LEARNING FROM PAST MISTAKES: ADVICE FOR HSC MATHEMATICS AND EXTENSIONS STUDENTS 2006 Robert Yen, Hurlstone Agricultural High School

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1 LEARNING FROM PAST MISTAKES: ADVICE FOR HSC MATHEMATICS AND EXTENSIONS STUDENTS 2006 Robrt Yn, Hurlston Agricultural High School Evry yar, th HSC xaminrs (markrs) publish a rport on th prformancs of studnts in ach xam, outlining thir common strngths, waknsss and rrors as wll as providing marking guidlins and advic for studnts and tachrs. Ths Nots from th Marking Cntr can b downloadd from th Board of Studis wbsit Th Mathmatical Association of NSW (MANSW) also publishs booklts of past HSC xams with workd solutions (phon or wbsit you may b abl to purchas ths through your school. Evry Fbruary at Macquari Univrsity, MANSW conducts an HSC Fdback and Advic day for tachrs, whr snior xaminrs analys studnts' rsponss in th prvious yar's xams. Th rst of this articl summariss th advic givn by HSC xaminrs from past xams. Th following 2 xam tips ar basd on succssful tchniqus shown by studnts who achivd highly in Mathmatics, Mathmatics Extnsion and Mathmatics Extnsion 2.. Show working in your answrs Writing a bald answr in an xam with no working shown is th most common mistak mad by HSC studnts. Last yar s Qustion 6 (a) on using Simpson s rul with a tabl of valus was worth 3 marks. Th solution involvd using th corrct formula, substituting appropriat valus, thn valuating th answr. Studnts who wrot a bald wrong answr scord 0 out of 3, whil thos who gav a wrong answr with som corrct working scord or 2. This xampl illustrats th importanc of showing working. Som HSC qustions vn say justify your answr or giv rasons for your answr. Whn marking a qustion, th xaminr is looking for vidnc that you know your mathmatics. Each mark is awardd as you pass ach stag in thsolution, so it pays to st out your work as natly as possibl, listing ach stp down th pag. Evn if you mak a mistak at th start of a solution, you can still arn marks for th rst of th answr if your working is corrct. Th numbr of marks allocatd to th qustion is a good guid to how many stps ar rquird. If using a formula, show th valus bing substitutd as this may arn you that first mark. Bfor you writ th final answr, writ th unroundd answr first (g ) as this may arn you a mark vn if you round it incorrctly. Finally, you should not scribbl ovr or ras incorrct or rough working. A nat lin crossd through it may hlp xaminrs undrstand your thinking. Somtims, vn crossdout working can arn marks. 2. Rad and answr th whol qustion Somtims, studnts don't rad an HSC qustion compltly and answr what thy think th qustion is asking, lav out som important part of th answr or wast tim doing mor than what was rquird. Exampls from last yar s xams: Qustion What som studnts did instad

2 4 (b) (iii) Hnc sktch th graph of y = f(x), showing th turning points and th points whr th curv mts th x-axis. 6 (c) (i) Find th points of intrsction of th two curvs. Extnsion : (d) Using th substitution u = 2x 2 +, find 5 x x dx. Extnsion : 6 (a) (iii) Find th probability that Mgan arns at most 6 points during th ightn-wk sason. Extnsion 2: 6 (a) (i) For ach intgr n 0 prov by induction that Did not show th x-intrcpt of 3 or found th non-horizontal points of inflxion (not ncssary). Found thir x-valus only, not thir y- valus. Lft thir answr in trms of u and forgot to convrt back to x. Incorrctly intrprtd at most 6 points as lss than 6 and not including 6. Provd for n and nglctd n = 0. Whn rading an xam qustion, idntify xactly what nds to b found, highlighting any ky words, spcially thos in CAPITALS or italics. Aftr complting th solution, doubl-chck that you hav actually answrd th qustion by r-rading it, spcially if it rquirs a wordd rspons. If askd to find th quation of a tangnt or normal, don t just find its gradint. Whn finding stationary points, rmmbr to find y-valus also and tst th natur of th points, including horizontal points of inflxion. Finally, bwar of doublbarrlld qustions whr two qustions ar askd in on, such as Qustion (c) in last yar s x Extnsion xam: Stat th domain and rang of y = cos Highlight th ky words of an HSC qustion Studnts ar oftn unfamiliar with th trminology found in HSC qustions. Hr ar som xampls from last yar s xams: Qustion 9 (b) (ii): Show that th limiting sum BD + EF + GH + is givn by 6 sc θ tan θ. Qustion 0 (a) (i): Explain why α + β = m and αβ = -b. Extnsion Qustion 7 (b) (iv): Hnc, or othrwis, dduc that g(θ) has an invrs function. Extnsion 2 Qustion 2 (d) (iii): Dscrib th locus of R as z varis. Hr ar som xampls of ky words found in HSC qustions, including som dfinitions from th Board of Studis glossary ( Compar: show how things ar similar or diffrnt Dscrib/idntify: provid charactristics or faturs Estimat/approximat: mak an ducatd guss or giv an inxact answr Evaluat/calculat: find th valu, th numrical answr Expand: rwrit th xprssion without brackts,.g. 3(2b + ) = 6b + 3 Explain/Giv a rason: dscrib why or how Exprss: rwrit in a particular way Hnc (or othrwis): find or prov a rsult using th answrs from th prvious part of th qustion (th or othrwis mans that thr ar altrnativ mthods but ths ar usually mor complicatd) Intrprt: draw maning from th mathmatical rsult Justify: support your argumnt or conclusion

3 Prov that / Show that / Dduc that: in qustions whr th answr is givn, prov that th answr is tru using formal mathmatics and rasoning Simplify: rduc to a shortr, natr form Solv: find th valu of th unknown variabl in an quation or inquality Vrify: show or chck that a rsult is tru by substituting back (not a formal proof) Writ down / Stat: writ th answr, formula or rsult without working or xplanation (usually in a on-mark qustion) To practis rading, highlighting and answring qustions involving such ky words, us past HSC xams and trial paprs. Whn askd to answr in words or prov a rsult, us th numbr of marks allocatd as a guid to how many things to writ about or how many stps to show. Don t writ a pag of writing or working for a 2-mark qustion. Studnts who writ long answrs try to includ vrything, oftn bcoming confusd and contradicting thmslvs. Just answr th qustion. 4. Know how to answr a show that qustion Oftn in an xam, you will find th phras show that or prov that followd by an answr or rsult. Som xampls from last yar s xams: Qustion 3 (c) (i): Show that th quation of th lin AD is y = 2x 2. Qustion (a) (i): Show that th volum of th cylindr is givn by V = 2πh(R 2 h 2 ). Extnsion Qustion 6 (a) (i): Show that th probability that Mgan arns on point for a givn wknd is Why ar you givn th answr in th qustion? Firstly, to hlp you stay on track with your proof and work towards gtting th corrct rsult. Scondly, th answr will most likly b usd in th nxt part of th qustion, so vn if you fail to find it, you can still continu with th rst of th qustion. Many studnts hav troubl answring show that qustions. Som us th answr in thir working, but this can lad to back-to-front or circular proofs that ar incorrct. Som studnts don t achiv th rquird answr, but mak dodgy changs to thir working to mak it look lik thy hav provd it (what HSC xaminrs call fudging ). Wors still, som studnts claim that th xam s answr is wrong! You should trat a show that qustion lik a normal qustion, and work towards gtting th answr as th nd rsult. Don t us th answr in your working but hop that your answr matchs it. Furthrmor, bcaus th answr is givn, you must show xtra working or xplanation to arn full marks. If you don t gt th answr, thn backtrack to find whr you mad th mistak and corrct vry lin of working from that point. Don t fudg your answr or th xaminrs will catch you out. 5. Ralis whn th parts of a qustion ar rlatd Most HSC qustions hav rlatd parts, whr th answr to part (i) lads to th solution of (ii) and so on. This is always th cas with th coordinat gomtry qustion in th Mathmatics papr (usually Qustion 2) or any qustion involving a proof. Studnts nd to look out for such qustions, vn whn th phras hnc show is not statd, and us past HSC xams and trial paprs to practis answring thm. Qustion 2 (c) of last yar s Extnsion xam askd for th drivativ of 3x (cos x 3 sin x) in part (i), whil part (ii) askd hnc, or othrwis, find x 3 sin x dx. Many studnts did not ralis that part (ii) was a on-mark qustion involving th answr to part (i), -0 3x sin x, and instad thy trid to intgrat from scratch, oftn wasting tim and making mistaks. A similar qustion appard in Qustion 3 (b) of th xam.

4 In a rlatd-parts qustion, you ar bing ld stp-by-stp through its solution, with hints kindly providd along th way. Othrwis, it would sm odd to b askd to find thr or four unrlatd things in th sam qustion. Studnts who don t ralis that a qustion s parts ar rlatd oftn us incorrct or complicatd mthods that ar longr and pron to rror, or rinvnt th whl and solv th first part of th problm again. Tak not of th marks allocatd to ach part of a qustion. 6. Aim for dp undrstanding rathr than rcip larning. HSC studnts ar oftn guilty of rot-larning a mathmatical mthod without undrstanding th thory or maning bhind it, lik a bginnr chf who blindly follows th instructions of a cooking rcip. Studnts who do this oftn gt a stp wrong or hav difficulty solving an unusual problm that rquirs adjusting th mthod thy hav larnd. For xampl, Qustion 5 (a) of th Extnsion xam askd for th intgral sin 2 2xdx. Studnts who had mmorisd sin 2 x dx x sin 2x did not know how to adapt this to 2 4 answr th qustion what to do with th, and th 2x. 2 4 Qustion 4 (b) of th Mathmatics xam involvd graphing f(x) = (x + 3)(x 2 9). Studnts who mmorisd a curv-sktching rcip answrd most of this qustion wll, but had troubl whn thy wr askd th mor thortical on-mark qustion, For which valus of x is th graph of y = f(x) concav down, bcaus thy had only larnd th procss and not th logic bhind it. A dp undrstanding of mathmatics rathr than larning a rcip distinguishs a good studnt from an ordinary studnt. 7. Choos th simplst and quickst mthod Evry qustion in an HSC xam has bn dsignd to b answrd within a spcific tim, with th numbr of marks allocatd suggsting how long it should tak. Howvr, studnts ar oftn guilty of using a mor complicatd mthod whn a simpl on would do th job ffctivly. Qustion 3 (a) of last yar s xam askd studnts to valuat th sum n 5 3 2n, which rducd to Instad of simply adding to gt 27 for this on-mark qustion, som studnts actually usd an arithmtic sris formula! Of cours, this still workd but it took mor tim and had a gratr chanc of rror. In Qustion (), th absolut valu inquality x 3 is bst solvd by rwriting it as - x 3, but som studnts complicatd things by splitting it into x 3 and -(x 3). Othr common rrors: using th sin or cosin ruls with right-angld triangls instad of th basic sin, cosin and tangnt ratios using A = 2 ab sin C for th ara of a triangl whn A = 2 bh is asir using trigonomtry whn Pythagoras thorm or similar triangls is quickr listing th many possibl outcoms of a probability problm instad of using a tabl, tr diagram or systmatic counting listing all of th trms in a sris instad of using a formula using long division of polynomials instad of th rmaindr and factor thorms (Extnsion )

5 Aftr rading an HSC qustion, don t jump in impulsivly and start writing straight away, not knowing whr it will lad. Think about th most fficint mthod of solution. Considr drawing a diagram or graph, making a list or tabl, looking for a numbr pattrn, using logic, common sns, guss-and-chck, som thory or formula, or th answr to a prvious part of th qustion. If your working-out to a problm is taking too long, thn it s probably wrong or inappropriat (prhaps unlss it s an Extnsion 2 qustion).. St out your gomtrical proofs corrctly Many studnts hav difficulty writing a gomtrical proof clarly and logically, oftn using inappropriat trminology, notation and rasoning. Examinrs oftn find 2-pag answrs for a 2-mark qustion. Whn solving a gomtry problm, rad th qustion, draw a big diagram, idntify what nds to b found, sktch a rough proof on th diagram, thn writ ach stp and its rasoning (in brackts) formally, prsnting your proof as concisly as possibl. In Qustion 5 (b) of th Mathmatics xam and th Extnsion xam, and Qustion 7 (a) of th Extnsion xam, xaminrs rportd that studnts did not nam angls corrctly or mak thir rasons clar. It is important to copy any diagrams givn so that som rough working could b don on it rathr than on th qustion papr. Common rrors mad in gomtry proofs includ badlydrawn diagrams or a lack of on, not justifying statmnts with rasons and incorrct trminology. Don't say Z angls whn you man altrnat angls, buttrfly angls whn you man angls in th sam sgmnt of a circl (Extnsion ), or th windsurfr thorm whn you man th altrnat sgmnt thorm (Extnsion ). Such slang trms may b usful larning aids but thy ar nithr widly usd nor accptabl trms. Not vryon will know what you man and you will los marks! 9. Diffrntiat and intgrat proprly You should ralis by now that most of th Mathmatics HSC cours is takn up with calculus topics, with th only non-calculus topics bing sris, probability and gomtry, so any srious study program should includ thorough rvision of thm. Common mistaks in this ara includ: diffrntiating instad of intgrating, or vic-vrsa (Extnsion 2 Qustion (c)) suprficial undrstanding or misus of th product, quotint and chain ruls (Qustions 2 (b), 6(b) (iii), Extnsion Qustion 2 (a)) diffrntiating and intgrating xponntial, logarithmic and trigonomtric functions not tsting stationary points or points of inflxion (Qustions 4 (b) (ii), (a) (ii), 0 (a) (iv)) difficultis with th trapzoidal or Simpson s ruls: finding th valu of h, th wightings, using th x-valus instad of th y-valus (Qustion 6(a)) graphing y = f'(x) from y = f(x) (Qustion 7 (b) (iii)) You should know: how to find, tst and draw points of inflxion accuratly th maning of incrasing/dcrasing and th first drivativ tst Evry yar, som HSC studnts los a coupl of marks by not rfrring to th list of standard intgrals at th back of th xam papr. This list is to hlp you rmmbr important ruls so familiaris yourslf with it during your study tim.

6 0. Us your calculator corrctly Larn to valuat long xprssions such as usd in a sris problm (Qustion 7 (a) (iii)) or cos 0.6 usd whn applying th cosin rul (Qustion 4 (a) (ii)). For trigonomtry, know whn to st your calculator in dgrs or radians mod and how to convrt btwn th two. Radian masur is usd for arc lngth, aras of sctors and sgmnts, som trigonomtric quations, th calculus of trigonomtric functions and its applications to motion. Mak sur you undrstand th indx and logarithm laws and ar abl to us thm to simplify xprssions and quations involving log x and x. In last yar s xam, som studnts had troubl using th chang of bas law to valuat log 3 7 on thir calculator (Qustion 5 (a)). Whn valuating a natural logarithm on a calculator, rmmbr to us th ln ky for log x, not th log ky (which valuats log 0 x).. Study th Prliminary cours and your basic Yars 9-0 skills Up to 20% of th Mathmatics xams may tst topics from th Prliminary cours, so don t forgt to rvis Yar work as wll. If you ar studying for th Extnsion and Extnsion 2 xams, rmmbr to practis your Mathmatics and Extnsion skills, rspctivly, as wll. Studnts ar xpctd to know all of th formulas of th cours, so mak an ffort to mmoris thm prcisly. Th xaminrs always find instancs of formulas writtn with + and signs in th wrong placs, spcially in coordinat gomtry, trigonomtry and sris. With financial applications of sris, studnts should not b larning spcific formulas for tim paymnts or suprannuation, as thy ar not part of th syllabus. As vry financial problm is diffrnt, any such formula usd in an xam would nd to b provd first. Studnts who us ths formulas in an xam oftn fac difficultis whn th qustion is unconvntional and rquirs adjusting th formula. Also brush up on your basic Yars 9-0 skills, bcaus th xaminrs ar always amazd by th numbr of silly mistaks mad by HSC studnts in simpl numbr and algbra, such as in xpanding and factorising algbraic xprssions, substitution, solving linar and quadratic quations, simplifying surds, adding and multiplying fractions in probability problms, and graphing lins on a numbr plan. Exampls from last yar s xams: Qustion 4 (a) (ii): whn using th cosin rul to find an unknown sid, studnts usd incorrct signs or forgot to squar root Qustion (b): som wrot th quation of a smi-circl as y = 2 4 x = 2 x. Extnsion Qustion (f): solving -4 = 2m as m = - 2. Extnsion 2 Qustion (c): xpanding Examinrs find it frustrating whn thy s a studnt scor marks for solving a complx part of a problm but los marks for making a carlss rror on th asist part of th solution. 2. Draw graphs and diagrams accuratly Each yar, HSC xaminrs ar amazd by th numbr of studnts who fail to bring ssntial quipmnt such as a rulr to th xam, rsulting in poorly-drawn graphs and to 6 6.

7 diagrams. Ths half-hartd attmpts ar small, mssy and drawn frhand without much attntion to dtail. All diagrams should b clar and larg, taking up half a pag. With numbr plan graphs, th x- and y-axs should b ruld, lablld and markd with a consistnt scal, and asymptots, turning points and x- and y-intrcpts shown whr appropriat. Studnts should b familiar with th graphs and proprtis of all functions. Last yar, som Extnsion studnts could not graph y 2x + 3 proprly (Qustion (b)). Also, w Spnd tim rvising how to draw barings (in trigonomtry) and tr diagrams (in probability) proprly, as ths skills ar tstd frquntly. Common mistaks in using tr diagrams includ not knowing what to do with th probabilitis on th branchs, and adding or multiplying fractions incorrctly. 0 STEPS FOR SUCCESS IN HSC MATHEMATICS EXAMS. Find out all you can about th format of th xam (rviw past HSC paprs). 2. B prpard. 3. Us th rading tim and numbr of marks to plan your approach to th xam. Not th hardr qustions you may nd to spnd mor tim on thm. 4. Spnd th first minut of ach qustion planning, thinking and choosing th bst stratgy to us. You don't nd to b writing all of th tim. What you'r writing may b wrong and a wast of tim. 5. Sprad out your work. Draw big diagrams. 6. Pac yourslf kp an y on th tim. 7. Mak sur you hav answrd th qustion. Chck that it sounds rasonabl, spcially if it involvs mony or masurmnt. Did you includ th corrct units?. Don't gt boggd down mov on if you'r gtting nowhr. If your working-out to a hard qustion is taking too long, thn it's probably wrong. Rtrac your stps, start again, or com back to it latr. 9. If you mak a mistak, draw a nat lin through it. Don't scribbl ovr it or us corrction fluid. You may still gt marks for it if it is right. 0. At th nd of th xam, chck your work and go back and attmpt hardr or uncrtain qustions. Rfrncs Board of Studis NSW (2006) HSC Nots from th Marking Cntr: Mathmatics, Mathmatics Extnsion and Mathmatics Extnsion 2, BOS wbsit HSC Examinrs Day (Fbruary 2006), run by MANSW at Macquari Univrsity. Th Mathmatical Association of NSW, Inc (2006). Highr School Crtificat 2005 Mathmatics Examination Paprs and Solutions. Yn, R. (995). Tn common mistaks mad by HSC Maths studnts, Rflctions, 20(4), p.7.