Whats up? Reflect on todays VCE Seminar Think about what you want - - PDF document

Further Mathematics Past-student perspective

Hayley Short Recent VCE Graduate

What’s up?

• Reflect on today’s VCE Seminar…

– Think about what you want to achieve from today / Further / VCE / life in general – Take a moment to say hi!

Hayley Short

• Graduated in 2013
• Studying at Monash University

– Bachelor of Business (Accounting)

• Seminars Director with Engage
• Harry Potter Nerd…

Motivation

You’re almost there!

• Know what motivates you:

– Competition? – A specific end goal? – Constructive fear?

Study Techniques

How can I study effectively to score well on exams?

Engage Education Foundation

Studying Process

ReadText Books ReadText Books HighlightKey Points HighlightKey Points Chapter Summary Chapter Summary Questions fromtextbook Questions fromtextbook Readallnotes multipletimes Readallnotes multipletimes PracticeExams PracticeExams

But don’t stop there!

• Learn from your mistakes:

– Correct your work – Work out where you lost marks – Re-write those answers until they are the mark you want to achieve – Re-read that response to remind yourself – Find similar questions and make sure you can do them – Mark question types you tend to lose marks on

H O W I L E A R N T T H E N

www.engageeducation.org.au 1

H O W I L E A R N N O W

Studying - Procrastination

DISTRACTIONS!!!

Know What Works for YOU!

• Quiet or Music?
• At home or at school?
• Cold or hot?
• Hungry or full?
• Rewards or punishment?
• Facebook or a run?
• Cold shower or no snacks?

Time Management

It’s super important. But actually.

Engage Education Foundation

What to do now?

• Exam 1 is on October 30
• That is just 19 days away!
• How can you maximise the time you have

left?

Two Weeks to Exam

www.engageeducation.org.au 2

One Week to Exam Exam Week Study timetable

• Be honest
• Be specific
• Commit to it
• Make ALL subjects a priority

Time Management – Other Tips

• Have an overall plan
• Write ‘to do’ lists every day
• Make time for yourself

Exams

Unavoidable Not that scary really

Engage Education Foundation

Know the exam

• What questions come up every year?
• VCAA Exam 1 2014: Variable data

includes distance, sex, number of children, type of car, postcode How many are categorical? 0, 1, 2, 3 or 4?

• Where can you find exams?

Before the exam

• Pack the night before (make sure you have

a fully charged calculator!)

• Check the details (where is it, what time?)
• Read over your bound reference
• Don’t speak to people who will stress you
• ut:

– “Did you see the exam that said…?” – “I heard the examiners are including a section about…”

In the exam

• Don’t waste your reading time- test how to

best use these precious 15 minutes.

• Read carefully:

– Questions with instructions (number of decimal places) – How many marks is that?

• Highlight or underline key information

www.engageeducation.org.au 3

The Engage Wiki– Makesureyoucheckitout!

• 12subjects
• 150,000wordsofnotes
• 25+hoursofvideo
• Toptipsfromtopscores(videoseries)
• CheatsheetsforeverySACandtheEXAM
• TopicspecificVCAAquestions(likecheckpoints)
• 75FREEpracticeexams
• ATARcalculator
• VCEExplainedvideoseries

30shortvideosansweringallyourquestions e.g.Howismystudyscorecalculated

More subjects

Freepracticeexams– goto wiki.ee.org.au

Question Time

Favourite Harry Potter book?

Questions?

• Email us at info@ee.org.au
• Email me at hayley.short@ee.org.au
• Post on the Engage Facebook page

– The Engage Team will answer your questions

Some final words…

• READ the question carefully
• ANSWER what the question asks
• LEARN from your mistakes
• Ask for help:

– Your teacher – Your tutor – Your friends – Me!

DO NOT BE MR BEAN!

Further Maths

Emily Condos Sunday, October 11th 2015

FURTHERMATHEMATICS CORE– DataAnalysis

www.engageeducation.org.au 4

DATA

Categorical Numerical Discrete Continuous DatacanbeclassifiedasCategoricalorNumerical. Categoricaldatainvolvesnamesorlabels. Numericaldatainvolvesnumbers. Datawhichinvolvescounting iscalledDiscrete. Datawhichinvolvesmeasuring iscalledContinuous. eg AFLteamyousupport eg numberofbrothers egheight

CLASSIFYINGDATA

Somecategoricaldatacanappearnumerical: eg theTVchannelsusenumbersfortheirnames Channel7andChannel9 also,youallprobablyhavestudentnumbersatyourschool eg 53772. However,thesenumbersarenotusedforcalculations,sotheyare CategoricalData.

CLASSIFYINGDATA

Discretedata canonlytakecertainvalues withinarange remember: COUNTING whilst Continuousdata canpossiblytakeallvalueswithinarange. remember: MEASURING

NumericalData CategoricalData

CLASSIFYINGDATA

VCAAExam1,2006.

UNIVARIATEDATA

Datawhichdescribesonevariable. Whatyouhavetobeabletodo:

1.Displaythedata. 2.Interpretadatadisplay. 3.CalculateUnivariate statistics.

DISPLAYINGUNIVARIATEDATA

Histogram BarGraph SegmentedBarGraph BoxPlot DotPlot StemPlot Numberof children Numberof families 18 1 27 2 16 FrequencyTable

DISPLAYINGUNIVARIATEDATA

QuestionsaskingyoutodisplaydatainacertainwaycanonlybeaskedonExam2. VCAAExam2,2006.

DISPLAYINGUNIVARIATEDATA

QuestionsaskingyoutodisplaydatainacertainwaycanonlybeaskedonExam2. VCAAExam2,2008.

DISPLAYINGUNIVARIATEDATA

QuestionsaskingyoutodisplaydatainacertainwaycanonlybeaskedonExam2. VCAAExam2,2010. ThevalueforCanadainthisquestionwas inatableonthepagebefore.Itwas16.

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DISPLAYINGUNIVARIATEDATA

QuestionsaskingyoutodisplaydatainacertainwaycanonlybeaskedonExam2.

Youarenotaskedtodrawbig, complexdiagramsinthissection sodonotspendyourrevisiontime doingthistypeofactivity.

UNIVARIATEDATA

Datawhichdescribesonevariable. Whatyouhavetobeabletodo:

1.Displaythedata 2.Interpretadatadisplay 3.CalculateUnivariate statistics

INTERPRETINGUNIVARIATEDATA DISPLAYS

1.Readtheheadingtothequestioncarefully. 2.Readthelabelsontheaxesortheheadingsofatablecarefully. 3.Readtheunitsontheaxescarefully. 4.LookattheKeyifthereisone. VCAAExam1,2006. VCAAExam1,2007.

SegmentedBarGraphs

VCAAExam1,2010.

SegmentedBarGraphs

VCAAExam2,2008.

SegmentedBarGraphs

VCAAExam1,2009.

INTERPRETINGUNIVARIATEDATA DISPLAYS

Shapeofthedistribution

NEGATIVELYSKEWEDSYMMETRICALPOSITIVELYSKEWED

StemLeaf 07 123 224579 3023688 447899 5278 613 StemLeaf 076688 1237899 224579 3023 446 527 61 StemLeaf 07 12 224 3023 447899 5278889 61357

INTERPRETINGUNIVARIATEDATADISPLAYS

Stemplots

StemLeaf 07 12 224 3023 447899 5278889 61357 StemLeaf 07 12 224 3023 447899 5278889 61357

Key:2|4=24 Key:2|4=2.4 Thisstemplot displaysthevalues 7,12,22,24,30,32,33,44,47, 48,49,49,52,57,58,58,58,59, 61,63,65,67. Thisstemplot displaysthevalues 0.7,1.2,2.2,2.4,3.0,3.2,3.3,4.4, 4.7,4.8,4.9,4.9,5.2,5.7,5.8,5.8, 5.8,5.9,6.1,6.3,6.5,6.7.

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CALCULATINGUNIVARIATESTATISTICS

MODE – thevaluewhichoccursmostoften – thevaluewhichhasthehighestcolumn – thevaluewhichhasthegreatestfrequency VCAAExam1,2007.

StemLeaf 07 12 224 3023 447899 5278889 61357 Key:2|4=240

Whatisthemode?

CALCULATINGUNIVARIATESTATISTICS

MEDIAN – themiddlevalueoftheordereddistribution 50%ofthedistributionislessthanthemedian 50%ofthedistributionisgreaterthanthemedian LOWERQUARTILE (Q1 orQL) – middleofthelowerhalfofthedistribution 25%ofthedistributionislessthanthemedian UPPERQUARTILE (Q3 orQL) – middleoftheupperhalfofthedistribution 25%ofthedistributionisgreaterthanthemedian INTERQUARTILERANGE (IQR) =UPPERQUARTILE– LOWERQUARTILE RANGE =MAXIMUMVALUE– MINIMUMVALUE 12192326303334 MEDIAN LOWER QUARTILE UPPER QUARTILE MAX MIN Median=26 IQR=33– 19=14 Range=34– 12=22 7values. Middleisatthe

th 8 2 1 7

• position.

123149213246305338341372406410 MEDIAN LOWER QUARTILE UPPER QUARTILE MAX MIN Median=(305+338) 2=321.5 IQR=372– 213=161 Range=410– 123=287 10values. Middleisatthe

th 5 . 5 2 1 10

• position.

VCAAExam1,2006. 21values. Middleisatthe

th 11 2 1 21

• position.

Thatisa2. VCAAExam1,2009. VCAAExam1,2011. VCAAExam1,2006.

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INTERPRETINGUNIVARIATEDATADISPLAYS

BoxPlots

Oneveryimportantfactaboutboxplotsisthat25%ofthevaluesliein eachsection. Thisiseasytoconsiderforasymmetricboxplot,butisalsotruefora skewedboxplot. VCAAExam1,2007. VCAAExam1,2011. VCAAExam1,2007.

INTERPRETINGUNIVARIATEDATADISPLAYS

BoxPlots

AnotherimportantaspectofboxplotsareOUTLIERS. Outliersarevalueswhichareaconsiderabledistancebeloworabovethe

• therscores.Theyareoftentheresultofdatainputerror.

Mathematically,wetest 12192326303334 MEDIAN LOWER QUARTILE UPPER QUARTILE MAX MIN IQR=33– 19=14 IQR 1.5=21 So,tobeanoutlier,avaluewouldhavetobe lessthan19– 21=– 2

• r

greaterthan33+21=54. Clearly,noneofthesevaluesareoutliers. VCAAExam1,2008.

CALCULATINGUNIVARIATESTATISTICS

MEAN – alsocalledtheAVERAGE – addupallthevaluesanddividethetotalbythenumberofvalues STANDARDDEVIATION – calculatetheSDusingyourCAScalculator – measuresthespreadofvaluesaboutthemean – especiallysignificantforabellshapeddistribution The68–95–99.7%ruleforabellshaped curvestatesthatapproximately: (a) 68%ofdataliewithin1standard deviation eithersideofthemean (b) 95%ofthedataliewithin2standard deviations eithersideofthemean (c) 99.7%ofdataliewithin3standard deviations eithersideofthemean.

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VCAAExam1,2006. VCAAExam1,2011. LearntousetheStatsListEditor

• nyourCAScalculatortofind:

Mean ***StandardDeviation*** and Median,Q1,Q3,min,maxetc. Whenadistributionissymmetrical,likeabellshapeddistribution,both themeanandthemedianaregoodmeasuresofthecentralvalue. Ifthedistributionisskewed,themedianbecomesthebettermeasureofa centralvalue.Themeanlosesitsreliability. VCAAExam2,2010.

Z scores

AnotherstatisticwhichcomesfromthebellshapedcurveisaZscore. Zscores(alsoknownasstandardisedscores)areusedtocomparevaluesfrom differentdistributions. Azscoreof0indicatesthevalueisequaltothemean. Azscoreof1indicatesavalueequalto1standarddeviationabove themean. Azscoreof1indicatesavalueequalto1standarddeviationbelow themean andsoon... Azscoreiscalculatedusingtheformula

( )

VCAAExam2,2006.

BIVARIATEDATA

Datawhichdescribestwovariables. Bivariate datacanbedisplayedusing

BacktoBackStemPlots ParallelBoxPlots

anumericalvariable and acategoricalvariable with2categories. anumericalvariable and acategoricalvariable withatleast2categories.

BIVARIATEDATA

InExam2,studentsareoftenaskedtocomparethetwoormoredatastatisticsin bivariate andmultivariatedata. Whenyoudosoitisimportantto:

• 1. Makeacomparisonstatement

eg “…isgreaterthan…“

and

2.Supportyourstatementwithspecificvalues.

VCAAExam2,2011. VCAAExam2,2006.

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Scatterplots

Usedtodisplaytherelationshipbetweentwonumericalvariables NumberofswimmersatPrahranPoolandthecorrespondingdailytemperature. Beforegraphing,identifytheindependentvariableandthedependentvariable . Reasoning:Itisreasonabletoexpectthatthenumberofvisitorsattheswimming poolonanydaywilldependonthetemperatureonthatday(andnottheotherway around). Answer:Dailytemperatureistheindependentvariable; Numberofvisitorsisthedependentvariable.

Visitors 130 145 180 200 85 121 130 60 75 80 Temperature(C) 24 27 29 30 18 22 23 17 18 16

Number of visitors at Prahran pool Daily temperature

Eachdotonascatterplot represents2values. eg thecircleddotrepresents30monthsoldand92cmtall. Therelationshipbetweentwovariablesiscalledthecorrelation. Whendescribingthecorrelation,youneedtocommenton Strength …strong,moderateorweak Direction …positiveornegative Form …linearornonlinear Outliers. Fromthisscatterplot,ageandheightcouldbedescribedashavinga Moderatetostrong,positive,linearcorrelation.Oneoutlierispresent. Positive scatterplot isrisinglefttoright. Negative – scatterplot isfallinglefttoright. Strong – dotsare linedup. Moderate – dotsare tightlyclustered aroundanimaginary line. Weak – dotsare widelyclustered aroundanimaginary line. Apositivecorrelationbetweentwovariables,asabove,canbe describedbyasentencethatreads: “Astheageincreases,theheightincreases.” Remember,theheightisthedependentvariable,soitdependson thechangesintheindependentagevariable,ifthereisagood correlation. Hereisascatterplot fromwhichitappearsthatthereisamoderate,positive, nonlinearcorrelationbetweenincomeandaverageage.Ofwomanatfirst marriage.Nooutliersarecompletelyobvious. So,againbecausethescatterplot showsapositiverelationshipwecould write: “Astheincomeincreases,theaverageageofwomenatfirstmarriage increases.” Hereisascatterplot fromwhichitappearsthatthereisamoderate,negative, nonlinearcorrelationbetweentelevisionhoursperweekandhomework hoursperweek. Thistime,becausethescatterplot showsanegativerelationshipwecould write: “Asthetelevisionhoursperweekincreasesthehomeworkhoursperweek decrease.” Againnotethatthedependentvariabledependsonthechangesinthe independentvariableandnottheotherwayaround. Correlation,evenwhenitisstrong,doesnotmeanthatonevariable CAUSESanother.Itsimplytellsusthatthereisarelationshipbetween thetwovariableswhentheyoccurtogether. BecarefulofMultiplechoiceoptionswhichimplyonevariablecauses theother!

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r value

CalledtheCorrelationCoefficient. Givesanumericalvaluetothecorrelationbetweentwovariables. Perfectpositivelinearcorrelation…r =1 Norelationship…r =0 Perfectnegativelinearcorrelation…r = 1 Learntouse yourCAScalculator tofind: r – value Thervalueisusedtomake thedescriptionofthe correlationbetweentwo variablesmoreobjective. However,thervalueisonly validifthecorrelationislinear innature. Ifitisnonlinear,wehaveto transformitbeforewecan usestatisticslikethervalue.

r value r2 value

CalledtheCoefficientofDetermination. Iscalculatedbysquaringthervalue,andsoitisalwaysastatistic between0and1(0%to100%). Again,isonlyvalidinassessingbivariate datawithalinearrelationship. Isusedin2ways: Ittellsus 1.Theproportionofvariationinonevariablewhichcanbeexplained bythevariationintheothervariable. OR 2.Howwellthelinearrulelinkingthetwovariablescanpredictthe valueofthedependentvariablewhenwearegiventhevalueofthe independentvariable.

Scatterplots

Animportantstatisticaldevicewecalculateforascatterplot isalineofbestfit. WecallitaRegressionLine. Thelinesummarisesthepointsinthescatterplot. ItisonlyrelevanttofitalineofbestfitifthecorrelationisLinear. VCAAExam2,2010.

Scatterplots

Thelinehastheequationy =m x +c ORy =c+m x.

Scatterplots

Thelinehastheequationy =m x +c ORy =c+m x.

m istheslopeoftheline.Itindicatestherateatwhichthedependentvariableis

increasingordecreasinginrelationtotheindependentvariable.

c istheyintercept.Itindicatestheapproximatevalueofthedependentvariablewhen

theindependentvariableequals0.

Scatterplots

Theequationcanbeusedtocalculatevaluesforwhichdatadoesnotexist(predictions). Ifthepredictedvalueisinsidetherangeoftheexistingdata,itiscalledan INTERPOLATION. Ifthepredictedvalueisoutsidetherangeoftheexistingdata,itiscalledan EXTRAPOLATION. InterpolationsaremorereliablepredictionsthanExtrapolations.

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VCAAExam2,2010.

Scatterplots

FINDINGTHEEQUATION

Thereare3waystofindtheequationofthelinewhichyouhaveto learn.

1.3MEDIANMETHOD

a)fromtheGraph b)fromtheoriginaldata

2.LEASTSQUARESREGRESSION

a)fromtheoriginaldata

3.FromSummaryStatistics

FINDINGTHEEQUATION

3MEDIANMETHOD(fromthegraph)

The3MedianMethodisonewayoffindingalineofbestfitfordatawhichislinear in nature. Itrequiresatleast6 datapointstogetameaningfulline. Itisnotadverselyaffectedbyasmallnumberofoutliers.

Dividethepointsinto3groupsusingverticallines. (a) Ifthenumberofpointsisdivisibleby3, dividetheminto3equalgroups,eg 3,3,3or7,7,7. (b) Ifthereis1extrapoint,puttheextrapoint inthemiddlegroup,eg 3,4,3or7,8,7. (c) Ifthereare2extrapoints,put1extrapoint ineachoftheoutergroups,eg 4,3,4or8.

• Calculatethemedianxvalueofthe

pointsineachofthe3groups(when movinglefttoright).

• Calculatethemedianyvalueofthe

pointsineachofthe3groups(when movingbottomtotop).

FINDINGTHEEQUATION

3MEDIANMETHOD

DrawingtheLine

• Placearulersothatitpassesthrough

theloweranduppermedians.

• Maintainingthesameslope,slidethe

ruleronethirdofthewaytowardsthe middlemedian. FindingtheEquation Theequationofthelineforthe3Medianmethodisintheformy =mx+c whereand

FINDINGTHEEQUATION

3MEDIANMETHOD(fromtheoriginaldata) LEASTSQUARESREGRESSIONEquation

EachoftheseequationsfortheLineofBestFit canbefoundbyenteringthedataintolists

• nyourCAScalculatorandusingthe

appropriateoptions. Theequationswillbeslightlydifferentforeachmethodfor thesamesetofdatabecausetheyusedifferentapproaches. The3MedianMethodisbasedonmedians;theLeast SquaresRegressionEquationisbasedonmeans. Sincemediansarelessaffectedbyoutliersthanmeans,the 3MedianLineisabetterchoiceifthereareoutliersinthe data. Bothequationsareintheformy =m x +c ORy =c +mx.

Learntouse yourCAScalculator tofind: 3Medianequation & LeastSquares Regressionequation

FINDINGTHEEQUATION

FromSummaryStatistics

Tousetheformulae,youwouldbegiven: themeanoftheindependentvariable(xvariable) themeanofthedependentvariable(yvariable)

Sx

thestandarddeviationoftheindependentvariable

Sy

thestandarddeviationofthedependentvariable

r

Pearson’sproduct–momentcorrelationcoefficient.

x y

FindingtheEquation Theequationofthelineforthe3Medianmethodisintheformy =mx+c whereand VCAAExam1,2006.

EVALUATINGTHELINEARITYOFTHEDATA

Becauseourstatisticsarenotrelevantifthedistributionofdataisnotlinear, itisimportantto a) Checkhowlinearourdatais b) Tryandimprovethelinearityofourdata sothatwecanhavegreaterconfidenceinthepredictionsandconclusions wemakefromthedata.

Thecloserthervalueisto1orto1,themorelinearthedatais. Thecloserther2valueisto1,themorelinearthedatais.

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EVALUATINGTHELINEARITYOFTHEDATA

ResidualAnalysis

ResidualAnalysisisusedtofurthercheckwhetherthenatureofthedatais linearornot. Aresidual istheverticaldifferencebetweeneachdatapointandthe regressionline. eachResidual=ActualPredicted

y valuey value

(fromthe

• riginaldata)

(calculatedfrom theequation) Thisgivesasetofpoints,positiveandnegative,whichwecangraphandanalyse.

EVALUATINGTHELINEARITYOFTHEDATA

ResidualAnalysis

VCAAExam2,2007.

EVALUATINGTHELINEARITYOFTHEDATA

ResidualAnalysis

Oneof3“patterns”usuallyemerges: Theresidualsare randomlyaboveand belowthex axis. Conclusion: Datais probablyLINEAR. Theresidualsshowa curvedpattern(),witha seriesofnegative,then positiveandbackto negativealongthexaxis. Conclusion: Datais probablyNONLINEAR. Theresidualsshowa curvedpattern(),witha seriesofpositive,then negativeandbackto positivealongthexaxis. Conclusion: Datais probablyNONLINEAR.

TRANSFORMINGTHELINEARITYOFTHEDATA

Forthisshapeof ResidualAnalysis, y vlog10 x OR y v transformationsmay improvelinearity. Ifdataappearstobenonlinear,atransformationmayimprovethelinearity.

x 1

Forthisshapeof ResidualAnalysis, y vx2 transformationmay improvelinearity. Improvementsinthe rvalueorr2value wouldmeasure whetherthelinearity hadbeenimprovedor not. Onceatransformation hasbeendone,you includethe transformedvariablein theequation.

TRANSFORMINGTHELINEARITYOFTHEDATA

Ifyouhavetheoriginalscatterplot ofthedata,adecisionaboutwhich transformationissuitablecouldbemadefromthefollowingtable. r2 =0.75412=0.6 60%ofthevariationin heightcanbeexplainedby variationsinage.

TIMESERIES

ATimeSeriesisanexampleofbivariate datawheretheindependent variableisalwayssomeunitoftimeeg days,monthsoryears.

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TIMESERIES

Duringyour preparationfor Exams… DON’T spendalotof timedrawinggraphs. DO taketimeto understandthe processes.

TIMESERIES Trends TIMESERIES FittingaTrendLine

ATrendLineislikealineofbestfitforascatterplot. Thesamemethods,3MedianMethodORLeastSquaresRegression,areusedto:

• Findanddrawtheline

and/or

• Tocalculateitsequation.

VCAAExam1,2011. X X X

TIMESERIES FittingaTrendLine

ATrendLinehasanequationwhichisalsointheform y =m x +c ORy =c+m x. ATimeCodeisusuallyappliedtothetimevariabletomakecalculations withtheequationpossible.

VCAAExam1,2008.

TIMESERIES Smoothing

Timeperiod Rainfall(mm) 3pointmedian 5point median 1 15 2 17 16 3 16 17 17 4 21 19 19 5 19 21 19 6 23 19 7 9

MedianSmoothing Themediansarefoundfrom successivegroupsofmedians andlinedupagainstthe middleofthegroup.

VCAAExam1,2008. VCAAExam1,2006. VCAAExam1,2010.

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TIMESERIES Deseasonalising

Thisnewdatahastheseasonalinfluencesremovedandunderlyingtrendscanbe revealed– eg increasingordecreasingovertime.

TIMESERIES Deseasonalising

Thereasonforcollectingdataoveraperiodoftimeis toattempttoforecastthefutureusingcurrenttrends. Deseasonalised ActualValue ValueSeasonalIndex

=

Seasonalised Deseasonalised Seasonal ValueValue Index

(Forecastprediction)

=

x

VCAAExam1,2006. VCAAExam1,2009.

FURTHERMATHEMATICS Matrices

TERMINOLOGY TERMINOLOGY

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BASICCALCULATIONS MATRIXMULTIPLICATION

thisis 13 X3 4

13 X3 4becausethetwonumbersinthecentredonotmatch

MATRIXMULTIPLICATION MATRIXMULTIPLICATION SOLVINGSYSTEMSOFEQUATIONS SOLVINGSYSTEMSOFEQUATIONS

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SOLVINGSYSTEMSOFEQUATIONS TRANSITIONandSTATEMATRICES TRANSITIONandSTATEMATRICES

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VCAAEXAM2,2008

FURTHERMATHEMATICS LinearGraphs&Relations

Thegeneralequationofastraightlineis y=mx +c, wheremisthegradient(slope)andcistheyintercept. Inreallifesituations,thegradientrepresents thechange(increaseordecrease)iny, asxincreasesby1unit. Thesteepertheline,thegreatertherateofchange.

THELINEAREQUATION

VCAAExam2,2008.

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Theyinterceptisthevalueofywherethe graphcutstheyaxis. Whenappliedtoreallifesituations,the yinterceptoftenrepresentstheinitial (ororiginal)valueofsomething. Parallellineshavethesamegradient, butdifferentyintercepts.

THELINEAREQUATION

VCAAExam1,2008. VCAAExam2,2007.

THELINEAREQUATION

VCAAExam1,2007.

THELINEAREQUATION

VCAAExam1,2007. VCAAExam1,2010.

SKETCHINGLINEARGRAPHS

GradientInterceptmethod

Thismethodisusediftheequationisiny=mx +c form.

• 1. Thefirstpointplottedisthey–intercept,givenby

thevalueofc.Plotitonasetofaxes.

• 2. Thegradientisgivenbym.

Writethegradientasafractionandidentifythe valuesoftheriseandtherun.

• 3. Toobtainthesecondpoint,startfromthey

interceptandmoveup(ordown)andacross,as suggestedbythegradient.

• 4. Jointhetwopointstogetherwithastraightline

andlabelthegraph.

VCAAExam1,2010.

SKETCHINGLINEARGRAPHS

x andyinterceptmethod

Thismethodisusediftheequationisinax +by=cform,

• r

ifyouarerequiredtoshowboththex andyintercepts.

• 1. Ifapointisontheyaxis,itsxcoordinateis0.

Tofindtheyintercept,substitute0forxandsolve theresultantequation.

• 2. Ifapointisonthexaxis,itsycoordinateis0.

Tofindthexintercept,substitute0foryandsolve theresultantequation.

• 3. Plotthexinterceptandtheyinterceptonasetof

axes.

• 4. Jointhetwopointstogetherwithastraightline

andlabelthegraph.

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VCAAExam1,2006.

SKETCHINGLINEARGRAPHS

Sketchingalineoverarequiredinterval

Ifagraphneedstobesketchedbetweentwogiven xvalues,itsendpointsmustbeshown.Sinceonlytwo pointsareneededtosketchaline,wecanobtainthe coordinatesoftheendpointsandjointhemtogether. Toconstructagraphofastraightlinebetweenaandb, followthesesteps:

• 1. Rearrangetheequationtomakeythesubject.
• 2. Substituteeachofthetwogivenxvalues

(thatis,aandb)intotheequationand findcorrespondingvaluesofy.

• 3. Plottheendpointsonasetofaxes.
• 4. Jointhetwopointstogetherwithastraightlineand

labelthegraph.

APAIROFLINEARGRAPHS

VCAAExam1,2007.

Whenwould2equationshaveno simultaneoussolution? A:Iftheyareparallel.

APAIROFLINEAREQUATIONS

VCAAExam1,2008.

APAIROFLINEARGRAPHS

BreakEvenAnalysis isanapplicationofSimultaneousEquations.

• 1. Makingaprofitdependsonthecostsassociated

withthebusiness(labour,rawmaterialsandplant) anditsrevenue(themoneyitearnsthroughsales).

Profit=RevenueCosts

• 2. Thebreakevenpoint occurswhenCosts=Revenue.
• 3. Graphically,thebreakevenpointisthepointofintersection
• fthestraightlinesrepresentingcostsandrevenue.
• 4. Neitheraprofitnoralossismadeatthebreakevenpoint.

VCAAExam2,2008.

BreakEvenAnalysis

VCAAExam1,2010.

LINESEGMENTGRAPHS

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VCAAExam1,2006.

STEPGRAPHS

VCAAExam1,2011.

NONLINEARGRAPHS POWERGRAPHStoLINEARGRAPHS

Whendealingwiththefollowinggroupofquestions, 1.Checktheaxescarefully 2.Usevalueswhichhavebeengiventoyouaspoint’scoordinates.

VCAAExam1,2007. VCAAExam1,2008.

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VCAAExam1,2010.

greaterthanorequalto >greaterthan lessthanorequalto <lessthan 1.Linearinequations areintheformax +by<c,wherethe<signcanalsobe>,or. 2.Thesolutiontoalinearinequation isaregion(orhalfplane)eitheraboveorbelow thegraphofthecorrespondinglinearequation. 3.Todeterminethesolutionregionfollowthesesteps. (a)Plotthecorrespondinglinearequation(boundary). (b)Pickatestpointabovethisequationline. (c)Ifthepointsatisfiesthelinearinequation,thenthesolutionregionisabovethe line,otherwiseitisbelowtheline.Shadetheregionnotrequired. (d)Inequations withthesigns<or>donothavetheequationlineaspartofthe solutionregionandareindicatedwithadashedline. (e)Inequations withthesignsordohavetheequationlineaspartofthesolution regionandareindicatedwithasolidline. 4.Alwaysincludeakey indicatingtheregionrequired.

LINEARINEQUATIONS

VCAAExam2,2010.

LINEARPROGRAMMING

LinearProgrammingisanapplicationoflinearinequations whereanobjectivefunction,

• ftenassociatedwithprofitorcosts,ismaximisedorminimisedgivenasetof

constraintsthatlimitthepossiblesolutions.

• 1. Linearprogrammingproblemsaremadeupofthreecomponents:

(a) asetofdecisionvariables(xandy) (b)asetofconstraintsonthesevariables,expressedasasetofsimultaneouslinear inequations (c) anobjectivefunctionofthesevariables,whichiseitherminimisedormaximised.

LINEARPROGRAMMING

• 2. Tosolvethegenerallinearprogrammingproblem:

(a) definethedecisionvariables (b) definetheconstraintinequations (c) graphtheconstraintsasasetoflinearinequations (d) determinetheintersections(vertices)ofthesolutionregion (e) definetheobjectivefunctionanddecidewhetheritistobeminimisedor maximised (f) the‘best’(oroptimal)valueoftheobjectivefunctionisatoneoftheverticesof thesolutionregion,sosubstituteeachvertexinturnandcalculatetheobjective function. (g) selectthevaluesofxandywhichoptimise(minimiseormaximise)theobjective function.

LINEARPROGRAMMING

• 2. Tosolvethegenerallinearprogrammingproblem:

(b) definetheconstraintinequations (c) graphtheconstraintsasasetoflinearinequations.

VCAAExam2,2006.

LINEARPROGRAMMING

Youcanbeaskedtowritetheinequations,whichactasconstraints(ie the conditionswhichlimitthepossibleanswerstotheoverallquestion).

www.engageeducation.org.au 22

LINEARPROGRAMMING

Youcanbeaskedtoindicatethefeasibleregionbyshading.

LINEARPROGRAMMING

(0,8) (2.9,5.8) Solve(20x+25y=200andy=2x,{x,y}) Key: Feasible region (0,8) (2.9,5.8) Key: Feasible region

LINEARPROGRAMMING LINEARPROGRAMMING

Youcanbeaskedtowritetheobjectiveequationie thefocus(object)oftheoverall question.Thisisoftenaboutmaximisingprofitorminimisingcosts. Thesolutiontotheobjectiveequationcanonlybefoundinthefeasibleregion, mostcommonlyatthevertices. However,becarefulaboutwhetheronlywholenumberanswersmakesense

• rwhetherdecimalanswerscanbeincluded.

Iftheverticesonlyinvolvewholenumbers,thisconcerndoesn’toccur. (0,8) (2.9,5.8) Key: Feasible region

LINEARPROGRAMMING

(0,8) (2.9,5.8) Key: Feasible region

LINEARPROGRAMMING

Noticeyouarebeing askedwhatisthe maximumprofit,notat whatvaluesdoyoufind themaximumprofit. READTHEQUESTION

VCAAExam1,2008.

LINEARPROGRAMMING

Theoptimalsolutionisregularlyfoundattheverticesofthefeasibleregion. Testtheobjectivefunctionateachofthevertices.

FURTHERMATHEMATICS BusinessrelatedMaths

www.engageeducation.org.au 23

ACCURACYOF BUSINESSCALCULATIONS

PERCENTAGESin BUSINESSCALCULATIONS

TocalculateX%ofanamountweuse

amount X

• 100

TocalculateamountAasapercentageofAmountBweuse

% 100

• B

amount A amount Discount($)=Originalprice Sale price PercentageDiscount=

Discount

Adiscount isanamountofmoneybywhichthepriceofanitemisreduced. Ifexpressedasapercentageoftheoriginalprice,itiscalledapercentagediscount.

VCAAExam1,2006

100 ($) Price Original ($) Discount

• CapitalGain

Acapitalgain (orloss)isthedifferencebetweenthecostpriceandthesellingprice

• fanitem.

Capitalgain($)=SalepriceCostprice

Commission

Acommission isanamountofmoneywhichasalesmanearnsasapercentageof thesaleshemakes.

Commission($)=

($) 100 Sales Total Rate Commission

• VCAAExam1,2007

PERCENTAGEINCREASESandDECREASES

Forexample: A3%increase multiplyby1.03 A3%decrease multiplyby0.97 A15%increase multiplyby1.15 A15%decrease multiplyby0.85 A60%increase multiplyby1.60 A60%decrease multiplyby0.40

Inflation

Inflation isameasureoftheaverageincreaseinthepriceofgoodsandservices fromoneyeartothenext. Itisanexampleofpercentageincreasecalculatedyearafteryear.

VCAAExam1,2011

$160 1.25 0.90 0.75 1.20=$162

MonTuesWedThursFri&Sat

PERCENTAGEINCREASESandDECREASES

VCAAExam1,2011

Goods&ServicesTax(GST)

FinalPriceofanitemorservice =thepreGSTamount +theGST (whatthecustomer(10%ofthe paysfortheitem) baseprice) Theretailercollectsthesaleprice,keepsthepurchasepriceandpassestheGSTontothe GovernmentTaxOffice.

www.engageeducation.org.au 24

VCAAExam1,2006

Goods&ServicesTax(GST)

TocalculatetheGSTfromaFinalPriceofaitemorService,we dividetheamountby11.

Not10!

StampDuty StampDutyisataxleviedondocumentsinvolvedinthetransferofrealestate, businesses,insurancepolicies,mortgagesandmotorvehicles.

VCAAExam1,2006

SIMPLEINTEREST

Simpleinterestcanbecalculatedinthecaseof: *aninvestment – theinvestorisrepaidtheamountheinvestedplusinterest *aloan– theborrowermustrepaytheamountborrowedplusinterest Theformulaforcalculatingsimpleinterestis

T r P I

• 100

I=Simpleinterestearnedonaninvestmentorchargedforaloan($) P=Principal(amountofmoneyinvestedorloaned)($) r=Rateofinterestpertimeperiod(%perperiod) T=Time,thenumberofperiods(years,months,days)overwhichtheagreementoperates.

NBTheinterestrate,r,andtimeperiod,T,mustbestatedandcalculatedinthe sametimeterms. Totalamountofloanorinvestment =Principal+Interest(chargedorearned)

A=P+I

VCAAExam1,2006 VCAAExam1,2007

TheSIMPLEINTEREST formula isusedtocalculateintereston

BANKACCOUNTS.

VCAAExam1,2006

Usetheminimumbalance amountforthemonthasthe Principal. TheSIMPLEINTEREST formula isusedtocalculateintereston

PERPETUITIES.

VCAAExam1,2006

Aperpetuityisaninvestmentwhichneverdecreasesbecauseasinterestisearnedit isgiventotheinvestor. Examplesarepensionsandscholarships. TheSIMPLEINTEREST formula isusedtocalculateintereston

PERPETUITIES.

VCAAExam1,2006

TheSIMPLEINTEREST formula isusedtocalculateintereston

HIREPURCHASEpurchases.

Peoplebuyonhirepurchasewhentheycannotaffordtobuythegoodsforcash. Adepositisusuallypaidandthebalanceispaidoverafixedperiodoftime. Theretailerarrangesacontractwithafinancialinstitutionandthepurchaserpays regularinstallmentsincludinginterestataflatratetothefinancialinstitution.

Aflatrate isthesameassimpleinterestrate.

Themainstagesofinterestandtotalpricecalculationsare: 1.Loanamount=priceofgoods– depositpaid

• 2. Flatrateinterestontheloaniscalculatedusingthesimpleinterestformula.

3.Instalment amount= 4.Totalcostofgoods=priceofgoods+interest

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Themainstagesofinterestandtotalpricecalculationsare: 1.Loanamount=priceofgoods– depositpaid

• 2. Flatrateinterestontheloaniscalculatedusingthesimpleinterestformula.

3.Installmentamount= 4.Totalcostofgoods=priceofgoods+interest

VCAAExam1,2006 VCAAExam1,2009 VCAAExam1,2008

COMPOUNDINTEREST

Compoundinterestdiffersfromsimpleinterestinthatitiscalculatedonthe currentamountofaninvestmentorloan,ratherthanontheoriginalamount. Theformulaforcalculatingcompoundinterestis

T r P A ) 100 1 (

• A=Totalamountofaninvestmentor aloan($)

P=Principal(amountofmoneyinvestedorloaned)($) r=Rateofinterestpertimeperiod(%perperiod) T=Time,thenumberofperiods(years,months,days)overwhichtheagreementoperates.

NBTheinterestrate,r,andtimeperiod,T,mustbestatedandcalculatedinthe sametimeterms. AmountofInterest($)=Totalamountofloanorinvestment– Principal

I = A– P

COMPOUNDINTEREST

VCAAExam1,2007 VCAAExam1,2011 VCAAExam1,2008

COMPOUNDINTEREST EFFECTIVEINTERESTRATE

Inasimpleinterestloan,theamountborrowedreducesoverthetermofthe loan,butthecustomerisstillpayinginterestonthetotalinitialloanamount. Theeffectiveinterestrateistheequivalentreducingbalanceinterestratetaken

• verthecontractperiod.

Theformulaforcalculatingtheeffectiveinterestrateis wherenisthenumberofpayments. ThismakestheEffectiveinterestratealittlelessthan2× Simpleinterestrate.

EFFECTIVEINTERESTRATE

VCAAExam1,2008

REDUCINGBALANCELOANS

AReducingBalanceLoan,alsoknownasanAnnuity,involvestheborrowingofaPrincipal, whichisrepaidwithregularrepaymentswhichcontainanamountoftheprincipal+an amountofinterest,whichiscalculatedonthecurrentbalanceoftheloan. YouwillusetheFINANCE($)appontheCAScalculatortodoReducingBalanceLoan calculations.

TheFINANCEappscreenshowsthefollowingheadings: N= Nisthenumberoftimeperiodswheninterestisadded I%=Iistheinterestrateperannum PV=PVisthePrincipalValue PMT=PMTisthepaymentcontributionpertimeperiod FV= FVistheFinalValueofinvestmentattheendofthetime P/Y=P/Yisthenumberofpaymentcontributionsperyear C/Y=C/Yisthenumberofinterestcompoundingperyear PMT:ENDBEGIN ENDmeansthecalculationsaredoneattheendof eachtimeperiod

PV +

(moneycomingTOyou)

PMT–

(moneygoingAWAYfromyou)

FV –

(moneygoingAWAYfromyou)

www.engageeducation.org.au 26

ANNUITYINVESTMENTS

AnAnnuityinvestmentinvolvestheregular(eg monthlyorquarterly)paymentofmoneyinto anaccountoveranextendedperiodoftime.Thisinvestmentearnsinterestcontinuously

• verthelifeoftheinvestment,basedonthecurrentbalance.Theinterestisaddedtothe

Principalasitisearned.SuperannuationisanexampleofanAnnuityinvestment. YouwillusetheFINANCE($)appontheCAScalculatortodoReducingBalanceLoan calculations.

TheFINANCEappscreenshowsthefollowingheadings:

N= Nisthenumberoftimeperiodswheninterestisadded I%=Iistheinterestrateperannum PV=PVisthePrincipalValue PMT=PMTisthepaymentcontributionpertimeperiod FV= FVistheFinalValueofinvestmentattheendofthetime P/Y=P/Yisthenumberofpaymentcontributionsperyear C/Y=C/Yisthenumberofinterestcompoundingperyear PMT:ENDBEGIN ENDmeansthecalculationsaredoneattheendof eachtimeperiod

PV –

(moneygoingAWAYfromyou)

PMT–

(moneygoingAWAYfromyou)

FV +

(moneycomingTOyou)

VCAAExam1,2010

DEPRECIATION

Theestimatedlossinvalueofassetsiscalleddepreciation. Theoriginalcostofanitemissometimescalleditsprimecost. Theestimatedvalueofanitematanypointintimeiscalleditsbookvalue. Whenthebookvaluebecomeszero,theitemissaidtobewrittenoff. Attheendofanitem’susefuloreffectivelifeitsbookvalueisthencalleditsscrapvalue.This maybezerobutoftenisnot.

3MethodsofcalculatingDepreciation 1.FlatRate depreciation(alsoreferredtoasStraightLine depreciation andPrimeCost depreciation) 2.Reducingbalancedepreciation (alsocalleddiminishingvaluedepreciation.) 3.UnitCostDepreciation

Totaldepreciation=primecost currentvalue Rateofdepreciation=totaldepreciation numberofyears %rateofdepreciation=annualamountofdepreciation primecost

100%

FLATRATEDEPRECIATION

BVT =P– d T

BVT =bookvalue($)aftertime,T P=costprice($)orprimecost($) T=timesincepurchase(years) d=fixedamountperyear ORpercentageofPperyear

VCAAExam1,2010 FlatRateDepreciationisamethodwhichinvolvessubtractingafixedamounteachtime period(normallyeachyear). ThefixedamountcanbeanumberofdollarsORapercentageofthecostprice.

REDUCINGBALANCEDEPRECIATION

BVT =bookvalue($)aftertime,T P=costprice($)orprimecost($) T=timesincepurchase(years) r=rateofdepreciation

Ifanitemdepreciatesbythereducingbalance methodthenitsvaluedecreasesbya fixedpercentagerateeachtimeinterval,generallyeachyear.Thisrateisapercentageof thepreviousbookvalueoftheitem. VCAAExam1,2006

www.engageeducation.org.au 27

UNITCOSTDEPRECIATION

UnitCostDepreciationisbasedondepreciatingthevalueofanitembasedonits productioneg depreciatingaphotocopierbasedonthenumberofcopiesitmakesor depreciatingamachineonthenumberoftoysitproducesordepreciatingavanonthe numberofkilometres ittravels. VCAAExam1,2009

FURTHERMATHEMATICS Geometry&Trigonometry

GEOMETRY Angles

VCAAExam1,2010

GEOMETRY Triangles

VCAAExam1,2011

GEOMETRY Quadrilaterals GEOMETRY Polygons

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RIGHTANGLETRIANGLES

IntheExaminer’scommentsforthis question,itwasstatedthatmany studentsassumedACBwasright angled.Also,manyassumedXY=AY. Neitherassumptionwastrue. VCAAExam2,2006

PYTHAGORAS’THEOREM

a c (hypotenuse) b

c2 = a2 + b2 a2 = c2 – b2

VCAAExam2,2006

PYTHAGORAS’THEOREM

VCAAExam2,2007

PYTHAGORAS’THEOREM SOH CAH TOA

VCAAExam1,2010

MEASUREMENTFORMULAE

VCAAExam1,2010

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MEASUREMENTFORMULAE VOLUME MEASUREMENTFORMULAE VOLUME

VCAAExam1,2010

MEASUREMENTFORMULAE

VCAAExam2,2007

Inpart(a)wewereaskedtoshowAW=34cm.

SIMILARTRIANGLES SCALEFACTOR Length SCALEFACTOR Area&Volume

VCAAExam1,2011 VCAAExam1,2010

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NONRIGHTANGLEDTRIANGLES NONRIGHTANGLEDTRIANGLES TheSINERULE

ProblemsinwhichtheSineRuleshouldbeusedwillinvolve

2sidesand2angles.

OR

NONRIGHTANGLEDTRIANGLES TheSINERULE

VCAAExam1,2011

NONRIGHTANGLEDTRIANGLES TheCOSINERULE

ProblemsinwhichtheCosineRuleshouldbeusedwillinvolve

3sidesand1angle.

AND Tofindasidelength Tofindanangle

NONRIGHTANGLEDTRIANGLES TheCOSINERULE

VCAAExam2,2008

AREAFORMULAEforTRIANGLES

VCAAExam1,2010

BEARINGS

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VCAAExam1,2006

ADVICE:

VCAAExam2,2008

ANGLESofELEVATIONandDEPRESSION

Theangleofelevation istheangleabovethehorizon. Theangleofdepression istheanglebelowthehorizon.

VCAAExam2,2008

ANGLESofELEVATIONandDEPRESSION TRIANGULATION CONTOURMAPS

c c

B

20m– 15m=15m

Run Rise Slope

VCAAExam2,2011

www.engageeducation.org.au 32

FURTHERMATHEMATICS Networks&DecisionMathematics

UNDIRECTEDGRAPHSandNETWORKS

Thegraphmayberepresented inanadjacencymatrix formas:

• 2

1 2 1 1 1 1 1 1 1

ABCD D C B A

UNDIRECTEDGRAPHSandNETWORKS

Aloop isconsideredasasingleedgebutitcontributes2tothedegreeoftheendpoint.

UNDIRECTEDGRAPHSandNETWORKS

EULER’SFORMULA www.engageeducation.org.au 33

UNDIRECTEDGRAPHSandNETWORKS Paths&Circuits UNDIRECTEDGRAPHSandNETWORKS Paths&Circuits

SHORTESTPATH

UNDIRECTEDGRAPHSandNETWORKS Trees DIRECTEDGRAPHS Reachability DIRECTEDGRAPHS Routes

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DIRECTEDGRAPHS Connectivity

C2 C ConnectivityMatrix C+C2=

• 1

4 2

DIRECTEDGRAPHS Dominance DIRECTEDGRAPHS NetworkFlow

AIM:findthemaximumflowpossiblethroughanetwork

Ineachcut,onlycount theflowsthatare headingtowardsthe sinksideofthecut. eg Inthediagram,for Cut1weshouldignore flowBD;inCut2we shouldignoreflowEB.

NetworkFlow

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CRITICALPATHANALYSIS

AIM:findtheshortesttimetocompleteaseriesoftasks.

CRITICALPATHANALYSIS BIPARTITEGRAPHS

AIM:tooptimisetheallocationoftasks

ABipartiteGraphisadirectedgraphwhichhastwodistinctgroups. Connectionsexistbetweenthetwogroups,notwithinagroup.

BIPARTITEGRAPHS– TheHungarianAlgorithm www.engageeducation.org.au 36

MATHSQUEST12 Novak,Bakogianis,Boucher,Nolan,Phillips

FURTHERMATHEMATICS NumberPatterns

ARITHMETICSEQUENCES

1.Anarithmeticsequenceisasequenceofnumbersforwhichthedifference betweensuccessivetermsisthesame. 2.Thefirsttermofanarithmeticsequenceisreferredtoas‘a’. 3.Thecommondifferencebetweensuccessivetermsisreferredtoas‘d’. 4.tn isthenth term;forexample,t6 referstothe6thterminthesequence. Totestwhetherasequenceofnumbersisanarithmeticcalculatet2 t1 and t3 t2 andsoonthroughthesequence.Iftheanswersarethesamethenitisan arithmeticsequence.

eg 8,15,22,29,…

Tofindthenth termofanarithmeticsequencewecanusetheformula

tn =a+(n 1)d

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VCAAExam1,2006 VCAAExam1,2007 VCAAExam1,2008

FINDINGTHERULEOFANARITHMETICSEQUENCES

Substituteyourvaluesintotn =a+(n 1)d andthensimplifytheequation.

VCAAExam1,2011

FINDINGa andd ifyouknowtwoterms

Finda andd inanarithmeticsequenceifthe4th termis132andthe11th term is188.

VCAAExam1,2007

ADDINGTHETERMSOFARITHMETICSEQUENCES

Thesumofanarithmeticsequenceiscalledanarithmeticseries. So 2,5,8,11,14isanexampleofanarithmeticsequence. 2+5+8+11+14isanexampleofanarithmeticseries. Tofindthesumofagivennumberoftermsofanarithmeticsequencewecanuse

• neoftwoformulae.

Ifweknowthefirstandlasttermandhowmanytermsthereare,wecanuse

where Sn =thesumofn terms n =thegivennumberofterms a =thefirstterm l =thelasttermie thenthterm.

Ifweknowthevaluesofa andd andhowmanytermsthereare,wecanuse

VCAAExam1,2011 VCAAExam1,2008

GEOMETRICSEQUENCES

1.Ageometricsequenceisasequenceofnumbersforwhichthefirsttermis multiplied byanumber,knownasthecommonratio,tocreatethesecondterm whichismultipliedbythecommonratiotocreatethethirdterm,andsoon. 2.Thefirsttermofageometricsequenceisreferredtoas‘a’. 3.Thecommonratiobetweensuccessivetermsisreferredtoas‘r’. 4.tn isthenth term;forexample,t6 referstothe6thterminthesequence. Totestwhetherasequenceofnumbersisangeometriccalculatet2 t1 and t3 t2 andsoonthroughthesequence.Iftheanswersarethesamethenitisageometric sequence.

eg 4,12,36,108,…

Tofindthenth termofangeometricsequencewecanusetheformula

tn =ar (n – 1)

VCAAExam1,2008 VCAAExam1,2006 VCAAExam1,2010

ADDINGTHETERMSOFGEOMETRICSEQUENCES

Thesumofangeometricsequenceiscalledangeometricseries. Tofindthesumofn termsofageometricsequence,wecanusetheformula

where Sn =thesumofn terms n =thegivennumberofterms a =thefirstterm l =thelasttermie thenthterm. VCAAExam1,2010

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INFINITEGEOMETRICSEQUENCES

Ifthecommonratio,r,isavaluebetween1and1,thenitispossibletofindthe sumofitsinfinitenumberofterms. Thisiscalledthesumtoinfinity. Theformulaforitis

VCAAExam1,2010

GRAPHSOFARITHMETICSEQUENCES

tn n tn n

ifd ispositive ifd isnegative GRAPHSOFGEOMETRICSEQUENCES

tn n tn n

a > 1, r > 1

tn n tn n

r < -1 a > 1, r > 1

• 1 < r < 1

a < 1, r > 1

tn n

VCAAExam1,2010

DIFFERENCEEQUATIONS

Anysequenceofnumberscanberepresentedlikethis

...,tn – 2 ,tn – 1 ,tn ,tn +1 ,tn +2 , …

Afirstorderdifferenceequationconsistsof

• anequationlinking2consecutiveterms(eg tn andtn+1 ORtn1 andtn )
• astartingterm(thestartingtermmightbedenotedbyt0 ort1 ).

eg tn =3tn– 1 +5t0 =2

• rtn+1 =tn +20t1 =5

eg tn =3tn– 1 +5t0 =2 wouldgive2,then3 2+5=11,then3 11+2= 38,andsoon… ie

2,11,38,109,332,…

eg

tn+1 =tn +20t1 =5

wouldgive5,then5+20=25,then25+20= 45,andsoon… ie

5,25,45,65,85,…

VCAAExam1,2006

n fn f(n+1)=5+f(n) 1 f1=–1 f(2)=5+–1=4 2 f2=4 f(n+1)– f(n)=5 f(n+1)=5+f(n)

ARITHMETICSEQUENCESas FIRSTORDERDIFFERENCEEQUATIONS

Anarithmeticsequencewithacommondifferenceofd

tn =a+(n 1)d

givesthesamevaluesasafirstorderdifferenceequationoftheform:

tn +1 =tn +b

wherebisthecommondifferenceandfor b>0itisanincreasingsequence b<0itisadecreasingsequence. Sod andb arerepresentingthesamething, namelythecommondifferenceinanarithmeticsequence.

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GEOMETRICSEQUENCESas FIRSTORDERDIFFERENCEEQUATIONS

Ageometricsequencewithacommonratioofr

tn =ar (n – 1)

givesthesamevaluesasafirstorderdifferenceequationoftheform:

tn +1 =a tn

whereaisthecommonratioandfor a>1itisanincreasingsequence 0<a<1itisadecreasingsequence a <0itisasequencealternatingbetweenpositiveandnegativevalues. Sor anda arerepresentingthesamething,namelythecommonratioinangeometric sequence.

PERCENTAGEINCREASESandDECREASES

Forexample: A3%increase multiplyby1.03 A3%decrease multiplyby0.97 A15%increase multiplyby1.15 A15%decrease multiplyby0.85 A60%increase multiplyby1.60 A60%decrease multiplyby0.40

VCAAExam1,2007

OTHERSEQUENCESas FIRSTORDERDIFFERENCEEQUATIONS

VCAAExam1,2006

FINDINGTHE FIRSTORDERDIFFERENCEEQUATION FROMAGRAPHOFVALUESOR ASEQUENCEOFNUMBERS

Tofindthefirstorderdifferenceequationfollowthesesteps: 1.Writedownthevaluesofthetermsinorderasasequence. 2a.Ifthegraphisastraightlineorhasacommondifference(b)byt2 – t1 ,t3 – t2 etc. Thenwriteyourequationtn+1 =tn +bt1 =…ORtn =tn1 +bt0 =… 2b.Ifthegraphisacurvedline,findthecommonratio(a)byt2 ÷ t1 ,t3 ÷ t2 etc. Thenwriteyourequationtn+1 =a tn t1 =…ORtn =a tn1 t0 =… ALWAYSCHECKTHEQUESTIONTOSEEIFITSTIPULATESWHICHFORMIS REQUIREDBETWEENtn+1 andtn ORtn andtn1 . ALWAYSREMEMBERTOINCLUDEAFIRSTTERMVALUE

VCAAExam1,2006 VCAAExam1,2007

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SECONDORDERDIFFERENCEEQUATIONS

AFibonacciSequenceisoneinwhicheachnewtermiscreatedbyaddingthe previoustwoterms. AsecondorderdifferenceequationforaFibonaccisequenceissetoutusing thefollowingnotation:

fn+2 =fn +fn+1 givenf1 andf2

OR

fn =fn– 1 +fn– 2 givenf0 andf1

Forexample,theclassicFibonaccisequenceis1,1,2,3,5,8,13,21,34,55,… Thesecondorderdifferenceequationforthiswouldbe fn+2 =fn +fn+1 f1 =1,f2 =1 OR fn =fn– 1 +fn– 2 f0 =1,f1 =1

VCAAExam1,2006 VCAAExam2,2008

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