Mathematics for the International Student: Pre-Diploma SL and HL

608 Pages·2008·11.576 MB·English

by  Pamela Vollmar et al.

HAESE HARRIS PUBLICATIONS
&
Specialists in mathematics publishing
Mathematics
for the international student
Pre-Diploma SL and HL (MYP 5 Plus)
second edition
Presumed Knowledge for SL and HL courses
Pamela Vollmar
Edward Kemp
Michael Haese
Robert Haese
Sandra Haese
Mark Humphries
Chris Sangwin
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
MATHEMATICSFORTHEINTERNATIONALSTUDENT
Pre-DiplomaSLandHL(MYP5Plus)secondedition
PresumedKnowledgeforSLandHLcourses
PamelaVollmar B.Sc.(Hons.),PGCE.
EdwardKemp B.Sc.,M.A.
MichaelHaese B.Sc.(Hons.),Ph.D.
RobertHaese B.Sc.
SandraHaese B.Sc.
MarkHumphries B.Sc.(Hons.)
ChrisSangwin M.A.,M.Sc.,Ph.D.
HaeseMathematics
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
Telephone: [phone], Fax: [phone]
Email: [email protected]
Web: www.haesemathematics.com.au
NationalLibraryofAustraliaCardNumber&ISBN 978-1-876543-89-1
©Haese&HarrisPublications2008
PublishedbyRaksarNomineesPtyLtd
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
FirstEdition 2006
SecondEdition 2008 Reprinted 2008,2010,2011,2012
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,
recording or otherwise, without the prior permission of the publisher. Enquiries to be made to Haese
Mathematics.
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 (URLs) given in this book were valid at the time of printing.
Whiletheauthorsandpublisherregretanyinconveniencethatchangesofaddressmaycausereaders,no
responsibilityforanysuchchangescanbeacceptedbyeithertheauthorsorthepublisher.
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FOREWORD
Pre-DiplomaSLandHL(MYP5Plus)secondeditionisanattempttocover,inonevolume,thePresumed
Knowledge required for the IB Diploma courses ‘Mathematics SL’and ‘Mathematics HL’. This book may
also be used as a general textbook at about 10th Grade level in classes where students complete a rigorous
courseinpreparationforthestudyofmathematicsatahighlevelintheirfinaltwoyearsofhighschool.
Feedback from teachers using the first edition suggested that while it provided satisfactory preparation for
prospective Mathematics SL students, several sections needed to be more rigorous to prepare students
thoroughlyforMathematicsHL.Thefirsteditionhasbeenrevisedthroughoutandthehighlightedtopicsin
thetableofcontentsshowataglancethemainareasthathavebeensubstantiallyrevisedandextended.
IntermsoftheIBMiddleYearsProgramme(MYP),thisbookdoesnotpretendtobeadefinitivecourse.In
responsetorequestsfromteacherswhouse‘MathematicsfortheInternationalStudent’atIBDiplomalevel,
wehaveendeavouredtointerprettheirrequirements,asexpressedtous,forabookthatwillpreparestudents
for Mathematics SLand Mathematics HL.We have developed this book independently of the International
Baccalaureate Organization (IBO) in consultation with experienced teachers of IB Mathematics.The text is
notendorsedbytheIBO.
Itisnotourintentionthateachchapterbeworkedthroughinfull.Teachersmustselectcarefully,accordingto
the abilities and prior knowledge of their students, to make the most efficient use of time and give as
thoroughcoverageofcontentaspossible.
ThreeadditionalchaptersappearontheCDasprintablepages:
Chapter23:Countingandprobability
Chapter24:Locus
Chapter25:Networks
Thesechapterswereselectedbecausethecontentcouldberegardedasextensionbeyondwhatmightbeseen
asanessentialprerequisiteforIBDiplomamathematics.
WeunderstandtheemphasisthattheIBMYPplacesonthefiveAreasofInteractionandinresponsethere
arelinksontheCDtoprintablepageswhichofferideasforprojectsandinvestigationstohelpbusyteachers
(seep.5).
Frequent use of the interactive features on the CD should nurture a much deeper understanding and
appreciationofmathematicalconcepts.Theinclusionofournew Self Tutor software(seep.4)isintendedto
helpstudentswhohavebeenabsentfromclassesorwhoexperiencedifficultyunderstandingthematerial.
Thebookcontainsmanyproblemstocaterforarangeofstudentabilitiesandinterests,andeffortshavebeen
madetocontextualiseproblemssothatstudentscanseethepracticalapplicationsofthemathematicstheyare
studying.
Wewelcomeyourfeedback. Email:[email protected]
Web:www.haesemathematics.com.au
PV,EK,PMH,RCH,SHH,MH,CS
Acknowledgements
The authors and publishers would like to thank all those teachers who have read proofs and offered advice
andencouragement.
Among those who submitted courses of study for MiddleYears Mathematics and who offered to read and
comment on the proofs of the textbook are: Margie Karbassioun, Kerstin Mockrish, Todd Sharpe, Tamara
Jannink, Yang Zhaohui, Cameron Hall, Brendan Watson, Daniel Fosbenner, Rob DeAbreu, Philip E.
Hedemann,AlessandraPecoraro,Jeanne-MariNeefs,RayWiens,JohnBush,JaneForrest,DrAndrzejCichy,
William Larson, Wendy Farden, Chris Wieland, Kenneth Capp, Sara Locke, Rae Deeley, Val Frost, 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. Special thanks to
HeatherFarish.Toanyonewemayhavemissed,weofferourapologies.
Thepublisherswishtomakeitclearthatacknowledgingtheseindividualsdoesnotimplyanyendorsementof
thisbookbyanyofthem,andallresponsibilityforthecontentrestswiththeauthorsandpublishers.
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USING THE INTERACTIVE CD
TSrsetthuvuedidseieinnonttnetssrtaaconcaldtneivapreveraevCcitDsthiicetiescte.oidxTnethcbaeeolpoCftoksDratiatanusldsgceohhpotheoiannlsdactenhlnadetskstseteauxendtpdyot.fhuenthdCeeDrbtoaaoktkeh,otahmleleior.woiwnng teprashc(cid:129)noArisenaestxofeInteracssMMtioeenin(cid:129)cccsYYltoouatdinnsetSPPiMMcsddsepl55aeefck ddaTPPgiiuesaattlltiiuu(cid:129)ooognnIIrssrNNattpTThiEEnhhRRgAAanCCdTTeegIIeVVoEEmmmeSStrTTyUUsoDDftEEwaaNNareTT(cid:129)CCiDDttPPnSSstrrruLLiiceetio--ccnaaDDsfonnssriigrppddaphlli©csooHHcal2cmmul0ato1LLrs1aa(cid:129)demsonnoisttralatuiominss(cid:129)
Byclickingontherelevanticon,arangeofnewinteractivefeatures ffoorr uussee wwiitthh tthhee
IIBB MMiiddddlleeYYeeaarrssPPrrooggrraammmmee
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 7 Self Tutor
Sketch each of the following functions on the same set of axes as y = x2. In each
case state the coordinates of the vertex.
a y =(x¡2)2+3 b y =(x+2)2¡5
a We draw y =x2 and translate it b We draw y =x2 and translate it
µ ¶ µ ¶
2 ¡2
by . by .
3 ¡5
y y
@\=\(!\-\2)X\+\3
@\=\!X
@\=\!X (cid:10)2 x
(cid:11)3 x @\=\(!\+\2)X\-\5 (cid:10)5
(cid:11)2
The vertex is at (2, 3). The vertex is at (¡2, ¡5).
SeeChapter17,Quadraticfunctions,p.421
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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:
SATISFYING PAPER PROPORTIONS
LINKS Areasofinteraction:
clickhere Approachestolearning/Environments/Humaningenuity
Linkstoprintablepagesofideasforprojectsandinvestigations
Chapter3:Radicalsandsurds SATISFYING PAPER PROPORTIONS
p.77 Approachestolearning/Environments/Humaningenuity
Chapter5:Coordinategeometry WHERE DOES THE FIGHTER CROSS THE COAST?
p.130 Humaningenuity
Chapter6:Congruenceandsimilarity THE USE OF MODELLING
p.152 Approachestolearning
Chapter7:Transformationgeometry TRANSFORMING ART
p.167 Environments/Humaningenuity
Chapter8:Univariatedataanalysis DECODING A SECRET MESSAGE
p.209 Humaningenuity
Chapter9:Quadraticequations MINIMISING THE COSTS
p.231 Environments/Humaningenuity
Chapter10:Trigonometry WHERE ARE WE?
p.264 Approachestolearning/Humaningenuity
Chapter11:Probability WHAT ARE YOUR SURVIVAL PROSPECTS?
p.292 Communityservice/Healthandsocialeducation
Chapter13:Formulae HOW MUCH DO WE HAVE LEFT?
p.323 Humaningenuity
Chapter14:Relations,functionsand FIBONACCI
sequences p.353 Humaningenuity
Chapter16:Exponentialfunctions EARTHQUAKES
andlogarithms p.410 Environments/Humaningenuity
Chapter18:Advancedtrigonometry IN TUNE WITH TRIGONOMETRY
p.460 Humaningenuity
Chapter20:Matricesandlinear HILL CIPHERS
transformations p.504 Approachestolearning/Humaningenuity
Chapter22:Introductiontocalculus ARCHIMEDES’ NESTED CYLINDER, HEMISPHERE AND CONE
p.548 Approachestolearning/Humaningenuity
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6 TABLEOFCONTENTS
TABLE OF CONTENTS
Contenthasbeenrevisedthroughoutandthehighlighted
areasshowthetopicsthathavebeensubstantiallyrevised
andextendedinthissecondedition.
GRAPHICSCALCULATOR F Moredifficultproblems(Extension) 98
INSTRUCTIONS 9 Reviewset4A 100
Reviewset4B 101
A Basiccalculations 10
B Basicfunctions 12
5 COORDINATEGEOMETRY 103
C Secondaryfunctionandalphakeys 15
D Memory 15 A Distancebetweentwopoints 105
E Lists 18 B Midpoints 108
F Statisticalgraphs 20 C Gradient(orslope) 110
G Workingwithfunctions 21 D Usingcoordinategeometry 116
E Equationsofstraightlines 118
1 SETSANDVENNDIAGRAMS 29 F Distancefromapointtoaline 127
G 3-dimensionalcoordinategeometry
A Numbersets 30
(Extension) 129
B Intervalnotation 32
Reviewset5A 130
C Venndiagrams 33
Reviewset5B 131
D Unionandintersection 36
E ProblemsolvingwithVenndiagrams 40
6 CONGRUENCEAND
F Thealgebraofsets(Extension) 42
SIMILARITY 133
Reviewset1A 43
Reviewset1B 44 A Congruenceoffigures 134
B Constructingtriangles 135
2 ALGEBRAICEXPANSION C Congruenttriangles 137
ANDFACTORISATION 45 D Similarity 146
E Areasandvolumesofsimilarfigures 150
A Revisionofexpansionlaws 46
Reviewset6A 152
B Revisionoffactorisation 48
Reviewset6B 153
C Furtherexpansion 50
D Thebinomialexpansion 51
7 TRANSFORMATIONGEOMETRY 155
E Factorisingexpressionswithfourterms 54
F Factorisingquadratictrinomials 55 A Translations 157
G Factorisationbysplitting 57 B Reflections 158
H Miscellaneousfactorisation 60 C Rotations 160
Reviewset2A 61 D Dilations 162
Reviewset2B 62 Reviewset7A 167
Reviewset7B 168
3 RADICALSANDSURDS 63
8 UNIVARIATEDATAANALYSIS 169
A Basicoperationswithradicals 65
B Propertiesofradicals 67 A Statisticalterminology 171
C Multiplicationofradicals 70 B Quantitative(numerical)data 176
D Divisionbyradicals 72 C Groupeddiscretedata 179
E Equalityofsurds 74 D Continuousdata 181
Reviewset3A 77 E Measuringthecentre 184
Reviewset3B 78 F Cumulativedata 191
G Measuringthespread 194
4 PYTHAGORAS’THEOREM 79 H Box-and-whiskerplots 196
I Statisticsfromtechnology 200
A Pythagoras’theorem 81
J Standarddeviation 202
B TheconverseofPythagoras’theorem 85
K Thenormaldistribution 206
C ProblemsolvingusingPythagoras’theorem 88
Reviewset8A 209
D Circleproblems 93
Reviewset8B 211
E Three-dimensionalproblems 96
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TABLEOFCONTENTS 7
9 QUADRATICEQUATIONS 213 14 RELATIONS,FUNCTIONSAND
A Quadraticequationsoftheform xX(cid:12)(cid:13)(cid:12)k 215 SEQUENCES 325
B Solutionbyfactorisation 216 A Relationsandfunctions 326
C Completingthesquare 220 B Functions 329
D Problemsolving 222 C Functionnotation 331
E Thequadraticformula 227 D Compositefunctions 334
Reviewset9A 231 E Transforming y(cid:12)(cid:13)(cid:12)(cid:14)(x) 335
Reviewset9B 232 F Inversefunctions 337
G Themodulusfunction 340
10 TRIGONOMETRY 233 H Wherefunctionsmeet 343
A Trigonometricratios 235 I Numbersequences 344
B Trigonometricproblemsolving 240 J Recurrencerelationships 350
C 3-dimensionalproblemsolving 246 Reviewset14A 354
D Theunitcircle 250 Reviewset14B 355
E Areaofatriangleusingsine 252
15 VECTORS 357
F Thesinerule 255
G Thecosinerule 257 A Directedlinesegmentrepresentation 358
H Problemsolvingwiththesine B Vectorequality 360
andcosinerules 259 C Vectoraddition 361
I Trigonometricidentities(Extension) 261 D Vectorsubtraction 365
Reviewset10A 264 E Vectorsincomponentform 367
Reviewset10B 265 F Scalarmultiplication 371
G Vectorequations 373
11 PROBABILITY 267 H Parallelismofvectors 374
A Experimentalprobability 269 I Thescalarproductoftwovectors 376
B Probabilitiesfromtableddata 271 J Vectorproof(Extension) 380
C Representingcombinedevents 272 Reviewset15A 382
D Theoreticalprobability 274 Reviewset15B 384
E Compoundevents 277
16 EXPONENTIALFUNCTIONS
F Usingtreediagrams 280
G Samplingwithandwithoutreplacement 283 ANDLOGARITHMS 385
H Mutuallyexclusiveandnon-mutually A Indexlaws 386
exclusiveevents 285 B Rational(fractional)indices 389
I Venndiagramsandconditionalprobability 287 C Exponentialfunctions 391
Reviewset11A 292 D Growthanddecay 393
Reviewset11B 293 E Compoundinterest 395
F Depreciation 398
12 ALGEBRAICFRACTIONS 295
G Exponentialequations 400
A Simplifyingalgebraicfractions 296 H Expansionandfactorisation 401
B Multiplyinganddividingalgebraic I Logarithms 404
fractions 300 Reviewset16A 410
C Addingandsubtractingalgebraic Reviewset16B 411
fractions 302
D Morecomplicatedfractions 305 17 QUADRATICFUNCTIONS 413
Reviewset12A 307 A Quadraticfunctions 414
Reviewset12B 308 B Graphsofquadraticfunctions 416
C Axesintercepts 425
13 FORMULAE 309
D Axisofsymmetryandvertex 429
A Formulasubstitution 310 E Quadraticoptimisation 433
B Formularearrangement 313 Reviewset17A 435
C Formulaconstruction 315 Reviewset17B 436
D Formulaebyinduction 318
E Moredifficultrearrangements 320 18 ADVANCEDTRIGONOMETRY 437
Reviewset13A 323 A Radianmeasure 438
Reviewset13B 324 B Trigonometricratiosfromtheunitcircle 441
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8 TABLEOFCONTENTS
C Themultiplesof30ºand45º 444 23 COUNTINGANDPROBABILITY CD
D Graphingtrigonometricfunctions 448 A Theproductandsumprinciples CD
E Modellingwithsinefunctions 451 B Countingpermutations CD
F Trigonometricequations 454 C Factorialnotation CD
G Negativeandcomplementaryangle D Countingwithcombinations CD
formulae 457 E Probabilitiesusingpermutations
H Additionformulae 458 andcombinations CD
Reviewset18A 461 F Thehypergeometricdistribution CD
Reviewset18B 462 Reviewset23A CD
Reviewset23B CD
19 INEQUALITIES 463
A Signdiagrams 464 24 LOCUS CD
B Intervalnotation 468 A Locus CD
C Inequalities 471 B Circles CD
D Thearithmeticmean-geometricmean C Ellipses CD
inequality(Extension) 473 D Otherlocusproblems(Extension) CD
Reviewset19A 476 Reviewset24A CD
Reviewset19B 476 Reviewset24B CD
20 MATRICESANDLINEAR 25 NETWORKS CD
TRANSFORMATIONS 477
A Networkdiagrams CD
A Introductiontomatrices 478 B Isomorphismandadjacencymatrices CD
B Operationswithmatrices 480 C Directednetworks CD
C Matrixmultiplication 484 D Problemsolvingwithnetworks CD
D Thedeterminantofamatrix 487 Reviewset25A CD
E Multiplicativeidentityandinverse Reviewset25B CD
matrices 489
F Simultaneousequations 491 ANSWERS 555
G Lineartransformations 494
H Proofswith(cid:15)(cid:12)(cid:16)(cid:12)(cid:15)matrices(Extension) 503 INDEX 606
Reviewset20A 504
Reviewset20B 505
21 DEDUCTIVEGEOMETRY 507
A Circletheorems 509
B Furthercircletheorems 513
C Geometricproof 517
D Cyclicquadrilaterals 521
Reviewset21A 526
Reviewset21B 527
22 INTRODUCTIONTOCALCULUS 529
A Estimatinggradientsoftangentstocurves 530
B Gradientsusingquadratictheory 531
C Gradientsusinglimittheory 532
D Differentiation 535
E Optimisation 540
F Areasundercurves 543
G Integration 545
H Thedefiniteintegral 547
Reviewset22A 549
Reviewset22B 550
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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
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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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