Te Pāngarau me te Tauanga, Kaupae 1 2014iii) Arne decides that he wants to go overseas for a...

© Mana Tohu Mātauranga o Aotearoa, 2014. Pūmau te mana. Kia kaua rawa he wāhi o tēnei tuhinga e tāruatia ki te kore te whakaaetanga a te Mana Tohu Mātauranga o Aotearoa.

MĀ TE KAIMĀKA ANAKE

TAPEKE

Te Pāngarau me te Tauanga, Kaupae 1, 201491028M Te tūhura i ngā pānga i waenganui i ngā papatau,

ngā whārite me ngā kauwhata

9.30 i te ata Rātū 18 Whiringa-ā-rangi 2014 Whiwhinga: Whā

Paetae Paetae Kaiaka Paetae KairangiTe tūhura i ngā pānga i waenganui i ngā papatau, ngā whārite me ngā kauwhata.

Te tūhura i ngā pānga i waenganui i ngā papatau, ngā whārite me ngā kauwhata mā te whakaaro whaipānga.

Te tūhura i ngā pānga i waenganui i ngā papatau, ngā whārite me ngā kauwhata mā te whakaaro waitara hōhonu.

Tirohia mehemea e ōrite ana te Tau Ākonga ā-Motu (NSN) kei tō pepa whakauru ki te tau kei runga ake nei.

Me whakautu e koe ngā pātai KATOA kei roto i te pukapuka nei.

Whakaaturia ngā mahinga KATOA.

Ki te hiahia koe ki ētahi atu wāhi hei tuhituhi whakautu, whakamahia te (ngā) whārangi kei muri i te pukapuka nei, ka āta tohu ai i ngā tau pātai.

Tirohia mehemea kei roto nei ngā whārangi 2 – 19 e raupapa tika ana, ā, kāore hoki he whārangi wātea.

HOATU TE PUKAPUKA NEI KI TE KAIWHAKAHAERE HEI TE MUTUNGA O TE WHAKAMĀTAUTAU.

910285

1SUPERVISOR’S USE ONLY

9 1 0 2 8 M

2

Te Pāngarau me te Tauanga 91028M, 2014

MĀ TE KAIMĀKA

ANAKE

PĀTAI TUATAHI

(a) KeitetūhurahuraaJosieingātauiratauingāwhakaaturangahokomaha.

Layer

1234

Apa

(i) Whakaotihiatepapatauitetahamatau.

(ii) Kitetukutukuiraronei,tuhiatekauwhataewhakaatuanaitemahaongākēneiiaapa.

5 10

5

10

15

n

A

(iii) EkīanaaJosiekeipapahoromēnākawhakatūhiapēneihiangākēnekitētahiaparārangi-tahi,ā,kawhakatauiakitetāpiriiētahirārangieruaanōkimuri,kianohoaietorungārārangikiiaapa,pēneiitehoahoairaro.

Tuhiatewhāritemōtetapekeongākēne,A,itētahiapa,n,otētahiwhakaaturanga,inahangaiaiaapakingārārangie3.

A =

Layer

1234

Apa

Temahaongāapaongākēne,

n

Temahaongākēnekiteapa,

A

1 4

2

3

4

7

Ki te hiahia koe ki te tuhi anō i tēnei kauwhata,

whakamahia te tukutuku i te whārangi 14.

QUESTION ONE

(a) Josieisinvestigatingnumberpatternsinsupermarketdisplays.

Layer

1234

Layer

(i) Completethetableontheright.

(ii) Onthegridbelowplotthegraphthatshowsthenumberofcansineachlayer.

5 10

5

10

15

n

A

(iii) Josiesaysarrangingcansinasingle-rowlayerlikethisistoounstableanddecidestoaddanother2rowsbehind,sothereare3rowsineachlayer,asshownbelow.

Givetheequationforthenumberofcans,A,inanylayer,n,ofadisplay,whereeachlayerismadeof3rows.

A =

Layer

1234

Layer

Numberoflayersofcans,n

Numberofcansinlayer,A

1 4

2

3

4

7

If you need to redraw this graph, use the

grid on page 15

3

Mathematics and Statistics 91028, 2014

ASSESSOR’S USE ONLY

4


MĀ TE KAIMĀKA

ANAKE

(iv) 15apateteiteiotētahiwhakaaturanga.E4kēnehokikeiiarārangioteapaorunga.

Kiahiangākēnemōteapaorarotonuotēneiwhakaaturangamēnāe3ngārārangikiiaapa?

Mewhakaatuekoetewhakamahingaotōwhārite.

(b) (i) Tuhiatekauwhatao =+y x x72

2

5–5

–5

10

5

x

y

(ii) KeitemōhioaJosiekoTn n72

2

=+

tewhāritemōtetapekeongākēne,T,itētahi

whakaaturangametenaparārangi-tahi.

KapēheatererekēotekauwhataotepāngaiwaengaiTmenkitekauwhataituhikoei(b)(i)?

Homaiētahitakemōtōwhakautu.

Ki te hiahia koe ki te tuhi anō i tēnei kauwhata,

whakamahia te tukutuku i te whārangi 14.

(iv) Adisplayis15layershigh.Italsohas4cansineachrowofthetoplayer.

Howmanycanswouldsheneedforthebottomlayerofthisdisplayifeachlayerismadeof3rows?

Youmustshowuseofyourequation.

(b) (i) Sketchthegraphof =+y x x72

2

5–5

–5

10

5

x

y

(ii) JosieknowsthatTn n72

2

=+

istheequationforthetotalnumberofcans,T,inadisplay

withnsingle-rowlayers.

HowwouldthegraphoftherelationshipbetweenTandndifferfromthegraphyoudrewin(b)(i)?

Givereasonsforyouranswer.

If you need to redraw this graph, use the

grid on page 15

5



6


MĀ TE KAIMĀKA

ANAKE

PĀTAI TUARUA

(a) (i) Tuhiatewhāritemōtekauwhataewhakaaturiaanairaronei.

5–5

–10

40

50

60

20

30

10

x

y

(ii) Mōēheauaraotexhetōraroay?

(b) KeitetūhurahuraaRajaingāraupapatauotemomo

y = x2 – x+q

inakoqhetaupūmau.

Mēnākatāiaeiahekauwhataotēnei,katohuteqiteahairungaitekauwhata?

QUESTION TWO

(a) (i) Givetheequationofthegraphshownbelow.

5–5

–10

40

50

60

20

30

10

x

y

(ii) Forwhatvaluesofxisynegative?

(b) Rajaisinvestigatingnumbersequencesoftheform

y = x2 – x+q

whereqisaconstantnumber.

Ifhedrewagraphofthis,whatwouldqrepresentonthegraph?

7



8


MĀ TE KAIMĀKA

ANAKE

(c) IwhakamahiaRajaitanatātaitaikitetuhiitekauwhataoy = x2 – xkingātuakairaronei.

4 6 8 10x

y

40

50

60

70

80

90

100

110

120

20

30

10

–10 –8 –6 –4 –2 0 20

(i) Heahangātaunga1oteakitu(tepūwāhitinohahakarawa)otekauwhataorunga?

Memaumaharakitewhakaatuiōmahinga.

(ii) Whakaatuā-tuhituhi,kapēheatekauwhataotey = –2x2 + 2xewhakatairiteaikitekauwhataorunga.

(iii) Mēnākanekehiatekauwhataitewāhanga(c)(i)orungakia5ngāwaeinewhakatemataumete15waeinewhakarunga,kaahatewhāritehou?

Whakarūnāhiatōwhakautu.

1takotoranga

(c) Rajausedhiscalculatortodrawthegraphofy = x2 – xontheaxesbelow.

4 6 8 10x

y

40

50

60

70

80

90

100

110

120

20

30

10

–10 –8 –6 –4 –2 0 20

(i) Whataretheco-ordinatesofthevertex(lowestpoint)ofthegraphabove?

Remembertoshowyourworking.

(ii) Describe,inwords,howthegraphofy = –2x2 + 2xcomparestothegraphabove.

(iii) Ifthegraphfrompart(c)(i)abovewasmoved5unitstotherightand15unitsup,whatwoulditsnewequationbe?

Simplifyyouranswer.

9



10


MĀ TE KAIMĀKA

ANAKE

PĀTAI TUATORU

(a) E$450tenamaaArnekeitanapūteataurewaākonga. Keiteutuatananamamāte$15iiawiki.

(i) Mō ngā wiki tuatahi e 5,tuhiatekauwhataotetapekemeututonueArne,A,kitemahaongāwikieutuanaiaitepūteataurewa,w.

5 10 15 20 25 30 35 40

Number of weeks, w

Total still to pay, A

30

60

90

120

150

180

210

240

270

300

330

360

390

420

450

480

0

Tetapekeenamatonuana,A

Temahaongāwiki,w

(ii) TuhiatewhāritemōtetapekeAenamatonuaiaArneimuriitanautuite$15iiawiki.

A =

Ki te hiahia koe ki te tuhi

anō i tēnei kauwhata,

whakamahia te tukutuku i te whārangi

16.

QUESTION THREE

(a) Arneowes$450onhisstudentloan. Heispayingitoffattherateof$15eachweek.

(i) For the first 5 weeks,sketchthegraphofthetotalamountArnestillhastopay,A,againstthenumberofweekshehasbeenpayingofftheloan,w.

5 10 15 20 25 30 35 40

Number of weeks, w


30

60

90

120

150

180

210

240

270

300

330

360

390

420

450

480

0

Totalstilltopay,A

Numberofweeks,w

(ii) GivetheequationfortheamountAthatArneowesafterhemakeseachweeklypaymentof$15.

A =

If you need to

redraw this graph, use the grid on page 17

11



12


MĀ TE KAIMĀKA

ANAKE

(iii) KeitewhakatauaArneehiahiaanaiakiahaerekitāwāhiharareiai. Kotanahiahiakiaotikatoatanautuitanapūteataurewaitemutungaote25wiki. Katitiroiakitetapekeetoetonuanakiautuametewhakataukiteutuitepūtea

taurewakatoaitemutungatonuote25wikimēnākawhakapikiiaiterahingaeutuanaiakite$20itewiki.

Tāpiritiatēneiwāhangakitōkauwhata.

(iv) Ehiangāwikikapauimuaitanautuite$20itewiki?

(v) Homaitewhāritemōtekauwhataewhakaatuanaingāutunganui ake.

(b) Homaitewhāritemōtekauwhataoraronei.

y

x

5

5–5 10–10

–5

–10

10

(iii) Arnedecidesthathewantstogooverseasforaholiday. Hewantstofinishrepayingallofhisloanattheendof25weeks. Helooksatwhathehaslefttopayanddecideshecanrepaytheloanexactlyattheend

of25weeksifheincreasestheamountherepaysto$20aweek.

Addthissectiontoyourgraph.

(iv) Afterhowmanyweeksdidhestartpaying$20perweek?

(v) Givetheequationforthegraphrepresentingtheincreasedpayments.

(b) Givetheequationofthegraphbelow.

y

x

5

5–5 10–10

–5

–10

10

13



14


MĀ TE KAIMĀKA

ANAKE

KitehiahiakoekitetuhianōitōkauwhatamaiitePātaiTuatahi(a)(ii),tuhiakitetukutukuoraro.Kiamāramatetohukotēheatekauwhatakahiahiakoekiamākahia.

5 10

5

10

15

n

A

KitehiahiakoekitetuhianōitōkauwhatamaiitePātaiTuatahi(b)(i),tuhiakitetukutukuoraro.Kiamāramatetohukotēheatekauwhatakahiahiakoekiamākahia.

5–5

–5

10

5

x

y

IfyouneedtoredrawyourgraphfromQuestionOne(a)(ii),drawitonthegridbelow.Makesureitisclearwhichgraphyouwantmarked.

5 10

5

10

15

n

A

IfyouneedtoredrawyourgraphfromQuestionOne(b)(i),drawitonthegridbelow.Makesureitisclearwhichgraphyouwantmarked.

5–5

–5

10

5

x

y

15



16


MĀ TE KAIMĀKA

ANAKE

KitehiahiakoekitetuhianōitōkauwhatamaiitePātaiTuatoru(a)(i),tuhiakitetukutukuoraro.Kiamāramatetohukotēheatekauwhatakahiahiakoekiamākahia.

5 10 15 20 25 30 35 40

Number of weeks, w


30

60

90

120

150

180

210

240

270

300

330

360

390

420

450

480

0

Tetapekeenamatonuana,A

Temahaongāwiki,w

IfyouneedtoredrawyourgraphfromQuestionThree(a)(i),drawitonthegridbelow.Makesureitisclearwhichgraphyouwantmarked.

5 10 15 20 25 30 35 40

Number of weeks, w


30

60

90

120

150

180

210

240

270

300

330

360

390

420

450

480

0

17



18


MĀ TE KAIMĀKA

ANAKETAU PĀTAI

He puka anō mēnā ka hiahiatia.Tuhia te (ngā) tāu pātai mēnā e hāngai ana.

19



QUESTION NUMBER

Extra paper if required.Write the question number(s) if applicable.

Level 1 Mathematics and Statistics, 201491028 Investigate relationships between tables,

equations and graphs

9.30 am Tuesday 18 November 2014 Credits: Four

Achievement Achievement with Merit Achievement with ExcellenceInvestigate relationships between tables, equations and graphs.

Investigate relationships between tables, equations and graphs, using relational thinking.

Investigate relationships between tables, equations and graphs, using extended abstract thinking.

Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.

You should attempt ALL the questions in this booklet.

Show ALL working.

If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.

Check that this booklet has pages 2 – 19 in the correct order and that none of these pages is blank.

YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.

© New Zealand Qualifications Authority, 2014. All rights reserved.No part of this publication may be reproduced by any means without the prior permission of the New Zealand Qualifications Authority.

English translation of the wording on the front cover

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