Mathematics for the International Student: Year 7 IB MYP 2

496 Pages·12.302 MB·English

by  Pamela Vollmar et al.

HAESE HARRIS PUBLICATIONS
&
Specialists in mathematics publishing
Mathematics
for the international student
7
MYP 2
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
MATHEMATICS FOR THE INTERNATIONALSTUDENT 7 (MYP2)
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.haesemathematics.com.au
NationalLibraryofAustraliaCardNumber&ISBN978-1-876543-41-9
©Haese&HarrisPublications2008
PublishedbyRaksarNomineesPtyLtd
3FrankCollopyCourt,AdelaideAirport, SA5950,AUSTRALIA
FirstEdition 2008
Reprinted 2009(twice),2010,2011
CartoonartworkbyJohnMartin.ArtworkbyPiotrPoturajandDavidPurton.
CoverdesignbyPiotrPoturaj.
ComputersoftwarebyDavidPurton,ThomasJanssonandTroyCruickshank.
TypesetinAustraliabySusanHaese(RaksarNominees).TypesetinTimesRoman10\Qw_/11\Qw_
ThetextbookanditsaccompanyingCDhavebeendevelopedindependentlyoftheInternational
BaccalaureateOrganization(IBO).ThetextbookandCDareinnowayconnectedwith,orendorsed
by,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
apologise for any accidental infringement where copyright has proved untraceable. They would be
pleasedtocometoasuitableagreementwiththerightfulowner.
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 7th Grade (or Year 7) level in classes where
studentsareexpectedtocompletearigorouscourseinMathematics.ItisthesecondbookinourMiddle
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.
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.haesemathematics.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. ©2011
Byclickingontherelevanticon,arangeofnewinteractivefeatures
canbeaccessed:
(cid:2) SelfTutor
INTERACTIVE
(cid:2) AreasofInteraction linkstoprintablepages
LINK
(cid:2) InteractiveLinks–tospreadsheets,graphingandgeometry
software,computerdemonstrationsandsimulations
N
E
W!
SELF TUTOR is a new exciting feature of this book.
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 5cm Self Tutor
Find the lengths of the unknown sides
and hence calculate the perimeter of the 7cm
figure:
1cm
15cm
5cm
We use the known lengths to calculate
the other side lengths:
7cm (cid:10)(cid:11)(cid:12)(cid:13)(cid:14)cm
Now P =5+8+15+1+10+7 cm (cid:12)(cid:15)(cid:16)(cid:15)(cid:13)(cid:12)(cid:17)cm
) P =46 cm 1cm
15cm
SeeChapter9,Lengthandarea,p.181
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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
subject areas in the curriculum and to promote an understanding of the interrelatedness of
different branches of knowledge and the coherence of
knowledgeasawhole.
Clickontheheadingto
accessaprintable‘pop-up’ In an effort to assist busy teachers, we offer the following
versionofthelink. printablepagesofideasforprojectsandinvestigations:
STAINED GLASS WINDOWS
LINKS Areasofinteraction:
clickhere Humaningenuity,Approachestolearning
Linkstoprintablepagesofideasforprojectsandinvestigations
Chapter2: Angles,linesand STAINEDGLASSWINDOWS
parallelism p.53 Humaningenuity,Approachestolearning
Chapter3: Propertiesofnumbers MATCHSTICKMATHEMATICS
p.74 Approachestolearning
Chapter6: Decimalnumbers p.133 LEAPYEARS Humaningenuity,Environment
Chapter7: Percentage p.155 ELECTIONS Approachestolearning
Chapter8: Algebra:Expressions BARYCENTRESINSPACE
andevaluation p.170 Humaningenuity
Chapter9: Lengthandarea p.195 POPULATIONDENSITY Healthandsocialeducation
Chapter11:Furthermeasurement HOWMUCHWATERISLOSTWHENATAPISLEFT
p.230 DRIPPING? Environments,CommunityandService
Chapter17:Linegraphs HOWARETAXIFARESCALCULATED?
p.347 Humaningenuity,Approachestolearning
Chapter18:Circles p.364 FLAGRATIOS Humaningenuity
Chapter22:Rates HOWMUCHOXYGENDOESAPERSONNEED?
p.439 Environments,Healthandsocialeducation
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6 TABLEOFCONTENTS
TABLE OF CONTENTS
1 WHOLE NUMBERS 9 5 ALGEBRA: PATTERNS AND
A Thenumbersystem 11 MODELS 95
B Roundingnumbers 12 A Geometricpatterns 96
C Estimation 14 B Numbercrunchingmachines 100
D Operatingwithnumbers 15 C Substitutingintoformulae 104
E Indexorexponentnotation 20 D Usingpatterns 106
F Squaresandcubes 22 E Practicalproblems 109
G Orderofoperations 24 F Numbersequences 111
Reviewset1A 28 Reviewset5A 112
Reviewset1B 29 Reviewset5B 113
2 ANGLES, LINES AND 6 DECIMAL NUMBERS 115
PARALLELISM 31
A Placevalue 116
A Pointsandlines 32 B Orderingdecimalnumbers 119
B Measuringandclassifyingangles 35 C Addingandsubtractingdecimal
C Angleproperties 38 numbers 121
D Geometricconstruction 41 D Multiplyinganddividingby
E Anglepairs 46 powersof10 122
F Parallellines 49 E Multiplyingdecimalnumbers 125
Reviewset2A 53 F Dividingdecimalnumbers 126
Reviewset2B 55 G Terminatingandrecurringdecimals 128
H Decimalapproximations 131
3 PROPERTIES OF NUMBERS 57 I Comparingsizes 133
A Divisibilitytests 58 Reviewset6A 134
B Factorsofnaturalnumbers 61 Reviewset6B 134
C Multiplesofnaturalnumbers 65
7 PERCENTAGE 135
D Directednumbers 68
E Rootsofwholenumbers 72 A Understandingpercentages 136
Reviewset3A 74 B Interchangingnumberforms 138
Reviewset3B 75 C Onequantityasapercentage
ofanother 141
4 FRACTIONS 77 D Findingpercentagesofquantities 143
A Manipulatingfractions 78 E Theunitarymethodinpercentage 144
B Operationswithfractions 82 F Percentageincreaseordecrease 145
C Problemsolving 88 G Findingapercentagechange 147
D Theunitarymethodwithfractions 90 H Businessapplications 149
E Squarerootsoffractions 91 I Simpleinterest 152
Reviewset4A 93 Reviewset7A 155
Reviewset4B 94 Reviewset7B 155
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TABLEOFCONTENTS 7
8 ALGEBRA: EXPRESSIONS AND G Gradientorslope 253
EVALUATION 157 Reviewset12A 254
A Buildingexpressions 158 Reviewset12B 255
B Keywordsinalgebra 162
13 EQUATIONS 257
C Simplifyingexpressions 164
D Algebraicproducts 166 A Equations 258
E Evaluatingalgebraicexpressions 168 B Maintainingbalance 260
Reviewset8A 171 C Inverseoperations 261
Reviewset8B 171 D Buildingandundoingexpressions 263
E Solvingequations 265
9 LENGTH AND AREA 173 F Equationswitharepeated
A Length 175 unknown 267
B Perimeter 178 Reviewset13A 270
C Area 183 Reviewset13B 271
D Areasofpolygons 186
14 POLYGONS 273
E Areasofcompositeshapes 192
Reviewset9A 195 A Classifyingtriangles 274
Reviewset9B 196 B Anglesofatriangle 275
C Anglesofisoscelestriangles 279
10 ALGEBRA (EXPANSION AND D Polygons 281
FACTORISATION) 199 E Quadrilaterals 284
A Thedistributivelaw 200 F Anglesofaquadrilateral 286
B Simplifyingalgebraicexpressions 203 G Interioranglesofpolygons 288
C Bracketswithnegativecoefficients 204 H Deductivegeometry(Extension) 290
D Theproduct (a(cid:2)b)(c(cid:2)d) 205 Reviewset14A 292
E Geometricapplications 206 Reviewset14B 293
F Factorisationofalgebraic
15 THE GEOMETRY OF SOLIDS 295
expressions 207
Reviewset10A 209 A Solids 296
Reviewset10B 210 B Netsofsolids 300
C Drawingrectangularsolids 305
11 FURTHER MEASUREMENT 211 D Constructingblocksolids 309
A Volume 212 Reviewset15A 312
B Volumeformulae 214 Reviewset15B 313
C Capacity 217
16 PROBLEM SOLVING 315
D Mass 221
E Time 224 A Writingequationsusingsymbols 316
Reviewset11A 230 B Problemsolvingwithalgebra 318
Reviewset11B 231 C Measurementproblems 320
D Moneyproblems 321
12 RATIO AND PROPORTION 233 E Miscellaneousproblemsolving 322
A Ratio 234 F Problemsolvingbysearch 324
B Writingratiosasfractions 236 G Problemsolvingbyworking
C Equalratios 237 backwards 326
D Proportions 242 Reviewset16A 328
E Usingratiostodividequantities 245 Reviewset16B 329
F Scalediagrams 247
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8 TABLEOFCONTENTS
17 LINE GRAPHS 331 Reviewset21A 423
A Propertiesoflinegraphs 332 Reviewset21B 424
B Estimatingfromlinegraphs 334
22 RATES 427
C Conversiongraphs 337
D Travelgraphs 339 A Rates 428
E Continuousanddiscretegraphs 341 B Comparingprices 429
F Graphinglinearrelationships 344 C Usingrates 431
Reviewset17A 348 D Averagespeed 433
Reviewset17B 349 E Density 435
F Convertingrates 437
18 CIRCLES 351 Reviewset22A 439
A Partsofacircle 352 Reviewset22B 440
B Circumference 355
23 ALGEBRAIC FRACTIONS 443
C Areaofacircle 358
D Cylinders 362 A Simplifyingalgebraicfractions 444
Reviewset18A 364 B Multiplyingalgebraicfractions 445
Reviewset18B 365 C Dividingalgebraicfractions 446
D Addingandsubtractingalgebraic
19 CHANCE 367 fractions 447
A Describingchance 368 Reviewset23A 452
B Assigningnumberstochance 370 Reviewset23B 453
C Experimentalprobability 371
CHALLENGE SETS 454
D Listingpossibleoutcomes 375
E Theoreticalprobability 376
ANSWERS 455
F Treediagrams 380
G Makingprobabilitygenerators 384
INDEX 493
Reviewset19A 385
Reviewset19B 385
20 STATISTICS 387
A Datacollection 390
B Categoricaldata 392
C Numericaldata 397
D Themean,medianandmode 401
Reviewset20A 404
Reviewset20B 406
21 SETS 407
A Sets 408
B Complementofaset 410
C Intersectionandunion 412
D Disjointsets 414
E Venndiagrams 416
F Problemsolvingwith
Venndiagrams 419
G Findingprobabilitiesfrom
Venndiagrams 421
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1
Chapter
Whole numbers
Contents: A The number system
B Rounding numbers
C Estimation
D Operating with numbers
E Index or exponent notation
F Squares and cubes
G Order of operations
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10 WHOLENUMBERS (Chapter1)
OPENING PROBLEM
A sixteenstoreyhotel with floors G, 1, 2, 3, ....., 15 has no accommodation
on the ground floor. On the even numbered floors (2, 4, 6, ......) there are
28 guest rooms. On the odd numbered floors there are 25 guest rooms.
Room cleaners work for four hours each day,
during which time each cleaner can clean 12
guest rooms. Each cleaner is paid at a rate of
E16 per hour.
Consider the following questions:
a How many floors are odd numbered?
b Intotal,howmanyguestroomsareonall
the odd numbered floors?
c If each guest room has three chairs, how
many chairs are on each even numbered
floor?
d How many guest rooms are in the hotel?
e How many cleaners are required to clean all guest rooms assuming the hotel was
‘full’ the previous night?
f What is the total cost of hiring the cleaners to clean the guest rooms of the hotel?
All over the world, people use numbers. They are a vital part of our lives, and have been
importanttohumansforthousandsofyears. Weneedtounderstandthepropertiesofnumbers
and the operations between them.
Over the ages, different people have created their own number systems to help them count.
The Ancient Egyptians, Romans, and Greeks all used different symbols for their numbers,
and helped to developed the more efficient systems we use today.
There are still many number systems in use around the world, but the most common is the
Hindu-Arabic system used in this course. An early form of this system was established in
ancient India around 3000 BC, and the first of the modern characters was developed about
2000 years ago. Use of the system slowly spread westwards, and in the 7th century AD it
was adopted by the Arabs.
RESEARCH HISTORY OF THE HINDU-ARABIC SYSTEM
Divide your class into small groups. Each group should write a report on
a particular aspect of the history of the Hindu-Arabic system. Topics you
should include are:
² the Indus valley civilization ² Brahmi numerals
² the Bakhshali manuscript ² Aryabhata
² Al-Uqlidisi ² the Codex Vigilanus
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