Pleasereadthisdocumentcarefully,andcompletethetasksset.

1.UnderstandinginMathsManyofyouwillhavesurvivedMathsuptonownotreallyunderstandingeverythingyouweredoing-orwhyyouweredoingit-quiteabitofthetime.Youmayhavebeenhappytosimplymemoriseamethodinresponsetoaparticularsituation,followaprocess,andgettheanswerinthebackofthebook.YoucangetthroughGCSElikethat,butatA-level,youwillfindthatbyChristmasyouarereallystruggling.

InYear12,manyofyouwillneedtobecomemoreawareofandpersonallyinchargeofyourunderstanding.Notsurewhattheteacherisonabout?Askaquestion!Wonderingwhathappensif…Askaquestion!Thinkofitlikethis‘Venn’Diagram.Itbecomesincreasinglydifficulttoexpandtheyellowbitwithoutexpandingthegreenbitfirst.

Sofromnowon,wheneveryou’redoingMaths,youneedtobeaskingyourself:Iftheansweris‘no’thenyouneedtodosomethingaboutit!Thatmightbeassimpleasaskingaquestioninclass.Itmightmeangoingawayandreadingthetextbookordoingsomemorepractice.Itmightbearareoccasionwhenyourteachersays“trustme–itwillbecomeclearasyoudoafewexamples”.Butifitdoesn’tbecomeclear,youneedtodosomething.

Mathematics

Stuffyoureallyunderstand

Stuffyoucando

TBSHSA-levelMathsBridgingUnit–Part1

DoIreallyunderstandthis?

Whatdoesthismean?WhatamIdoing?

Why?WhathaveIachieved?

CanIcheckit?WhatifIchangedthat…?

* Practise solving a quadratic equation by completing the square *Exam Qus on hypothesis testing with Binomial exp.

TASK1:Startyour‘To-Ask-and-Do’ListOnapieceofpaperatthefrontofyourfolderthatyoubringtoeveryMathslesson,youshouldkeepandconstantlyupdatea‘ToDo’listof:

• Questionsyouwanttoaskyourteacheraboutplacesyougotstuckduringindependentstudyorhomework.Thismightbe,forexample:

• Concepts,methodsortopicsyouknowyouneedtodomorepracticeofindependently

Startyourfirst‘To-Do’listnow.Callitwhatyoulike,butkeepitatthefrontofyourworkontheBridgingUnitAlgebraWork,andasyougothrough,writethingsyouneedtoworkonoraskabout,beingasspecificasyoucan.Forinstance,youmightdecideyouwanttodomoreindependentpracticeofsolvingsimultaneousequations,oryoumightwanttoasksomeonetoexplainhowtofigureoutthenumbersinthebracketswhenyoufactoriseaquadraticexpression.Youshouldalwaystrytobringalongaparticularquestionandyourattemptatitwhenyouwanttoaskamemberofstaffsomething.

* Q9 part (ii) on p213 – what does “verify” mean?

* June 2018 Paper 2 Q5 – what topic is this? Why?

2.CommunicationandPresentationinMaths

AtA-level,communicationbecomesevenmoreimportantinMaths.Mostofthetime,afinal‘answer’ismuchlessthefocusthanareadablesolutiontoaproblem,showingasequenceoflogicalstepsofdeduction-orworking-thatjustifyyouranswer.

Manyquestionsbeginwith“Showthat…”or“Prove…”sotheanswerisgivenfromtheoutsetandwhatyouneedtodoiscommunicateaclearlyreasonedargumentthatprovesitcorrect.Donotfear!Wewillteachyouhowtodothis,andwhiletherewillbesomewritingtodo,algebrawilldomostoftheheavylifting.Wealsohavesomegreatshorthand:∴ therefore ⇒ implies

∵ because ⇐ isimpliedby

⟺ bothimpliesandisimpliedby(i.e.isanequivalentstatement)

= isequalto ≠ isnotequalto

≡ isidentically(i.e.always)equalto < islessthan

∝ isdirectlyproportionalto ≥ isgreaterthanorequaltoUsingtheaboveattimes,youshouldaimtowriteMathsthatalwaysreads(thatis,someonecouldreaditoutloudanditwouldmakesenseinEnglish)becausewrittenMathsisreallyjustaspecialsupersetoftheeverydayEnglishlanguage.

PierredeFermat-all-timechampionofnotshowingyourworkings.

AmajorchallengeofA-levelMathsisinterpretingthelanguageusedinquestions.TheBESTwaytogetgoodatthisistostartspeakingandwritinginthatlanguagefromtheget-go.þ

Rule#1:WriteMathsthat‘reads’.

‘Modes’ofwork:ExpressionsvsEquationsThereareessentiallytwodifferentmodesofwritingMaths;youprobablyusethemandswitchbetweenthemalreadywithoutthinkingaboutit.Mode1:“Thisequalsthis,equalsthis,whichequalsthistoo,whichequalsthethingIwanted”Thatishowitshouldreadwhenyouaremanipulatinganexpression(numericaloralgebraic),butinthequoteabove,the‘equals’signsformachainacrossthepagewhichisgenerallynotnice.Yougettheaddedbenefitofitbeingeasiertocheckyourworkbycomparingeachlinedirectlywithwhatisaboveit.Forexample,lookatthisstudent’sworkonaGCSEproofquestion.

They’remissingtheword“squared”fromthesentenceattheend,butotherwisethisisperfect.Itreadsas:“Twonplusonemustbeanoddnumber.Twonplusoneworkedoutandthensquaredisequaltotwonplusonemultipliedbytwonplusone,whichequalsfourlotsofnsquaredplustwolotsofnplustwomorelotsofnplusone,whichisequalto….“andsoon.Youcanseewhyweusenotationinsteadofwords!Thatmiddle,mathematicalsentence,togetherwiththefirstandlastsentences,formaparagraphthattellsthestory,provingthedesiredresult.

Rule#2:WorkDOWNthepage.

Sometimes–especiallywhendoingcalculations–nowordsareneeded,butyoushouldstilljointhingsthatareequalupwith‘=’signsatthebeginningofeachlineasinthenextexample.

Thisstudenthasdonesomeextra‘thinking’andkeptthisnicelyseparatefromtheirmainpassageofwork,soitformsalovely,clear,readableandcheckablesolution.Anotherlovelythingthey’vedoneistoshoweachstepoftheworking.Manypeoplewouldmissoutthesecondandfourthlinesonthisquestion.Whilethatmightbesafeinthisexample,whenitformspartofalongerquestionwithmoregoingon,errorsareeasiertomake–especiallyifyouaretryingtodotwothingsatonce.Mostofthetime,itdoesn’ttakeanymoretimetowriteallthestepsthantothinkthem,butoncetheyareonthepageyoudon’thavetojuggletheMathsinyourhead,somistakesarelesslikely.Plus,thenextthingtodooftenoccurstoyouonlyafteryou’vedonethefirstthingandwrittenitdown.Thepageislikeanextensionofyourworkingmemory–liketheRAMinacomputer–ifyougetinthehabitofusingitmoreeffectively,youwilldoMathsmoreaccuratelyand,actually,quickertoo.Afterall,asNietzscheputit:“Hewhowouldlearntoflyonedaymustfirstlearntostandandwalkandrunandclimbanddance;onecannotflyintoflying.”

Rule#3:Showallthesteps–doonethingatatime.

Mode2:“Thisfactleadstothisfact,whichimpliesthisotherfact,whichmeansthisistrue.”Anequationisastatementsayingtwothingsareequal(the‘things’arecalledexpressions).WhenManipulatingEquations,wewriteanequationoneachline,andinsodoingwearestatingafact(withintheworldsetupinthequestion–probablynotauniversaltruth).Thenonthenextline,wewriteanotherequation(orfact)thatwecandeducefromtheequationabove.Andsoon.Sometimeswecombineseveralfactsintoonenewone.EquationsarethebreadandbutterofA-levelMaths.Youwillsoonbeabletosolveaquadraticequationorapairofsimultaneousequationsasonesmallstepinalargerproblemasreadilyasyoumightcurrentlyaddupacoupleofnumbers.Strictly,tomakeyoursolutiontoanequationread,youwant∴or⇒atthestartofeachline,butwemanipulateequationssooftenthatwetendtoleavetheseout.

Onecommonandhorriblemathematicalcrimeistomisusetheequalssign‘=’whereyoureallywant‘implies’or‘therefore’.Thisleadstothingslikethefollowingexample.ý

Ifyoureadthisthrough,thispersonisclaimingthateachandeveryexpressionthey’vewrittenisequal.Thatincludes8=9=3,whichisclearlynonsense!

DONOTDOTHIS!

Thisstudentcouldhaveplaced∴or⇒atthestartofallbutthefirstline,butweallknowwhattheymean.They’vegottwo‘=’signsinthelastline,butthisisprobablyoneofveryfewplaceswe’llforgivethat!

Similarcrimesarecommittedduring‘Mode1’worksometimes.Considerthefollowingexample.

Possiblyworsethanallthose‘=’signsononeline(theyarenotworkingdownthepage!)isthefactthatwhenwereadthatline,theyaresayingthat3times100isequalto175,whichisclearlynottrue!Thismaynotseemimportant,butitisVITALyousaywhatyoumeaninyourMaths–examinerscanonlymarkwhatisinfrontofthem,andifitisself-contradictory,you’llgetzero.Exercise:Writeoutthecalculationaboveintwoincreasinglybetterways.

(a) Writeeachcalculationsteponanewline,rewritingtheanswerfromthelineaboveinsteadofre-usingit.Hereisadifferentexampleworkedoutthisway:

(b) Writethewholeexpressionagainoneachnewline,precededbyanequalssign.Each

time,workoutthenextstepaccordingtoBIDMAS.Hereistheexampleagain:

ThisishowyouwillworkatA-level.Withpracticeitismoreefficient,lessrisky,andmuchmorereadable.Nowre-writetheexampleatthetopofthepagetheseways.

3.ClassicMistakesinMathsThereareanumberofcommonalgebraicpitfallsthatcatchA-levelmathematiciansout.Themostfrequentmustbethe‘signerror’:thelossofanegativesignorinadvertentswitchingbetweenapositiveandanegativewithinworking,butthereareothers.Duringthecourse,youwillneedtospotthesortsofsituationwhenyoumightmakecommonerrors,andfindwaystoavoidthem–andtocheckyourworktomakesureyouhave.Herearejustafewofexamples,thensomeexampleGCSEstudentresponsesforyoutoexamine–wherehavetheygonewrong?Couldtheyhavecheckedtheirwork?Whatshouldtheyhavedoneinstead?SignErrorsCanyouspotthestudent’serrorhere?Whatshouldtheanswerbe?

Hopefullyyouspottedtheyhaveworkedout−2×−2𝑝as−4𝑝insteadofas+4𝑝.Theymayhavecarefullyexpandedthesecondbracketbymultiplyingthecontentsby2ratherthan-2,sotheycouldmaketheirfirstlineofworkingcorrectwithbrackets:

5𝑝 + 15− (2− 4𝑝)Thisacknowledgestheyneedtosubtracttheentireresultofexpandingthesecondbracketfrom5𝑝 + 15.Sowhentheysubtract−4𝑝,they’llendupwith9𝑝astheyshould.Alternatively,theycouldhavejustdealtwiththenegativetimesanegativeastheyexpandedthebracket,andwritten5𝑝 + 15− 2+ 4𝑝beforesimplifying.Youwilldothissometimes(justlikeweallwrite‘there’insteadof‘their’sometimes)andtheonlywaytostopitistoseeitcomingandcheckyourwork.Oncethey’dwrittentheirfirstline,thisstudentneededtocarefullycompareittotheoriginalexpression.You’llgetreallyquickatthiswithpractice.

Whichbringsusto…

That’sarulethisstudenthastriedtofollow,butthey’vestillgonewrong.Canyouseewhere?

Despitethestudent’sowntick,thisresponsegot0intheexam.Hopefullyyouknowthatwhenyousquareanegativeyougetapositive.Thisstudentwantstosquare−5buttheyhavewritten−5!andBIDMASmeansthe5hereissquaredbeforethenegativesigngetsitshandsonit.Whichmeanstheyhavewrittensomethingworth−25.Sowhentheytypeditintotheircalculator,theygotanegativeanswerinsteadofapositiveone.Theymayhavemeanttosquare−5,butunlesstheywrite(−5)!theexaminercanonlyreaditasthey’vewrittenit,andthinktheymeantosquare5andmaketheanswernegative.Thatisalsowhattheircalculatorwilldo.Ifyoudon’tbelieveme,checkitforyourselfonyourcalculator.Thisisn’tapeculiarityofcalculators–whatthisstudentwrotedowniswrong.It’salwaysbestnottobewrongifyoucanavoidit.Onceagainhere,ifthey’dcheckedforamomentbythinking“Shouldmyanswerbepositiveornegative?”,theymighthavefixeditbythinking“Anegativesquaredisapositive,and2overapositiveispositive.Hangonaminute…”.Soaverygoodwaytocheckasyougoalongistothinkaboutwhatsortofanswer(size,sign,integerordecimaletc)youexpect,andbesuspiciousifyoudon’tgetthesortofthingyouthoughtyoushould.DodgyDistributionWecarryoutanumberofdifferentoperationsonnumbersduringcalculations,buttheseoperationsdonotallbehaveinthesameways.Forinstance,additionandmultiplicationarecommutativeoperations,whichmeanstheorderinwhichyouaddtwonumbersup(ormultiplytwonumberstogether)doesnotmatter.

So4+ 7isthesameas7+ 4.

Similarly,5×3 = 3×5.

Rule#4:Checkyourworkasyougoalong.

However,subtractionanddivisionarenotcommutativeoperations:

4− 7isnotthesameas7− 4.

Similarly,10÷ 2 ≠ 2÷ 10.Anotherpropertythatoperationscanhaveistobedistributive.Now,youwon’tneedtousethisterminyourA-levelexams,butknowingitcanhelpyouavoiderrorsbecausenotalloperationsaredistributive.Sometimesstudentstreatnon-distributiveoperationsasiftheyaredistributive,andsometimesvice-versa.Forourpurposes,anoperationisdistributiveif,whenyoucarryitoutonasumoftwonumbers,yougetthesameresultasifyouinsteadcarryouttheoperationonthetwoseparatenumbersfirstandthenaddthem.It’smuchclearerwithanexample:Supposewewanttomultiply(3+ 4)by5.Ifweaddupthe3and4firsttoget7andmultiplyby5,weget35.Ifinstead,wedothe‘×’by5tothe3and4separately,weget3×5+ 4×5whichgivesusthesameanswer.Somultiplicationlikethisisdistributive.Ifwewantedtoadd5insteadofmultiplyingby5,wewouldn’tadd5toBOTHthe3andthe4astheoverallresultwouldbetoadd10,soadditionisNOTdistributiveinthesameway.TheclassicexamplewherepeoplegowrongatA-levelissquaringandsquare-rooting.Squaringasumisnotassimpleassquaringthetwonumbersandaddingthemup,andthesamegoesforsquare-rooting.Supposeyouwanttoworkout 9+ 16.WhatyouCANNOTdoissquarerooteachofthe9and16separatelytoget3and4,thenadd.Youcanseethiswouldbewrongbyworkingitout.9+ 16 = 25sotheansweris5,not7.Thisisperhapsmorecommoninalgebraquestionsthanincalculations;acommonmistakemightbetothinkyoucanrewrite 𝑥! + 4as𝑥 + 2.Youshouldtrysomevaluesforxonyourcalculatortoconvinceyourselfthisisnottruealmostallthetime.Similarly,facedwithapairofsimultaneousequationslike

𝑥! + 𝑦! = 25𝑦 = 𝑥 − 1

adisappointinglycommonerroristothink“Iwanttogetridofthose‘squareds’,soI’lljustsquarerootthetopequation”andthinkyouget𝑥 + 𝑦 = 5.Youdon’t.Allyouwouldgetis 𝑥! + 𝑦! = 5whichismuchlesspleasantandusefulthanwhatyouhadtostartwith.Hopefullyyouknowthecorrectmethodherewouldbetousethebottomequationtosubstitute(𝑥 − 1)intothetopequationinplaceofy,givingyouanequationthatonlyinvolvesx.Thisiswhatwecall‘eliminating’y.

TASK3:ClassicMistakesYouwillneedtokeepinmindandreferbacktoallthevariousmistakesyou’vereadabouthereinordertoanswersomeofthefollowingquestions.You’llalsoneedsomeGCSEknowledge!SectionA–DistributionComprehension

1) Writeoutanexamplewithnumbersshowingthatsquaringisnotdistributive.(Youmaywishtousetheexplanationaboveforsquare-rootingasatemplate.)

2) Divisionisatrickybeastsometimes.Usethesetwoexamplestotrytoworkoutifandwhendivisionisdistributive:

!"!!"!"

and !"!!!

Writeasentenceortwoexplainingwhatyoufoundout.

SectionB–SpottheMistakesLookcarefullyateachofthefollowingstudentresponsestoGCSEquestionsandfindtheerror(orerrors!)Itmighthelptoworkthequestionoutyourself,orthinkofawaytocheckthestudent’sanswer.Explaininwordswhathasgonewrongandwhy,andtrytosuggesthowthiscouldbelookedoutforandavoided.1)Divideandconquer?Thereareatleasttwomistakeshere.Byallmeansworkoutwhattheyshouldhavedone,butwewantyoutotrytofigureoutwhatisactuallywrongwithwhatthey’vewritten.

2)PopQuiz,Hotshot!Thisstudenthasactuallyansweredadifferentquestion.Whichquestion?Whatshouldtheyhavedone,andwhatwastheirmistake?

HINT:Howisspeedshownonthisgraph?(HINT2:Whenisthecarmovingfastest?)Whydoesthequestionusetheword‘estimate’?3)Confused.comLookcloselyatthisone–allisnotasitappears.Itseemslikeasimplesliptostartwith,butwhatarethosecircleswith“+2”inthemallabout?

4)Loadsofworking,butnotallcorrectWe’veseenacorrectsolutiontothisearlier.Thismistakeisprobablyduetosomefairlyclassicrushing.What’stheerror?Howcouldtheyhaveimprovedtheirpresentationaswellastheiraccuracy?

5)Differenceoftwosquares!They’vedonelotsofgoodthings:checkingbyexpandingin(a),andbeingcarefulnottomakeasignerrorbyputtingthatbigbrackettherein(b),butthey’vemadesomanyslips.Where?Also,trytodopart(b)usingtheanswertopart(a),asthe‘Hence’suggests.

6)There’snosubstituteforsubstitution.Sothisisn’tastudentresponse–it’saGCSEquestionaboutamade-upstudentresponsethatmakesacommonerror.Whatwouldyouwritetoanswerthis1-markquestion?Furthermore,whatwouldyoudoifsomeonereadyouranswerandsaid“Whatdoyoumean?”or“Why?”Bysubstitutingsomenumbersintotheequation𝑦 = 𝑥! + 1,findawaytodemonstrateandjustifyyouranswertoanuncertainclassmate.Writeitdownsotheycouldreaditandbeconvinced.

7)Check,check,andcheckagain!Firstly,pleasecheckthefiveanswersthisstudenthasgivenbysubstitutingthemintothetwooriginalinequalities.Theydon’tallwork–whichdon’t?Showclearworkings.Suchashamethisstudentdidn’tcheckthat!Nowfindwherethisstudentwentwrongintheirworking.ThereareafewplacestheA-levelexamboardarepicky.Quadraticinequalitiesisoneofthem.AtA-level,you’dneedasketchedgraphorequivalentcalculationstojustifyyourchoiceofsolutionstoaquadraticinequalityliketheoneontherighthere.Whatwouldyoudo?

Wehopeyouhavefoundthisinteresting,challenging,orperhapsboth!

PleasebringyourwrittenresponsestoyourfirstlessoninSeptember,readytoshare.

PleasealsotrytofollowtheadviceabovewhentacklingyourAlgebraassignments.