Stage 5 Answers 5.1-5.2 - Numeracy Skills Frameworknumeracyskills.com.au/images/pdfs/Stage_5_Answers_5.1-5.2.pdf · NSW Department of Education Mathematics Stage 5 Diagnostic Task

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers1

MathematicsStage5DiagnosticTasksAnswerswithCommonErrorsAnswers5.1 CommonErrors5.1FinancialMathematics QUESTION1 a) $540b) $13.05(2decimalplaces)c) $847.60d)

i) $2942.31(2decimalplaces)ii) $6375

e) i) $908.65(2decimalplaces)

(using52weeksinayear)ii) $6100.96(2decimalplaces)

d) Incorrectidentifyingthefactortodivideby(i.e.,12or26or52)

(ii)Studentsmaymultiplyanswer(i)by2thinkingthatthereis2fortnightsinamonthratherthangoingbacktotheoriginalsalary.

e) Notdistinguishingthedifferencebetweenholidayloadingandholidaypay.

QUESTION2 a) $1250b) $61

Findingthevaluetocalculatecommissionon(i.e.Determiningcommissioncalculatedon$30000).

QUESTION3 a) $25466.33b)

i) $6672.50ii) Refund.$1077.50iii) Taxableincome=$39867.50

Medicarelevy=$698.10(2decimalplaces)

a) Addingthedeductionsinsteadofsubtractingb) ii)Notspecifyingtherefundamount.iii)mayworkoutthelevyusingthetaxamountinquestionspart(ii)

QUESTION4 a) $1560b) $3805.50c) $65.63(2decimalplaces)d)

i) $3900ii) $998.40iii) $4898.40iv) $6398.40

b)Workingouttheinterestandforgettingtoaddontheprincipalc)Incorrectlyconvertingrateortimeperiod.d)i) Notexcludingthedeposit.ii) UsingthewrongPrincipaliii) Notaddingtheprincipalamountiv) Notaddingtheinitialdeposittothetotal

pricepaid

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers2

5.1FinancialMathematics CommonErrorsQUESTION5 a)

i) $3675.13(2decimalplaces)ii) $675.13

b) $6486.25(2decimalplaces)

c)

i) $5728.86ii) $5373.45InvestmentA I=5000×0.03×6 =$900 A=$5900InvestmentBA=5000×1.1941 =$5970.50Therefore,InvestmentBisthebetterinvestmentby$70.50

a) andb)incorrectlydistinguishingbetweeninterestearnedandfinalvalue.

c)Notrecognisingthetableisonlyfor$1andyouneedtomultiplybytheprincipalamount.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers3

5.1Indices CommonErrorsQUESTION1 a)

i) a4ii) 25

b) i) 25ii) m9iii) 6x6iv) 5p4v) 32x3y5

a) Addingratherthanmultiplyingespeciallywith

(i)andgetting4aasananswerb) Multiplyingtheindicesinsteadofadding

iii) Multiplyingthebasenumberstoget4"thisisonlyacommonerrormadewhenthebasesandnumerical.

QUESTION2 a) y3b) 2xc) -6kd) 4xy2

Dividetheindicesratherthansubtract

QUESTION3 a) 56b) 23q9=8q9

c) #$%

&= ()$*

%*

a) Squaringthe5toget25-thisisonlyacommonerrormadewhenthebasesandnumerical.

b) andc)notraisingeverythinginsidethebrackettothepoweri.e.getting2𝑞/and#$

*

%*or#$

*

%

QUESTION4 a) 1b) 2c) 1

b)Thinkingthattheentireexpressionistothepowerofzeroandgivingtheanswer1

QUESTION5 a)

i) 0&*

ii) 0

"1

b) i) 2-4ii) -4-3

c) i)

0)

ii) − 0"1= − 0

0&"

iii) − 00-+ 0

(= /

0-

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers4

5.1LinearRelationships CommonErrorsQUESTION1 a) positiveb) (1.5,1)c) (

"

d) 41 ≈ 6.4(1decimalplace)

QUESTION2 a)

x 2 2 2 2 2 2y -2 -1 0 1 2 3

b)

x -2 -1 0 1 2 3y -2 -2 -2 -2 -2 -2

Gettingtheorderofthecoordinatesmixedup,doingthey-ordinateacrossandthex-ordinateup/downNotextendingthelinebeyondtheplottedpoints.Notusingarrowsonbothendsoftheline.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers5

5.1LinearRelationships CommonErrorsQUESTION3 a) x-intercept:

𝑦 = 0𝑥 + 3 = 0𝑥 = −3

y-intercept:𝑥 = 0𝑦 = 0 + 3𝑦 = 3

b)

a)Incorrectlysolvingtheequationswhenx=0andy=0

b)Notextendingthelinesufficientlytoshowthex-andy-intercepts.

Notusingarrowsonbothendsoftheline.

QUESTION4 3 = 3 2 − 23 = 6 − 23 ≠ 4No,(2,3)doesnotlieontheliney=3x–2

Substitutingthexandyvaluesintothewrongvariable.

QUESTION5 y=3xandy=3x+1

Includingy=-3xinthesolutionasitalsohasa3

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers6

5.1Non-LinearRelationships CommonErrorsQUESTION1 i)

x -2 -1 0 1 2 3

y 4 1 0 1 4 9

ii)

x -2 -1 0 1 2 3

y14

12

1 2 4 8

i) Caremustbetakenbystudentswhenputting(-2)2and(-1)2intothecalculatorasiftheydon’tusethebracketsitwillcomeoutwithanegativenumber

i)andii)pointsmustbejoinedusingasmoothcurvenotastraightline

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers7

5.1Non-LinearRelationships CommonErrorsQUESTION2 i) Aii) Ciii) B

QUESTION3 i) y=x2+1

x -2 -1 0 1 2 3

y 5 2 1 2 5 10

y=x2−1

x -2 -1 0 1 2 3

y 3 0 -1 0 3 8

y=x2+1crossestheyaxisaty=1y=x2–1crossestheyaxisaty=-1

ii) Thevalueoftheconstantisequaltowhere

thegraphcrossestheyaxis.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers8

5.1AreaandSurfaceArea CommonErrorsQUESTION1

a) 𝐴 = 𝑠& + 0&𝜋𝑟&

= 8×8 +12×𝜋×4

&= 89.13𝑐𝑚&(correctto2decimalplaces)

b) 𝐴 = G&𝑎 + 𝑏

=6.32 10 + 18.4

= 89.46𝑐𝑚&c) 𝐴 = 𝑏ℎ= 12×8= 96𝑐𝑚&

d) 𝐴 = 0/K/-K

𝜋𝑟&

=130360×𝜋×10

&= 133.45𝑐𝑚&(correctto2decimalplaces)

e) 𝐴 = 𝜋𝑅& − 𝜋𝑟&= 𝜋×9& − 𝜋×4&= 204.20𝑐𝑚&(correctto2decimalplaces)

f) 𝐴 = 0&𝑏ℎ − 𝑙𝑤

=12×25×14 − 4×3

= 163𝑐𝑚&

Acrossallquestionsinvolvingareaandsurfacearea

Ø MixingupareaandperimeterØ Mixingupsurfaceareaandvolume

a) Confusingtheuseofthecircumference

formulaandtheareaofacircleformulaorevenacombinationofboth(2𝜋𝑟&isacommonerrorfortheareaformula)orforgettingtohalvethecircle.

e) using18cmastheradiusofthelargercircle

insteadof9cm.

QUESTION2 a) 𝑇𝑜𝑝 = 22×6.5= 143𝑐𝑚&𝐹𝑟𝑜𝑛𝑡 = 22×10= 220𝑐𝑚&𝑆𝑖𝑑𝑒 = 6.5×10= 65𝑐𝑚&𝑆𝑢𝑟𝑓𝑎𝑐𝑒𝐴𝑟𝑒𝑎 = 2× 143 + 220 + 65 = 856𝑐𝑚&

b) Findingtheperpendicularheightofthetriangle:ℎ& = 13& − 5&ℎ = 12𝑐𝑚

𝐹𝑟𝑜𝑛𝑡 =12×10×12

= 60𝑐𝑚&𝑆𝑙𝑜𝑝𝑖𝑛𝑔𝑠𝑖𝑑𝑒 = 13×18= 234𝑐𝑚&𝐵𝑎𝑠𝑒 = 10×18= 180𝑐𝑚&𝑆𝑢𝑟𝑓𝑎𝑐𝑒𝐴𝑟𝑒𝑎 = 2×60 + 2×234 + 180= 768𝑐𝑚&

b)using13cmastheheightofthetriangle

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers9

5.1AreaandSurfaceArea CommonErrorsQUESTION3 a) 5b) 𝐹𝑟𝑜𝑛𝑡 = 36×42= 1512𝑐𝑚&𝑆𝑖𝑑𝑒 = 25×42= 1050𝑐𝑚&𝐵𝑎𝑠𝑒 = 36×25= 900𝑐𝑚&𝑆𝑢𝑟𝑓𝑎𝑐𝑒𝐴𝑟𝑒𝑎 = 2×1512 + 2×1050 + 900= 6024𝑐𝑚&

Notrecognisingthattheboxisopenandhasnottopandincludingitinthecalculation.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers10

5.1NumbersofAnyMagnitude CommonErrorsQUESTION1 a) 2430b) 0.238c) 1.29

a) Getting243,becausetheydon’tputthezerointoholdplacevalue.

b) Justcuttingthedecimaloffat3decimalplacesandnotroundingupi.e.getting0.237.

c) Roundingto3decimalplacesinsteadof3sigfigsi.e.1.288/1.287

QUESTION2 a)

i) 0.78cmii) 2.34miii) 1.86×10-6miv) 60μm

b) i) 7850Lii) 78.26mL

c) i) 7MB=7×1024KB=7168KBii) 5200GB=5200÷1024=5.078TB(correctto3decimalplaces)iii) 2.1TB=2.1×1024×1024×1024=2254858000KB(tothenearestthousand)

d) i) 7.8×60×1000=468000msii) 170000μs=170000÷1000000s

=0.17÷60min=0.0028min(correctto2significantfigures)

iii) 1fortnight=1×14days=14×24hours=336×60minutes=20160×60seconds=1209600s

iv) 897÷10=89.7decades

a),b)andd)mixingupthemultiplicationanddivisionbypowersof10i.e.multiplyingwhenitshouldbedivisionandviceversa.Orusingthewrongpowerof10c)Note:210=1024usingmultiplesof10forconvertingratherthanthepowersof2requiredfordigitalconversions.

Prefix Decimal/Analogvalue

Binary/Digitalvalue

k(kilo) 103

(1000)210

(1024)

M(mega) 106

(1000000)220

(1048576)G(Giga) 10

9

(1000000000)

230

(1073741824)

T(Tera) 1012

(1000000000000)

240

(1099511627776)

QUESTION3 a)

i) 2800=2.8×103ii) 1240000=1.24×106iii) 0.03456=3.456×10-2iv) 0.0000541=5.41×10-5

b) i) 2.6×102=260ii) 4.56×10-4=0.000456

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers11

5.1NumbersofAnyMagnitude CommonErrorsQUESTION4 a) 6.3x108,1.2x109,3.5x109b) 4.8x10-3,4.1x10-2,3.8x10-1

Comparingonlythedecimalpartofthenumberandnottakingthepowerof10intoconsideration

QUESTION5

a) 𝑡 = %^= 0."&×0K_

/×0K`= 507𝑠(correcttothe

nearestwhole)b) 28000×365=10220000words

=1.022×107words

QUESTION6 a) ±0.5kgb) 15.5kgto16.5kg

QUESTION7 a) 25+15=40b) 140+20=160(Answersmayvaryslightly)

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers12

5.1Trigonometry–RightAngledTriangles CommonErrorsQUESTION1 a)

i) ABii) BCiii) AC

b)

i) ("

ii) /"

iii) (/

c) i) 0.29ii) 0.21iii) 1.17

d) i) 82°ii) 57°iii) 37°

AcrossallquestionsinvolvingtrigonometryØ UsingthewrongratiosØ Gettingconfusedwhentheunknownisinthe

denominatorforexampleif:tan 30° = 0&e

rearrangingincorrectlytoget𝑦 = 12𝑡𝑎𝑛30°

Ø Notusingtheinversefunctionwhencalculatinganunknownangle(i.e.onthecalculatorstudentsneedtopressSHIFTbeforethetrigratio)

Ø Mixinguptheoppositeandadjacentsidesiscommon

c)incorrectrounding

QUESTION2 a) i) tan 52 = e

f

𝑦 = 8× tan 52𝑦 = 10.2cm(roundedto1decimalplace)

ii) cos 46 = #."j

𝑥 = 7.5 ÷ cos 46𝑦 = 10.8cm(roundedto1decimalplace)

b) i) tan 𝜃 = (

/

𝜃 = 53°ii) cos 𝜃 = -

0K

𝜃 = 53°c) tan 54 = G

"&K

ℎ = 520× tan 54ℎ = 716mtheheightofthecliffis716m.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers13

5.1PropertiesofGeometricFigures CommonErrorsQUESTION1 Twoshapesaresimilariftheirshapeisthesamebuttheirsizeisdifferent.

QUESTION2 AandCMatchinganglesareequal

QUESTION3 a) AC-PR,CD-RS,DB-SQ,BA-QPb) Ð A-Ð P,Ð C-Ð R,ÐD-Ð S,Ð B-ÐQ

QUESTION4 a)

i) scalefactor = imageoriginal

= 0K(= 2.5

ii) scalefactor = imageoriginal

= 00((= 0.25

b)

i) scalefactor = imageoriginal

= /-= 0.5

ii) 5× 0&= 2.5cm

Notrecognisingthattheobjectisonthelefthandsideandtheimageisontherighthandsidei.e.gettingthescalefactorbackwards.Oralwaysthinkingthatascalefactorhastobelargerthan1andhavingSF=4for(ii)

QUESTION5 i) scale= /(

f= 4.25

ii) 6.7×scale=6.7×4.25=28.5cm(correctto1decimalplace)

QUESTION6a)

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers14

5.1PropertiesofGeometricFigures CommonErrorsb)

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers15

5.1SingleVariableDataAnalysis CommonErrorsQUESTION1 Positivelyskewed,negativelyskewed,bi-modal,symmetric

QUESTION2 i)

GIRLS LEAF BOYS

9 0 56 1 2

542 2 56763 3 345

0 4 ii) boys

iii) Girls:mean=28.25,median=29,range=31

Boys:mean=22,median=25.5,range=30

Thegirlsachievedthebetterresultswithahighermedianandmeanalthoughtheirrangeislargeritisonlybyonemarkandisnegligiblewhenothermeasuresaretakenintoconsideration.

iv) Negativelyskewed

Somestudentsmaynotorderthestem-and-leafplotwhichwillaffecttheanswerstothequestionsthatfollow.

QUESTION3 a)

i) Datesourceandsizeofsamplenotmentioned.

ii) Fromacensus,thisdataismeaningful.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers16

5.1Probability CommonErrorsQUESTION1 i) (

"K

ii) &K"K+ "

"K= &"

"K= 0

&

Mixinguprelativefrequencywithfrequencyi.e.answering4for(i)

QUESTION2 54150×500 = 180

QUESTION3 a)

i) 89

ii) 5

iii) 0ff)

iv) --f)

b)

i) 54

ii) )Kf(K

iii) (/Kf(K

a) Notseeingthe5outsideoftheellipses

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers17

5.2Answers CommonErrors5.2FinancialMathematics

QUESTION1a)

iii) = $5624.32

iv) A=624.32

b)TypeA:𝐴 = 10000 1 + 0.05 "= 12762.815625≈ $12762.82

TypeB:

𝐴 = 10000 1 +0.0512

-K

= 12833.586785…≈ $12833.59

TypeBisthebetterinvestment.c)≈ 18.6%𝑑)𝐴 = 7500 1 + 0.07 qTryn=5,𝐴 = 7500 1.07 " = $10519.14Tryn=4,𝐴 = 7500 1.07 ( = $9830.97Therefore,itwilltake5years.

a) v) Mixingupsimpleandcompoundinterestand

usingtheincorrectformulavi) Notsubtractingtheprincipal

b) Incorrectlyconvertingrateortimeperiodc) Notconvertingtheratebacktoapercentaged) Tryingtosolvewithoutusingtrialanderror

QUESTION2a) $1412.49b)

i) $11339.90ii) 23000 − 11339.90= $11660.10Thecardepreciatedinvalueby$11660.10

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers18

5.2RatesandRatio CommonErrorsQUESTION1

a) $"."K0h

= $"."K-Kmin

= $0.09/min

b) &"KL

0h= &"KKKKmL

0h

= 250000mL/hc) -"km

0h= -"KKKm

-Kmin

=65000m3600s

= 18.1m/s

Usingincorrectconversionfactor,especiallywhenitcomestotime

QUESTION2

a) speed= &"Kkm"h

= 50km/h

b) rateofleak= &."L

-h

= 0.417L/h= 417mL/h

c) fuelconsumption= ("L

/KKkm

=15L

100km

= 15L/100km

c)calculatingperkmratherthanper100km

QUESTION3a) directb) inversec) direct

QUESTION4 a) 70°Fb) 35°Cc) positived) Yes,asonevariableincreasessodoesthe

other.

a) andb)readingfromthewrongaxis

QUESTION5 a) 𝐹 = 𝑘×𝑑b) 0

f𝑜𝑟(0.125)

c) 320km

Studentswillneedtoeitherusethetwopairsofinformationfrompartb)andc)orrecognisethat𝐹 = 0

f𝑑astheequationoftheline.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers19

5.2RatesandRatio CommonErrorsd)Sue’sfuelconsumption

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers20

5.2AlgebraicTechniques CommonErrorsQUESTION1a)

i) "j-

ii) &e

0"

iii) u

*

(

iv) 3𝑥

v) 00

&q

b) i) 2𝑦& + 12ii) 40𝑎& − 32𝑎 + 4iii) 𝑥& + 6𝑥 + 8iv) 2𝑘& − 𝑘 − 15

i) andii)NotcreatingacommondenominatorAdd/subtractingdenominatorsiii)andiv)notsimplifyingfullyiv)dividingthenumeratorsanddenominators

QUESTION2

i) 4𝑑/ − 4𝑑 = 4𝑑 𝑑& − 1 ii) 21𝑥𝑦 − 3𝑥 + 9𝑥& = 3𝑥 7𝑦 − 1 + 3𝑥 iii) ℎ& + 8ℎ + 12 = ℎ + 2 ℎ + 6

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers21

5.2Indices CommonErrorsQUESTION1a)

i) 0j*= 𝑥v&

ii) &w1 = 2𝑚v/

iii) 0(u*

= 0(𝑎v&

=𝑎v&

4 b)

i) 𝑥v/ = 0j1

ii) 4𝑚v" = (w`

c) 1v/ = 0

01= 1

−3×1 = −3𝑥v/isnotequivalentto−3𝑥

a) iii)raisingthe4tothenegativepowertoget4𝑎v&𝑜𝑟(4𝑎)v&

b) ii)puttingthe4inthedenominator,not

recognisingthatonlythemistoanegativepower,i.e.getting 0

(w`

QUESTION2a) i) 6𝑚v0Kii) 𝑥v0Kiii) 3𝑦/iv) 3𝑘0Kv) 64𝑚v-

b) i) 4𝑥K = 4,∴false

ii) 𝑎- ÷ 𝑎f = 𝑎v&,∴false

iii) 6𝑚 K = 1,∴true

iv) 3𝑥- ÷ 3𝑥- = 1,∴false

v) 4𝑐v/ = ($1,∴false

vi) 12𝑚0K×2𝑚v& = 24𝑚f,∴true

a) i)andiii)multiplyingtheindicesinsteadof

addingiv)dividingtheindicesinsteadofsubtractingv)notraisingeverythinginsidethebracketstothepowerof3i.e.getting4𝑚v-b) similarissuestoparta)

QUESTION3a) −3𝑚& 5𝑚& + 2𝑚(𝑛 = −15𝑚( − 6𝑚-𝑛

b) 0-j*

(j= 4𝑥

c) &Ku

1y*

0"uy1= (u*

/y

d) /je

&× &j

-= j*e

&

b)notsimplifyingfullye.g.getting0-j

(or(j

*

j

c)–f)notsimplifyingfully

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers22

5.2Indices CommonErrorse)

𝑏&𝑐/

4 ×12𝑏𝑐& = 3𝑏𝑐

f)(q1

/w* ÷0&q/w`

=4𝑛/

3𝑚& ×3𝑚"

12𝑛

=𝑚/𝑛&

3

f)dividingthenumeratorsandthedenominatorstogete.g./q

*

w1 𝑜𝑟/q*

wz1

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers23

5.2Equations CommonErrorsQUESTION1a) 𝑦 = f

"

b) 𝑦 = −12c) 𝑘 = 0"

0(

AcrossallquestionsinvolvingequationØ ErrorsinalgebraØ Notperformingtheinverseoperationto

bothsidesØ Tryingtosolveequationsbeforecollection

liketermsafterexpansionsØ Whendividingbothsidesbythesame

number,subtractingratherthandividinge.g.5𝑦 = 8"e

"= f

"

𝑦 = 3Ø Whenfacedwithadivisionquestione.g.

j"= 20,studentsmaysolveas

𝑥5 = 20𝑥 = 20 ÷ 5𝑥 = 4

QUESTION2a) 𝑦 = 00

/

b) 𝑦 = 10c) 𝑥 = 0K

/

d) 𝑚 = 3e) 𝑦 = f

/

c)multiplyingby5firstwillstillgivethesameanswer,butmakestheequationmoredifficulttosolve.

QUESTION3a)i) 𝑦 = ±2ii) 𝑚 = ±4iii) 𝑥 = ± 8

𝑥 = ±2.6b)i)𝑘 = ± 20

ii)𝑘 = ± &/

a)–b)notincludingthe±intheanswerb)squarerootingbeforedividingby6

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers24

5.2Equations CommonErrorsQUESTION4i) 𝑥 = −1ii) 𝑦 = 3𝑜𝑟𝑦 = 2iii) 𝑧 = 10𝑜𝑟𝑧 = −1

QUESTION5a) i) B = 7.5ii) L = 16

b) 𝐶 = 27℃

QUESTION6 a) 13,14and15b) 9cm

a)studentsmayonlysolvetheequationandnotgivetheanswertothequestion

QUESTION7a) 𝑥 > 5

b)𝑏 > 15

c)𝑏 ≥ 8

d)𝑦 < −1e)𝑚 ≥ − (

"

b)Leavingansweras15 < 𝑏orincorrectlyinterpretingthisas𝑏 < 15.e)notreversingtheinequalitysignwiththedivisionof-5

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers25

5.2Equations CommonErrorsQUESTION8a)

(212 , 4)

b) x=3andy=7c) x=4andy=1

a)notaccuratelygraphingandthereforthepointofintersectioncannotbeaccuratelyreadoffthegraphb)andc)errorsinsolvingequations

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers26

5.2LinearRelationships CommonErrorsQUESTION1a) Gradient Y-intercept

i) 3 -4

ii) -4 -5

iii) 12

1

iv) 1 32

b) 𝑦 = −2𝑥 + 6c) m=-4andy-int=2d)

i) 2ii) -1iii) 𝑦 = −𝑥 + 2

d)i)2ii)-1iii)𝑦 = −𝑥 + 2

ii)Maymixupanswersasthetermsintheequationarewritteninadifferentorder.iii)andiv)studentsmaystruggleastheequationsarenotwritteniny=mx+bform.Errorsmaybemadeintherearrangingoftheequation.d)ii)notrecognisingthegradientasnegative

QUESTION2

a) 𝐲 = −𝟐𝐱 + 𝟐𝐚𝐧𝐝𝐲 = 𝟏𝟐𝐱 + 𝟒

b)i)𝑦 = −4𝑥 − 2ii)𝑦 = − /

&𝑥 − 2

c)𝑦 = − /&𝑥 + 3

𝑦 = &/𝑥+2

𝑚0×𝑚& = −32×

23

= −1∴line1andline2areperpendicular

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers27

5.2Non-LinearRelationships CommonErrorsQUESTION1a)i)ii)

iii)Theco-efficientofthex2affectstheconcavity

andthewidthoftheparabola.Theconstanttermshiftstheparabolaupanddowntheyaxis(i.e.changingthey-intercept)

b) 𝑦 = 𝑎𝑥& − 1(anyvalueof‘a’isacceptablee.g.𝑦 = 2𝑥& − 1)

c) 𝑥 = ± 6(𝑥 = ±2.4)d) 𝑦 = −𝑥& + 1

Acrossallquestionsinvolvinggraphingnon-linearrelationships

Ø NosmoothcurveØ Coefficientinfrontofthebasicfunction

widensthecurveasitsvalueincreasesØ Mixingupthedirectionthattheconstant

shiftsthecurve

c)incorrectsolvingofequationsandforgettingthe±

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers28

5.2Non-LinearRelationships CommonErrorsQUESTION2a)

𝑦 = −2jisareflectionof𝑦 = 2jinthex-axis.𝑦 =2vjIsareflectionof𝑦 = 2jinthey-axisb)

Theconstantshiftstheexponentialfunctionup/downtheyaxiswhichalsochangeswheretheasymptoteis.

Acrossallquestionsinvolvinggraphingnon-linearrelationships

Ø NosmoothcurveØ Coefficientinfrontofthebasicfunction

widensthecurveasitsvalueincreasesØ Mixingupthedirectionthattheconstant

shiftsthecurvea)incorrectreflection,mixingup𝑦 = −2jand𝑦 =2vjb)noshiftingtheasymptotewiththechangeinthey-intercept

QUESTION3a) Centre(0,0)

Radius3unitsb) 𝑥& + 𝑦& = 4

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers29

5.2Non-LinearRelationships CommonErrorsQUESTION4a) i)Bii)Ciii)Eiv)Av)Dvi)F

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers30

5.2AreaandSurfaceArea CommonErrorsQUESTION1a) 915.4cm2b) 932.6cm2c) 143.1cm2

Acrossallquestionsinvolvingareaandsurfacearea

Ø MixingupareaandperimeterØ Mixingupsurfaceareaandvolume

a)usingtheincorrectformulafortheareaofacircle(𝐴 = 2𝜋𝑟&isacommonerrorstudentsusefortheareaofacircle)b)nothalvingtheareasforthecylinderorforgettingtoaddtherectangleonthebottom

QUESTION2a)

i) 691.2cm2ii) 789.2cm2

b) S/A=1078.8m2Cost=$48546c)

i) 75.4m2ii) 96m

a)ii)notrecognisingthatitisatintoputstationaryinandthereforhasnotop(i.e.onlyonecircularface)b)difficultywiththetrapeziumface,orincludingthetopinthecalculation(i.e.tilingthetopofthepool)c)i)calculatingthesurfaceareaoftheentiregreenhouseratherthanjusttheroof

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers31

5.2Volume CommonErrorsQUESTION1a) 1680m3b) 900cm3c) 424.1cm3d) 220cm3

AcrossallquestionsinvolvingvolumeØ Mixingupvolumeandsurfacearea

c)notconvertingallmeasurementstothesameunits

QUESTION2384KL

QUESTION3a) 249505.29mm3b) 317680mm3c) 21.5%

a)notconvertingallmeasurementstothesameunitsc)calculatingthepercentageofspacethatthecrackerstakesupinsteadoftheemptyspace(i.e.78.5%)

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers32

5.2RIGHT-ANGLEDTRIANGLES(TRIGONOMETRY)

CommonErrors

QUESTION1a) 13.67cmb) 32.04cm

AcrossallquestionsinvolvingtrigonometryØ UsingthewrongratiosØ Gettingconfusedwhentheunknownisin

thedenominatorforexampleif:tan 30° = 0&

e

rearrangingincorrectlytoget𝑦 =12𝑡𝑎𝑛30°

Ø Notusingtheinversefunctionwhencalculatinganunknownangle(i.e.onthecalculatorstudentsneedtopressSHIFTbeforethetrigratio)

QUESTION2a) 4701’b) 33024’

QUESTION3a) 13.2mb) 200mc) 2340d) i)

ii)419miii) 326041’

a)incorrectlyplacingthe15mastheheightitreachesupthewallratherthanthelengthoftheladderd)iii)stoppingatcalculatingtheangleandnotrecognisingthatthisisnotthebearing.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers33

5.2PropertiesofGeometricFigures CommonErrorsQUESTION1a) SSS,SAS,AAS,RHSb) i)Yes(SSS)ii)YES(RHS)c) i)RHS

ii)correspondinganglesincongruenttriangles

a)mixingupthecongruenttriangletestwiththetestforsimilartriangles,especiallytheequiangulartestforsimilarityc)notgivingreasonsorstatingreasonsthatarenotonthediagram

QUESTION2i) AASii) needtoproveoppositeanglesareequal

Noreasonsstatedornotstatingtheparallellinesthatformthealternateangles

QUESTION31. Equiangular2. Threesidesofonetriangleareproportionalto

thethreesidesoftheothertriangle3. Anangleofonetriangleiscongruenttoan

angleofasecondtriangleandthelengthsofthesidesincludingtheseanglesareproportional

4. Thelengthofthehypotenuseandasecondsideofarightangledtriangleareproportionaltohypotenuseandasecondsideoftheothertriangle

mixingupthetestforsimilartriangleswithcongruenttriangletestwith

QUESTION4a) Equiangularb) Threesidesofonetriangleareproportionalto

thethreesidesoftheothertriangle

QUESTION5y=4x=15

ErrorsinsolvingequationsIncorrectlysettingupequalitystatemente.g. e

0K=

"&

QUESTION6i) 32400ii) 1620iii) 180

i)using20×180°toget3600°insteadof18×180°

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers34

5.2SingleVariableDataAnalysis CommonErrorsQUESTION1a) Lower=2,Upper=12,Median=5.5

Q1=4,Q3=7b) 3c) 50%d) e) 25%

a)statingthemedianas5ad6ratherthanfindingthemeanof5and6d)notincludingascale

QUESTION2a) ii)b) i)

QUESTION3a) Year9=7Year12=8b) Year9=2Year12=3c) Year9=3Year12=7d) Year12spentthelongest,theyhaveahigher

medianand75%ofstudentsspentmorethan6hourscomparedwithYear9whohadonly25%ofstudentsspendmorethan4hours.

e)50%

Incorrectlyreadingthescale

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers35

5.2BivariateDataAnalysis CommonErrorsQUESTION1a) Dependent:Rainfall

Independent:monthb) Dependent:handspan

Independent:height

QUESTION2Linegraph,asyoucantracksmallchangesovertimemoreaccurately

QUESTION3a)

b) Septemberc) Januaryd) TherainfallisthehighestinJanuaryandthen

slowlyfallsthroughouttheyeartillSeptemberandthenslowlyincreasesagainthroughtoDecember

QUESTION4Strongpositive,nocorrelation,weakpositive,strongnegative

QUESTION5a) Linearandapositivecorrelationb) Approx.52c) Astheresultsinthetopictestsincrease,so

doestheresultinthefinalexam.

b)tryingtoestimatetheresultwithoutconstructingalineofbestfit.

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers36

5.2BivariateDataAnalysis CommonErrorsQUESTION6a)

b) Therelationshipislinearandindirect,the

furtheryoutravelthelessfuelyouhaveinthetank

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers37

5.2Probability CommonErrorsQUESTION1a) {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6}b) 0

0&

b)calculatingtheprobabilityofaheadora6andgetting #

0&

QUESTION2

a) (#

b) 0-()

c) 0&()

QUESTION3a)

b) 0

f

c) #f

QUESTION4

a) &)

b) /f

c)

(#

b)andc)notdecreasingthesamplespaceforeachsuccessiveevent.(whenyoueatthejellybeanitisnolongerinthebag)

QUESTION5a) Independentb) Independentc) Dependent

NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers38

5.2Probability CommonErrorsQUESTION6a) {3,4,5,6}b) 0

(

c) 0

a)listingthesamplespaceofthedieas{1,2,3,4,5,6}b)andc)usingtheincorrectsamplespace

QUESTION7a) Thesecondtossisindependentofthefirsttoss

thereforeverytimeyourolladiethereisa0-

chanceofobtainingaoneb) Eachtooisanindependentevent,everytime

youtossacointhereisanevenchanceofobtainingahead.