ear 11 Mathematics IAS 1 - Nulake 1.1 Sample.pdf · ear 11 Mathematics Contents uLake Ltd ua et d...

Year 11Mathematics

Contents

uLake Ltdu a e tduLake LtdInnovative Publisher of Mathematics Texts

IAS 1.1Robert Lakeland & Carl Nugent

Numeric Reasoning

• AchievementStandard .................................................. 2• PrimeNumbers ....................................................... 3• FactorsandMultiples ................................................... 4• RoundingandEstimation................................................ 7• StandardForm......................................................... 12• OrderofOperation..................................................... 15• Integers(+,x,÷,–)....................................................... 17• Fractions(+,x,÷,–)...................................................... 20• Percentages............................................................ 24• Ratio................................................................... 32• Proportion............................................................. 37• Rates.................................................................. 41• Powers................................................................. 44• CompoundingRates..................................................... 49• PracticeInternalAssessment1 ............................................ 55• PracticeInternalAssessment2............................................ 56• PracticeInternalAssessment3............................................ 57• Answers............................................................... 58

IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent

2 IAS 1.1 – Numeric Reasoning

NCEA 1 Internal Achievement Standard 1.1 – Numeric ReasoningThisachievementstandardinvolvesapplyingnumericreasoninginsolvingproblems.

◆ ThisachievementstandardisderivedfromLevel6ofTheNewZealandCurriculum,Learning Media.Thefollowingachievementobjectives,takenfromtheNumberStrategiesandKnowledge threadoftheMathematicsandStatisticslearningarea,arerelatedtothisachievementstandard: ❖ reasonwithlinearproportions ❖ useprimenumbers,commonfactorsandmultiples,andpowers(includingsquareroots) ❖ understandoperationsonfractions,decimals,percentages,andintegers ❖ useratesandratios ❖ knowcommonlyusedfraction,decimal,andpercentageconversions ❖ knowandapplystandardform,significantfigures,rounding,anddecimalplacevalue ❖ applydirectandinverserelationshipswithlinearproportion ❖ extendpowerstoincludeintegersandfractions ❖ applyeverydaycompoundingrates.

◆ Applynumericreasoninginvolves: ❖ selectingandusingarangeofmethodsinsolvingproblems ❖ demonstratingknowledgeofnumberconceptsandterms ❖ communicatingsolutionswhichwouldusuallyrequireonlyoneortwosteps. Relationalthinkinginvolvesoneormoreof: ❖ selectingandcarryingoutalogicalsequenceofsteps ❖ connectingdifferentconceptsandrepresentations ❖ demonstratingunderstandingofconcepts ❖ formingandusingamodel;

andalsorelatingfindingstoacontext,orcommunicatingthinkingusingappropriatemathematical statements. Extendedabstractthinkinginvolvesoneormoreof: ❖ devisingastrategytoinvestigateorsolveaproblem ❖ identifyingrelevantconceptsincontext ❖ developingachainoflogicalreasoning,orproof ❖ formingageneralisation;

andalsousingcorrectmathematicalstatements,orcommunicatingmathematicalinsight.

◆ Problemsaresituationsthatprovideopportunitiestoapplyknowledgeorunderstandingof mathematicalconceptsandmethods.Thesituationwillbesetinareal-lifeormathematicalcontext.

◆ Thephrase‘arangeofmethods’indicatesthatevidenceoftheapplicationofatleastthreedifferent methodsisrequired.

◆ Studentsneedtobefamiliarwithmethodsrelatedto: ❖ ratioandproportion ❖ factors,multiples,powersandroots ❖ integerandfractionalpowersappliedtonumbers ❖ fractions,decimalsandpercentages ❖ rates ❖ roundingwithdecimalplacesandsignificantfigures ❖ standardform.

Achievement Achievement with Merit Achievement with Excellence• Applynumericreasoningin

solvingproblems.• Applynumericreasoning,

usingrelationalthinking,insolvingproblems.

• Applynumericreasoning,usingextendedabstractthinking,insolvingproblems.

3IAS 1.1 – Numeric Reasoning


Prime Numbers

Prime NumbersAprimenumberisanumbergreaterthan1thathasnopositivedivisorsotherthan1anditself.Aprimenumberhasexactlytwofactors1anditself.Forexample17isaprimebecauseithasonlytwofactors1and17.Thesmallestprimenumberaswellastheonlyevenprimenumberis2,becauseitisdivisibleby1and2.Anumberngreaterthan1isdefinedasaprimenumberifitisonlydivisibleby1andn.Positivenumbersotherthan1thatarenotprimenumbersarecalledcompositenumbers.

A factor is a number that divides into another number without remainder. For example 2 is a factor of 6 because 2 divides into 6 without remainder.

Product of PrimesItispossibletowriteanypositivenumbergreaterthanoneasaproductofprimenumbers.

Thebestwaytodothisistouseafactortree.

Atthetopofthetreeyoustartwiththenumberyouwishtowriteasaproductofprimenumbers.

Youthenfindtwonumbersthatmultiplytogivethenumber.Onceoneofthebranchesofthetreehasaprimenumberatitsbranchendyoustopsimplifyingthatbranch.

Youcontinueworkingoneachbranchuntilonlyaprimenumberremains.

Ifyoumultiplyalltheprimenumbersattheendofeachofthebranchesyoushouldgetthenumberyoustartedwith.Aprimefactortreefor120isdrawnbelow.

80

20 4x

4 5 2 2xx

2 2x

So80=2x2x 5x 2x 2.Itdoesnotmatterwhattwonumbersyoufindtomultiplytogive80(i.e.40x2or8x5or10x8)youwillalwaysendupwiththesameprimefactorsattheend.

Example

a) Copythenumbers5,19,32,37,39,52andcircle thosethatareprime.b) Listthenexttwoprimenumbersafter61.c) Drawaprimefactortreefor150.

a) 5,19,32,37,39,52

b) Thenexttwoprimenumbersafter61are67 and71.c) Primefactortreefor150isasfollows.

Primefactorsare2x3x5x5.

150

15 10x

3 5 2 5xx

©uLake LtduLake Ltd Innovative Publisher of Mathematics Texts



To write any number in standard form first move the decimal point, so that the number has a value between 1 and 10

then multiply by an appropriate power of 10, so the number has the same value (found by counting the number of decimal places the decimal point has moved).

Writethefollowinginstandardform.a) 1320000 b) 0.00056 c) 2.1

a) 1.32x106 b) 5.6x10–4 c) 2.1x100as100=1

Writethefollowingasanordinarynumber.

a) 2.4x10–3 b) 8.647x102 c) 3.91x105

a) 0.0024 b) 864.7 c) 391000

Example

Example

To convert a number into standard form on your calculator use the scientific mode. Each calculator is a little different.

On the TI-84 Plus press MODE then choose SCI and then the number of decimal places you want to display. On the Casio 9750GII press SHIFT MENU then scroll down and select Display and choose SCI and then the number of decimal places you want to display.

Standard Form

Standard FormWeusestandard formasaconcisewayofwritingverylargeandverysmallnumbers.Anumberinstandardformiswrittenasanumberbetween1and10,multipliedbyapowerof10.Consider 3127686.2Youfirstmovethedecimalpointsothenumberhasavaluebetween1and10,i.e.3.1276862Younowmultiplyby1000000or106tomakethenumberequal3127686.2Instandardform3127686.2=3.1276862x106

To enter a number in standard form on a graphics calculator we use the EXP button (Casio 9750GII) or the EE button (TI-84 Plus),

e.g. for 3.127 686 2 x 106 we enter

3.1276862 Casio 9750GII

3.1276862 TI-84 Plus

EXP 6

2nd , 6EE

Withverysmallnumbersweworksimilarly.Consider 0.0045Movethedecimalpoint,sothenumberhasavaluebetween1and10,i.e.4.5.Younowneedtodividethisnumberby1000ormultiplyby0.001=10–3tomake4.5equal0.0045.Instandardform0.0045=4.5x10–3

Thedecimalpointneedstomove3placesleftfrom4.5sothepoweris–3.

0.0045=4.5x10–3

Thisnumbermustalwaysbebetween1and10.

65. 41500 66. 591 67. 12.75 68. 0.045

69. 0.592 70. 7 71. 12700000 72. 0.00000956

Merit–Writethefollowinginstandardform,calculatingtheanswerfirstifrequired.

A number displayed as 8.89E+05 means 8.89 x 105. A number displayed as 8.89E–05

means 8.89 x 10–5.




100. Acountry’snationaldebtis$1.5x1012.Ifthepopulationofthecountryis285million,howmuchisowedperindividual?

99. Abladeofgrassgrowsonaverageatarateof2.0x10–8metrespersecond.Howmuchwillabladeofgrassgrowinoneweekinmillimetres?

101.Usestandardformtocalculatethefollowinginformationabout75yearoldMartin.Roundallyouranswersto3significantfigures.

Oneestimateofthenumberofcellsintheaveragehumanbodyisseventy-threepointeightmillionmillion.

a) Writethisasanordinarynumber.

b) Writethisnumberinstandardform.

c) ThepopulationoftheEarthisapproximately 6150000000people. Calculatethetotalnumberofhumancellsfor

theentirepopulationofEarth.

d) Theaveragehumanweighs65.5kg.Whatistheaveragemassofonecellingrams?(1000g=1kg).

e) Themassofasinglehydrogenatomis1.66x10–24g.Calculatehowmanytimesheavierasinglehumancellisthanahydrogenatom.

Aperson’sheartbeats,onaverage,85beatsperminuteforeveryminuteoftheirlife.f) HowmanyminuteshasMartinlivedfor? Ignoreleapyearsandgiveyouranswerin standardform.

g) HowmanytimeshasMartin’sheartbeaten?

h) Eachbeatoftheheartpumpsabout67mL. Howmanylitreshastheheartpumpedin Martin’slife?




211. Keithisausedcarsalesmanandatthe beginningofamonthhas120vehiclesinstock.

a) Keithsells 245 ofhisstockinthemonth.How

manyvehiclesdoeshehaveleftattheendofthemonth?

b) 31 ofthevehiclesarecommercialvehicles,suchasvansandutes,and 8

5 ofthesecostlessthan$10000.HowmanyofKeith’scommercialvehiclescostlessthan$10000?

c) 32 ofthevehiclesarecarsandofthese 4

3 haveanenginecapacitylessthan2000cc.HowmanycarsdoesKeithhavewithanenginecapacitygreaterthan2000cc?

d) Keithsellsoneofhiscarsfor$27000.Thepurchaserputsdownadepositof 5

1 andpaysthebalanceoffover24monthsat0%interest.Howmuchdoesthepurchaserpaypermonth?

e) 45 ofthecustomerswhoboughtvehiclesonemonthpaidcashforthem.If20customerspaidcash,howmanyvehiclesintotaldidKeithsellduringthemonth?

f) Keithsells 245 ofhisstockinamonthand 5

2 oftheremainingstockthefollowingmonth.WhatfractionoftheoriginalstockofvehiclesremainafterthetwomonthsandhowmanycarsdoesKeithhaveleft?

To convert a decimal to a percentage move the decimal point two places to the right. On a Casio 9750GII to convert a fraction to a percentage on the calculator enter

On the TI-84 Plus to convert a fraction to a percentage enter

To simplify a fraction on the Casio 9750GII enter it into your calculator as a fraction and then press EXE. The calculator will automatically reduce

it down to its simplest form. If the simplified fraction is a mixed numeral press to convert it to an improper fraction. On the

TI-84 Plus to simplify a fraction like 35100

we enter

100. Percentagethereforemeans‘outof100’.Byrepresentingfiguresasapercentagewecaneasilymakeacomparisonbetweentwoormoresetsoffigures.

Thewordpercentageismadeupoftheprefix‘per’meaningoutofand‘centage’fromthesamerootas‘century’meaning

SHIFT

3 a b/c 5 x 1

EXE

0 0

ENTER

MATH

1Frac

3 ÷ 15 0 0

3 ÷ 5 x 1

ENTER

0 0

Percentages to Fractions

Percentages

Toconvertapercentagetoafractionwewritethepercentageasafractionoutof100andthensimplifythefractionifpossible.

Consider 35%

Asafraction = 35100

Simplify = 720

Decimal to a PercentageToconvertadecimaltoapercentagewemultiplyby100%.Consider 0.05Mult.by100% = 0.05x 100% = 5%

Fraction to a PercentageToconvertafractiontoapercentagewemultiplyby100%.

Consider 35

Mult.by100% = 35x100%

= 3005%

Simplify = 60%

F D




ExampleCalculatethefollowing.

247. Increase$210by23% 248. Decrease194by35%

Achievement–Answerthefollowingpercentagequestions.

a) Decrease$95by15%. b) Increase$255by16.5%

a) Startx(100±Change)% =Result 95x(100–15)%=Result 95x85% =Result Result =$80.75

b) Startx(100±Change)% =Result 255x(100+16.5)%=Result 255x116.5% =Result Result =$297.08(2dp)

249. Decrease$47.50by12.5% 250. Decreaseby30%ashirtthatcosts$67.50

Examplea) Anarticleisboughtfor$58andsoldfor$70.

Whatisthepercentageincrease?b) Findthepre-GSTamountwhenanarticle

sellingfor$250includesGSTof15%.

a) Startx(100±Change)%=Result

58x(100±Change)%=70

(100±Change)%=1.2069

(100±Change)=120.69 asa%

Change=20.7%(1dp)

Startx(100±Change)% =Result

Startx(100+15.0)% =250

Startx115.0% =250

Start =

250115.0%

=$217.39

251. Alechastopaya15%surchargeonameal costing$96.50.Howmuchwillhepay altogether?

252. Aplasmascreenusuallyretailsfor$3999.If apurchaserpayscashtheyareeligiblefora 12.5%discount.Howmuchdoestheplasma screencostwiththecashdiscount?

253. Duetoafluepidemic12%ofthepupilsina schoolareabsentoneday.Iftheschoolroll isnormally575pupils,howmanypupils arepresent?

254. Ahousesellsfor34%aboveitsgovernment valuationof$485000.Howmuchdoesitsell for?

255. Aschool’srollhasincreasedby6.5%overthe last10years.Iftenyearsagoithadarollof 1450,whatisitsrollnow?

256. Thevalueofacarhasdepreciatedinvalueby 55%.Ifitwasinitiallypurchasedfor$42500 whatisitworthnow?


37IAS 1.1 – Numeric Reasoning


Proportion

ProportionAproportionisapartconsideredinrelationtoawholeorastatementofequalitybetweentwoormoreratios.

i.e. ab

cd

==ab

cd

=

Directly Proportional Problems Directlyproportionalproblemsareproblemswhereachangeinonequantitycausesaproportionalchangeinanotherquantity.Twoquantitiesyandxareindirectproportionifbywhateverychanges,xchangesbythesameproportionormultiplier.Wewritey∝x,whichisreadasyisdirectlyproportionaltox,thismeansy=kx,wherekisaconstant.

E.g.Thecostofpensisdirectlyproportionaltothenumberofpensyoubuy.Iftwopenscost$1.50,howmanypenscanyoubuyfor$10.50? Firstwefindk,1.50=2k k=0.75(eachpencosts75cents)Tocalculatehowmanypenswecanbuyfor$10.50wedivide10.50bythecostofasinglepen0.75whichequals14pens.

Inversely Proportional Problems Inverselyproportionalproblemsareproblemsthataresimilartodirectlyproportionalproblemsexceptthatwhenxincreasesywilldecreaseandviceversa.Twoquantitiesyandxareinverselyproportionaliftheirproductalwaysremainconstant,i.e.xy=kor

y= kxwherekisaconstant.

E.g.Ifittakes4men6hourstodigadrain,howlongwillittake7mentodothesamejob? Firstwefindk,whichis4x6=24(totalnumberofmanhours).Tofindhowlongitwilltake7mentodigthedrainwedivide24(totalnumberofmanhours)

by7= 3 3

7hours.

The ratio of the number of men = the inverse ratio of the number of hours. i.e. 4 : 7 = x : 6

47=

x6

7x = 24

x = 3 3

7 hours

Directly proportional problems can also be set up as ratios, but make sure that the two ratios are written in the correct order.In the problem on the left,

if x = the number of pens then 1.52

= 10.5x

= 1.52

= 10.5x

Solving this equation gives 1.5x = 21 and then x = 14

Two quantities x and y are said to be inversely proportional if their product xy always remains constant.In the problem on the left, if x = the number of hours then 4 men x 6 hours = 7 men x x 24 = 7x

x = 3 3

7 hours

The more men the less time to complete the job, hence this is an inversely proportional problem. When x increases, y decreases.

A good test of an inverse proportional problem is to ask yourself,“If one quantity doubles, will the other half”, i.e. if x increases by a multiplier, y will decrease by the same divisor.




Page 9

30. 6.0(1dp)

31. 9(1sf)

32. 3.22(2dp)

33. 39(2sf)

34. 2.62(3sf)

35. 12(2sf)

36. 110(2sf)

37. 0.13(2sf)

38. 22.8m(1dp)

39. 47cm3(2sf)

40. 5.4m(1dp)

41. 5.6L(2sf) asonlythe45litresis measured(the8partsare counted).

42. 43cm2(2sf)

43. $353.50(2sf)

44. a) 21m2 b) 4 c) 55cm3(2sf) d) $170.95 e) $3897 f) Noshouldberoundedto 3sfi.e52.8m3. g) 18hours.

Page 10

45. 120(accept102)

46. 70(accept77)

47. 500

48. 9

49. 200

50. 60

51. 100

52. 250

53. $150

54. $10000

55. $100

56. $3

57. $12000

58. 1500cm3

59. $300

60. 1250km

AnswersPage 5

1. a) 1,2,4,7,14,28

b) 1,2,3,6,7,14,21,42

c) 1,19

2. a) 17,34,51,68,...

b) 21,42,63,84,...

c) 62,124,186,248,...

3. a) 1,3,7,21 and1,2,4,7,8,14,28,56

HCF=7

b) 1,3,5,9,15,45 and1,3,9,13,39,117

HCF=9

c) 1,5,19,95 and1,2,3,6,19,38,57,114

HCF=19

4. a) 12,24,36,48,60,72,...

20,40,60,...

LCM=60

b) 6,12,18,24,30,36,42,48, 54,60,66,...

11,22,33,44,55,66,..

LCM=66

c) 9,18,27,36,45,54,...

15,30,45,...

LCM=45

5. a) primenumber

b)

2x3x 23

c)

2x5x 2x 7

Page 6

6. 8,16,24,32,40,48,56,...

14,28,42,56,...

56seconds

138

2 69x

3 23x

140

10 14x

2 5 2 7xx

Page 6 cont...

7.

8. 4=2+2,5=2+3,6=3+3, 7=2+5,8=5+3, 9=2+7,10=5+5, 12=5+7,13=11+2, 14=11+3,15=2+13 16=13+3,18=7+11 19=2+17allcan.

14numberslessthan20.

9. HCFof448and616is 56sogreatestpossiblelength is56cm.

10. LCMof28and24whichis 168.So168seconds

11. 48=2x2x 2x2x 3

108=2x2x 3x3x 3

HCF=2x2x 3=12

12. LCMof40,48and60=240, so240minutes(4hours)

13. HCFof96,144and224, so16piecesofchicken.Page 8

14. 789.88

15. 0.0024

16. 50.0

17. 67500

18. 0.0098

19. 2000

20. 655.0

21. 0.0480

22. 27000

23. 44

24. 0.9

25. 5.23

26. 480

27. 8.4

28. 60

29. 7220

47

113

17

29

59

89

101

5

71


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