Mathematics for the International Student Year 9 IB MYP 4

576 Pages·15.518 MB·English

by  Pamela Vollmar et al.

HAESE HARRIS PUBLICATIONS
&
Specialists in mathematics publishing
Mathematics
for the international student
9
MYP 4
Pamela Vollmar
Michael Haese
Robert Haese
Sandra Haese
Mark Humphries
for use with
IB Middle Years
Programme
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symbol_pp
(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:9) swtimes
MATHEMATICSFORTHEINTERNATIONALSTUDENT9(MYP4)
PamelaVollmar B.Sc.(Hons.),PGCE.
MichaelHaese B.Sc.(Hons.),Ph.D.
RobertHaese B.Sc.
SandraHaese B.Sc.
MarkHumphries B.Sc.(Hons.)
Haese&HarrisPublications
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haeseandharris.com.au
NationalLibraryofAustraliaCardNumber&ISBN 978-1-876543-29-7
©Haese&HarrisPublications2008
PublishedbyRaksarNomineesPtyLtd
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
FirstEdition 2008
CartoonartworkbyJohnMartin.ArtworkbyPiotrPoturajandDavidPurton.
CoverdesignbyPiotrPoturaj.
ComputersoftwarebyDavidPurton,ThomasJanssonandTroyCruickshank.
TypesetinAustraliabySusanHaese(RaksarNominees).TypesetinTimesRoman10\Qw_/11\Qw_
ThetextbookanditsaccompanyingCDhavebeendevelopedindependentlyoftheInternational
BaccalaureateOrganization(IBO).ThetextbookandCDareinnowayconnectedwith,orendorsedby,
theIBO.
Thisbookiscopyright.ExceptaspermittedbytheCopyrightAct(anyfairdealingforthepurposesof
privatestudy,research,criticismorreview),nopartofthispublicationmaybereproduced,storedina
retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying,
recordingorotherwise,withoutthepriorpermissionofthepublisher.EnquiriestobemadetoHaese&
HarrisPublications.
Copyingforeducationalpurposes:WherecopiesofpartorthewholeofthebookaremadeunderPart
VBoftheCopyrightAct,thelawrequiresthattheeducationalinstitutionorthebodythatadministersit
has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the
CopyrightAgencyLimited.
Acknowledgements:ThepublishersacknowledgethecooperationofOxfordUniversityPress,Australia,
for the reproduction of material originally published in textbooks produced in association with
Haese&HarrisPublications.
While every attempt has been made to trace and acknowledge copyright, the authors and publishers
apologiseforanyaccidentalinfringementwherecopyrighthasproveduntraceable. Theywouldbepleased
tocometoasuitableagreementwiththerightfulowner.
Disclaimer:All the internet addresses (URL’s) given in this book were valid at the time of printing.
Whiletheauthorsandpublisherregretanyinconveniencethatchangesofaddressmaycausereaders,no
responsibilityforanysuchchangescanbeacceptedbyeithertheauthorsorthepublisher.
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FOREWORD
This book may be used as a general textbook at about 9th Grade (or Year 9) level in classes where
students are expected to complete a rigorous course in Mathematics. It is the fourth book in our Middle
Yearsseries‘MathematicsfortheInternationalStudent’.
IntermsoftheIBMiddleYearsProgramme(MYP),ourseriesdoesnotpretendtobeadefinitivecourse.
Inresponsetorequestsfromteacherswhouse‘MathematicsfortheInternationalStudent’atIBDiploma
level, we have endeavoured to interpret their requirements, as expressed to us, for a series that would
prepare students for the Mathematics courses at Diploma level. We have developed the series
independently of the International Baccalaureate Organization (IBO) in consultation with experienced
teachersofIBMathematics.NeithertheseriesnorthistextisendorsedbytheIBO.
Inregardtothisbook,itisnotourintentionthateachchapterbeworkedthroughinfull.Timeconstraints
will not allow for this. Teachers must select exercises carefully, according to the abilities and prior
knowledge of their students, to make the most efficient use of time and give as thorough coverage of
contentaspossible.
Toavoidproducingabookthatwouldbetoobulkyforstudents,wehavepresentedsomechaptersonthe
CD,asprintablepages:
Chapter26:Variation
Chapter27:Twovariableanalysis
Chapter26:Logic
Theabovewereselectedbecausethecontentcouldberegardedasextensionmaterialformost9thGrade
(orYear9)students.
WeunderstandtheemphasisthattheIBMYPplacesonthefiveAreasofInteractionandinresponsethere
are links on the CD to printable pages which offer ideas for projects and investigations to help busy
teachers(seep.5).
Frequent use of the interactive features on the CD should nurture a much deeper understanding and
appreciation of mathematical concepts. The inclusion of our new Self Tutor software (see p. 4) is
intendedtohelpstudentswhohavebeenabsentfromclassesorwhoexperiencedifficultyunderstanding
thematerial.
Thebookcontainsmanyproblemstocaterforarangeofstudentabilitiesandinterests,andeffortshave
beenmadetocontextualiseproblemssothatstudentscanseethepracticalapplicationsofthemathematics
theyarestudying.
Wewelcomeyourfeedback. Email:[email protected]
Web:www.haeseandharris.com.au
PV,PMH,RCH,SHH,MH
Acknowledgements
Theauthorsandpublisherswouldliketothankallthoseteacherswhohavereadproofsandofferedadvice
andencouragement.
AmongthosewhosubmittedcoursesofstudyforMiddleYearsMathematicsandwhoofferedtoreadand
commentontheproofsofthetextbookare:MargieKarbassioun,KerstinMockrish,ToddSharpe,Tamara
Jannink, Yang Zhaohui, Cameron Hall, Brendan Watson, Daniel Fosbenner, Rob DeAbreu, Philip E.
Hedemann,Alessandra Pecoraro, Jeanne-Mari Neefs, Ray Wiens, John Bush, Jane Forrest, DrAndrzej
Cichy,William Larson,WendyFarden,ChrisWieland,KennethCapp,SaraLocke,RaeDeeley,ValFrost,
Mal Coad, Pia Jeppesen, Wissam Malaeb, Eduardo Betti, Robb Kitcher, Catherine Krylova, Julie Tan,
Rosheen Gray, Jan-Mark Seewald, Nicola Cardwell, Tony Halsey, Ros McCabe, Alison Ryan, Mark
Bethune, Keith Black, Vivienne Verschuren, Mark Willis, Curtis Wood, Ufuk Genc, Fran O’Connor.
SpecialthankstoHeatherFarish.Toanyonewemayhavemissed,weofferourapologies.
The publishers wish to make it clear that acknowledging these individuals does not imply any
endorsement of this book by any of them, and all responsibility for the content rests with the authors and
publishers.
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USING THE INTERACTIVE CD
TheinteractiveCDisidealforindependentstudy.
Studentscanrevisitconceptstaughtinclassandundertaketheirown
revisionandpractice.TheCDalsohasthetextofthebook,allowing
studentstoleavethetextbookatschoolandkeeptheCDathome.
Byclickingontherelevanticon,arangeofnewinteractivefeatures
canbeaccessed:
(cid:2) SelfTutor
INTERACTIVE
(cid:2) AreasofInteraction linkstoprintablepages
LINK
(cid:2) PrintableChapters
(cid:2) InteractiveLinks–tospreadsheets,videoclips,graphingand
geometrysoftware,computerdemonstrationsandsimulations
N
E
SELF TUTOR is a new exciting feature of this book. W!
The Self Tutor icon on each worked example denotes an active link on the CD.
Simply ‘click’ on the Self Tutor (or anywhere in the example box) to access the worked
example, with a teacher’s voice explaining each step necessary to reach the answer.
Play any line as often as you like. See how the basic processes come alive using
movement and colour on the screen.
Ideal for students who have missed lessons or need extra help.
Example 2 Self Tutor
Simplify by collecting like terms:
a ¡a¡1+3a+4 b 5a¡b2+2a¡3b2
a ¡a¡1+3a+4 b 5a¡b2+2a¡3b2
=¡a+3a¡1+4 =5a+2a¡b2¡3b2
=2a+3 =7a¡4b2
f¡a and 3a are like terms f5a and 2a are like terms
¡1 and 4 are like termsg ¡b2 and ¡3b2 are like termsg
SeeChapter3,Algebraicexpansionandsimplification,p.73
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AREAS OF INTERACTION
The International Baccalaureate Middle Years Programme focuses teaching and learning
throughfiveAreasofInteraction:
(cid:2) Approachestolearning (cid:2) Environments
(cid:2) Communityandservice (cid:2) Healthandsocialeducation
(cid:2) Humaningenuity
TheAreasofInteractionareintendedasafocusfordevelopingconnectionsbetweendifferent
subjectareasinthecurriculumandtopromoteanunderstanding
Clickontheheadingto oftheinterrelatednessofdifferentbranchesofknowledgeandthe
accessaprintable‘pop-up’ coherenceofknowledgeasawhole.
versionofthelink.
In an effort to assist busy teachers, we offer the following
printablepagesofideasforprojectsandinvestigations:
CHESS BOARD CALCULATIONS
LINKS Areasofinteraction:
clickhere Approachestolearning/Humaningenuity
Linkstoprintablepagesofideasforprojectsandinvestigations
Chapter2:Indices CHESS BOARD CALCULATIONS
p.69 Approachestolearning/Humaningenuity
Chapter4:Radicals(surds) HOW A CALCULATOR CALCULATES
p.99 RATIONAL NUMBERS Humaningenuity
Chapter7:Mensuration WHAT SHAPE CONTAINER SHOULDWE USE?
p.174 Approachestolearning/Theenvironment
Chapter8:Quadraticfactorisation THE GOLDEN RATIO
p.191 Humaningenuity
Chapter11: Financialmathematics PAYING OFF A MORTGAGE
p.265 Healthandsocialeducation
Chapter13:Formulae INDUCTION DANGERS
p.300 Humaningenuity/Approachestolearning
Chapter15:Transformationgeometry WHAT DETERMINES COIN SIZES?
p.336 Humaningenuity
Chapter17:Simultaneousequations SOLVING 3 BY 3 SYSTEMS
p.365 Humaningenuity
Chapter19:Quadraticfunctions MAXIMISING AREAS OFENCLOSURES
p.401 Humaningenuity/Theenvironment
Chapter20:Treediagramsandbinomial WHY CASINOS ALWAYS WIN
probabilities p.416 Healthandsocialeducation
Chapter22:Otherfunctions:theirgraphs CARBON DATING
anduses p.450 Theenvironment
Chapter24:Deductivegeometry FINDING THE CENTRE OF A CIRCLE
p.498 Approachestolearning
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6 TABLEOFCONTENTS
TABLE OF CONTENTS
GRAPHICSCALCULATOR 4 RADICALS(SURDS) 87
INSTRUCTIONS 9 A Radicalsonanumberline 88
A Basiccalculations 10 B Operationswithradicals 89
B Basicfunctions 12 C Expansionswithradicals 93
C Secondaryfunctionandalphakeys 15 D Divisionbyradicals 96
D Memory 15 Reviewset4A 99
E Lists 18 Reviewset4B 100
F Statisticalgraphs 20
5 SETSANDVENNDIAGRAMS 101
G Workingwithfunctions 21
H Matrices 25 A Sets 102
I Twovariableanalysis 27 B Specialnumbersets 104
C Setbuildernotation 105
1 ALGEBRA(NOTATIONAND D Complementofsets 106
EQUATIONS) 29 E Venndiagrams 108
Reviewset5A 115
A Algebraicnotation 30
Reviewset5B 116
B Algebraicsubstitution 32
C Linearequations 34
6 COORDINATEGEOMETRY 117
D Rationalequations 38
E Linearinequations 40 A Thedistancebetweentwopoints 119
F Problemsolving 43 B Midpoints 122
G Moneyandinvestmentproblems 45 C Gradient(orslope) 124
H Motionproblems 47 D Usinggradients 128
I Mixtureproblems 48 E Usingcoordinategeometry 129
Reviewset1A 49 F Verticalandhorizontallines 131
Reviewset1B 50 G Equationsofstraightlines 132
H Thegeneralformofaline 136
2 INDICES 51 I Pointsonlines 138
J Wherelinesmeet 139
A Indexnotation 52
Reviewset6A 141
B Indexlaws 55
Reviewset6B 142
C Exponentialequations 61
D Scientificnotation(Standardform) 63
7 MENSURATION 145
E Rational(fractional)indices 66
Reviewset2A 69 A Error 147
Reviewset2B 70 B Lengthandperimeter 149
C Area 156
3 ALGEBRAICEXPANSIONAND D Surfacearea 162
SIMPLIFICATION 71 E Volumeandcapacity 167
Reviewset7A 174
A Collectingliketerms 72
Reviewset7B 175
B Productnotation 73
C Thedistributivelaw 75
8 QUADRATICFACTORISATION 177
D Theproduct (a(cid:10)(cid:11)(cid:10)b)(c(cid:10)(cid:11)(cid:10)d) 76
E Differenceoftwosquares 78 A Factorisationbyremovalofcommon
factors 178
F Perfectsquaresexpansion 80
B Differenceoftwosquaresfactorisation 180
G Furtherexpansion 82
C Perfectsquarefactorisation 182
H Thebinomialexpansion 84
D Factorisingexpressionswithfourterms 183
Reviewset3A 85
E Quadratictrinomialfactorisation 184
Reviewset3B 86
F Miscellaneousfactorisation 186
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TABLEOFCONTENTS 7
G Factorisationof axX(cid:10)(cid:11)(cid:10)bx(cid:10)(cid:11)(cid:10)c,(cid:10)(cid:10)a(cid:10)(cid:10)(cid:12)(cid:10)(cid:13) 186 14 COMPARINGNUMERICALDATA 303
Reviewset8A 191 A Graphicalcomparison 304
Reviewset8B 191 B Parallelboxplots 306
C Astatisticalproject 311
9 STATISTICS 193
Reviewset14A 312
A Discretenumericaldata 195 Reviewset14B 313
B Continuousnumericaldata 199
C Measuringthemiddleofadataset 201 15 TRANSFORMATIONGEOMETRY 315
D Measuringthespreadofdata 206 A Translations 318
E Box-and-whiskerplots 209 B Rotations 320
F Groupedcontinuousdata 212 C Reflections 324
G Cumulativedata 214 D Enlargementsandreductions 329
Reviewset9A 217 E Tessellations 333
Reviewset9B 217 Reviewset15A 337
Reviewset15B 338
10 PROBABILITY 219
A Experimentalprobability 221 16 QUADRATICEQUATIONS 339
B Probabilitiesfromdata 222 A Quadraticequationsoftheform xX(cid:10)(cid:14)(cid:10)k 341
C Lifetables 224
B TheNullFactorlaw 342
D Samplespaces 226
C Solutionbyfactorisation 343
E Theoreticalprobability 227
D Completingthesquare 346
F Using2-dimensionalgrids 229
E Problemsolving 349
G Compoundevents 230
Reviewset16A 351
H EventsandVenndiagrams 233
Reviewset16B 352
I Expectation 237
Reviewset10A 239 17 SIMULTANEOUSEQUATIONS 353
Reviewset10B 240
A Linearsimultaneousequations 354
11 FINANCIALMATHEMATICS 241 B Problemsolving 358
C Non-linearsimultaneousequations 362
A Businesscalculations 242
Reviewset17A 365
B Appreciation 248
Reviewset17B 365
C Compoundinterest 250
D Depreciation 255 18 MATRICES 367
E Borrowing 258
A Matrixsizeandconstruction 368
Reviewset11A 265
B Matrixequality 371
Reviewset11B 265
C Additionandsubtractionofmatrices 372
12 TRIGONOMETRY 267 D Scalarmultiplication 375
E Matrixmultiplication 376
A Usingscalediagrams 268
F Matricesusingtechnology 378
B Labellingtriangles 269
Reviewset18A 380
C Thetrigonometricratios 270
Reviewset18B 381
D Trigonometricproblemsolving 275
E Bearings 279 19 QUADRATICFUNCTIONS 383
F 3-dimensionalproblemsolving 282
A Quadraticfunctions 384
Reviewset12A 285
B Graphsofquadraticfunctions 387
Reviewset12B 286
C Usingtransformationstosketchquadratics391
13 FORMULAE 289 D Graphingbycompletingthesquare 393
E Axesintercepts 394
A Substitutingintoformulae 290
F Quadraticgraphs 397
B Rearrangingformulae 293
G Maximumandminimumvalues
C Constructingformulae 295
ofquadratics 399
D Formulaebyinduction 298
Reviewset19A 401
Reviewset13A 301
Reviewset19B 402
Reviewset13B 302
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8 TABLEOFCONTENTS
20 TREEDIAGRAMSAND 25 NON-RIGHTANGLEDTRIANGLE
BINOMIALPROBABILITIES 403 TRIGONOMETRY 501
A Samplespacesusingtreediagrams 404 A Theunitquartercircle 502
B Probabilitiesfromtreediagrams 405 B Obtuseangles 505
C Binomialprobabilities 411 C Areaofatriangleusingsine 507
Reviewset20A 416 D Thesinerule 508
Reviewset20B 417 E Thecosinerule 512
F Problemsolvingwiththesineand
21 ALGEBRAICFRACTIONS 419 cosinerules 514
A Evaluatingalgebraicfractions 420 Reviewset25A 516
B Simplifyingalgebraicfractions 421 Reviewset25B 517
C Multiplyinganddividingalgebraic
26 VARIATION CD
fractions 427
D Addingandsubtractingalgebraic A Directvariation CD
fractions 429 B Inversevariation CD
E Morecomplicatedfractions 432 Reviewset26A CD
Reviewset21A 433 Reviewset26B CD
Reviewset21B 434
27 TWOVARIABLEANALYSIS CD
22 OTHERFUNCTIONS: THEIR
A Correlation CD
GRAPHSANDUSES 435 B Pearson’scorrelationcoefficient,r CD
A Exponentialfunctions 436 C Lineofbestfitbyeye CD
B Graphingsimpleexponentialfunctions 437 D Linearregression CD
C Growthproblems 440 Reviewset27A CD
D Decayproblems 442 Reviewset27B CD
E Simplerationalfunctions 444
F Optimisationwithrationalfunctions 447 28 LOGIC CD
G Unfamiliarfunctions 449 A Propositions CD
Reviewset22A 450 B Compoundstatements CD
Reviewset22B 451 C Constructingtruthtables CD
Reviewset28A CD
23 VECTORS 453 Reviewset28B CD
A Vectorrepresentation 455
B Lengthsofvectors 456 ANSWERS 523
C Equalvectors 458
INDEX 573
D Vectoraddition 459
E Multiplyingvectorsbyanumber 463
F Vectorsubtraction 465
G Thedirectionofavector 467
H Problemsolvingbyvectoraddition 469
Reviewset23A 471
Reviewset23B 472
24 DEDUCTIVEGEOMETRY 473
A Reviewoffactsandtheorems 475
B Circletheorems 479
C Congruenttriangles 485
D Similartriangles 488
E Problemsolvingwithsimilartriangles 492
F Themidpointtheorem 494
G Euler’srule 496
Reviewset24A 498
Reviewset24B 499
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Graphics calculator
instructions
Contents: A Basic calculations
B Basic functions
C Secondary function and
alpha keys
D Memory
E Lists
F Statistical graphs
G Working with functions
H Matrices
I Two variable analysis
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10 GRAPHICSCALCULATORINSTRUCTIONS
In this courseit is assumedthat you have a graphicscalculator. If you learnhow to operate
your calculator successfully, you should experience little difficulty with future arithmetic
calculations.
There are many different brands (and types) of calculators. Different calculatorsdo not have
exactly the same keys. It is thereforeimportantthat you have an instructionbooklet for your
calculator, and use it whenever you need to.
However,tohelpgetyoustarted,wehaveincludedheresomebasicinstructionsfortheTexas
Instruments TI-83 and the Casio fx-9860G calculators. Note that instructions given may
need to be modified slightly for other models.
GETTING STARTED
Texas Instruments TI-83
Thescreenwhichappearswhenthecalculatoristurnedonisthehomescreen. Thisiswhere
most basic calculations are performed.
You can return to this screen from any menu by pressing 2nd MODE .
Whenyouare onthisscreenyoucantypein anexpressionandevaluateitusingthe ENTER
key.
Casio fx-9860g
Press MENU to access the Main Menu, and select RUN¢MAT.
This is where most of the basic calculations are performed.
When you are on this screen you can type in an expression and evaluate it using the EXE
key.
A BASIC CALCULATIONS
Most modern calculatorshave the rules for Orderof Operationsbuilt into them. This order
is sometimes referred to as BEDMAS.
This section explains how to enter different types of numbers such as negative numbers and
fractions, and how to perform calculations using grouping symbols (brackets), powers, and
square roots. It also explains how to round off using your calculator.
NEGATIVE NUMBERS
To enter negative numbers we use the sign change key. On both the TI-83 and Casio this
looks like (¡) .
Simply press the sign change key and then type in the number.
For example, to enter ¡7, press (¡) 7.
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