Maths Bridging Units Part 1 smaller · Pierre de Fermat - all-time champion of not showing your...
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Please read this document carefully, and complete the tasks set. 1. Understanding in Maths Many of you will have survived Maths up to now not really understanding everything you were doing - or why you were doing it - quite a bit of the time. You may have been happy to simply memorise a method in response to a particular situation, follow a process, and get the answer in the back of the book. You can get through GCSE like that, but at A-level, you will find that by Christmas you are really struggling. In Year 12, many of you will need to become more aware of and personally in charge of your understanding. Not sure what the teacher is on about? Ask a question! Wondering what happens if… Ask a question! Think of it like this ‘Venn’ Diagram. It becomes increasingly difficult to expand the yellow bit without expanding the green bit first. So from now on, whenever you’re doing Maths, you need to be asking yourself: If the answer is ‘no’ then you need to do something about it! That might be as simple as asking a question in class. It might mean going away and reading the textbook or doing some more practice. It might be a rare occasion when your teacher says “trust me – it will become clear as you do a few examples”. But if it doesn’t become clear, you need to do something. Mathematics Stuff you really understand Stuff you can do TBSHS A-level Maths Bridging Unit – Part 1 Do I really understand this? What does this mean? What am I doing? Why? What have I achieved? Can I check it? What if I changed that…?
Transcript of Maths Bridging Units Part 1 smaller · Pierre de Fermat - all-time champion of not showing your...
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.
