalwrittenexamination1hour15minsStudentsanswerallquestions.
40%ofAS16%ofAlevel
A21:PureMathematics
Externalwrittenexamination2hours30minsStudentsanswerallquestions.
36%ofAlevel
A22:AppliedMathematics
Externalwrittenexamination1hour30minsStudentsanswerallquestions.
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.)
Studentsshouldbeableto:
• 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
Studentsshouldbeableto:
• 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.)
Studentsshouldbeableto:
• 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
Studentsshouldbeableto:
• 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.)
Studentsshouldbeableto:
• 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
Studentsshouldbeableto:
• 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.)
Studentsshouldbeableto:
• 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
Studentsshouldbeableto:
• 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
Studentsshouldbeableto:
• 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.)
Studentsshouldbeableto:
• 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
Studentsshouldbeableto:
• 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
Studentsshouldbeableto:
• 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
Studentsshouldbeableto:
• 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.
ForAO2,candidatescharacteristically:
• independentlyconstructrigorousmathematicalargumentsinalmostallrelevantcontexts;
• makevaliddeductionsandinferencesinalmostallrelevantcontexts;
• assess,critiqueandimprovethevalidityofamathematicalargumentinalmostallrelevantcontexts;
• constructextendedchainsofreasoningtoachieveagivenresult,findandcorrecterrorsandexplaintheirreasoning,evaluatingevidenceinalmostallrelevantcontexts;and
• usemathematicallanguageandnotationcorrectlyinalmostallrelevantcontexts.
ForAO3,candidatescharacteristically:
• translateproblemsinmathematicalornon-mathematicalcontextsintomathematicalprocessesinalmostallrelevantcontexts;
• interpretsolutionstoproblemsintheiroriginalcontextand,whereappropriate,evaluatetheiraccuracyandlimitationsinalmostallrelevantcontexts;
• translatesituationsincontextintomathematicalmodelsinalmostallrelevantcontexts;
• usemathematicalmodelsinalmostallrelevantcontexts;and• evaluatetheoutcomesofmodellingincontext,recognisethelimitationsofmodelsand,whereappropriate,explainhowtorefinetheminalmostallrelevantcontexts.
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Grade
Description
ASEGrade
ForAO1,candidatescharacteristically:
• selectandaccuratelycarryoutsomeroutineprocedurescorrectly;and
• accuratelyrecallsomefacts,terminologyanddefinitions.
ForAO2,candidatescharacteristically:
• independentlyconstructrigorousmathematicalargumentsinsomerelevantcontexts;
• makevaliddeductionsandinferencesinsomerelevantcontexts;
• assess,critiqueandimprovethevalidityofamathematicalargumentinsomerelevantcontexts;
• constructextendedchainsofreasoningtoachieveagivenresult,findandcorrecterrorsandexplaintheirreasoning,evaluatingevidenceinsomerelevantcontexts;and
• usemathematicallanguageandnotationcorrectlyinsomerelevantcontexts.
ForAO3,candidatescharacteristically:
• translateproblemsinmathematicalornon-mathematicalcontextsintomathematicalprocessesinsomerelevantcontexts;
• interpretsolutionstoproblemsintheiroriginalcontextand,whereappropriate,evaluatetheiraccuracyandlimitationsinsomerelevantcontexts;
• translatesituationsincontextintomathematicalmodelsinsomerelevantcontexts;
• usemathematicalmodelsinsomerelevantcontexts;and• evaluatetheoutcomesofmodellingincontext,recognisethelimitationsofmodelsand,whereappropriate,explainhowtorefinetheminsomerelevantcontexts.
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A2GradeDescriptions
Grade
Description
A2AGrade
ForAO1,candidatescharacteristically:
• selectandaccuratelycarryoutalmostallroutineprocedurescorrectly;and
• accuratelyrecallalmostallfacts,terminologyanddefinitions.
ForAO2,candidatescharacteristically:
• independentlyconstructrigorousmathematicalargumentsinalmostallrelevantcontexts;
• makevaliddeductionsandinferencesinalmostallrelevantcontexts;
• assess,critiqueandimprovethevalidityofamathematicalargumentinalmostallrelevantcontexts;
• constructextendedchainsofreasoningtoachieveagivenresult,findandcorrecterrorsandexplaintheirreasoning,evaluatingevidenceinalmostallrelevantcontexts;and
• usemathematicallanguageandnotationcorrectlyinalmostallrelevantcontexts.
ForAO3,candidatescharacteristically:
• translateproblemsinmathematicalornon-mathematicalcontextsintomathematicalprocessesinalmostallrelevantcontexts;
• interpretsolutionstoproblemsintheiroriginalcontextand,whereappropriate,evaluatetheiraccuracyandlimitationsinalmostallrelevantcontexts;
• translatesituationsincontextintomathematicalmodelsinalmostallrelevantcontexts;
• usemathematicalmodelsinalmostallrelevantcontexts;and• evaluatetheoutcomesofmodellingincontext,recognisethelimitationsofmodelsand,whereappropriate,explainhowtorefinetheminalmostallrelevantcontexts.
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Grade
Description
A2EGrade
ForAO1,candidatescharacteristically:
• selectandaccuratelycarryoutsomeroutineprocedurescorrectly;and
• accuratelyrecallsomefacts,terminologyanddefinitions.
ForAO2,candidatescharacteristically:
• independentlyconstructrigorousmathematicalargumentsinsomerelevantcontexts;
• makevaliddeductionsandinferencesinsomerelevantcontexts;• assess,critiqueandimprovethevalidityofamathematicalargumentinsomerelevantcontexts;
• constructextendedchainsofreasoningtoachieveagivenresult,findandcorrecterrorsandexplaintheirreasoning,evaluatingevidenceinsomerelevantcontexts;and
• usemathematicallanguageandnotationcorrectlyinsomerelevantcontexts.
ForAO3,candidatescharacteristically:
• 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