Mathematics - Mathematics Pre-Diploma Studies SL (MYP 5 Plus) - First Edition

HAESE & HARRIS PUBLICATIONS

Specialists in mathematics publishing

Mathematics

for the international student

Pre-Diploma Studies SL (MYP 5)

Presumed Knowledge for Mathematical Studies SL

Keith Black
Pamela Vollmar

Michael Haese
Robert Haese
Sandra Haese

Mark Humphries

for use with
IB Middle Years

Programme

MATHEMATICS FOR THE INTERNATIONAL STUDENT

Pre-Diploma Studies SL (MYP 5)
Presumed Knowledge for Mathematical Studies SL

Keith Black B.Sc.(Hons.), Dip.Ed.
Pamela Vollmar B.Sc.(Hons.), PGCE.
Michael Haese B.Sc.(Hons.), Ph.D.
Robert Haese B.Sc.
Sandra Haese B.Sc.
Mark Humphries B.Sc.(Hons.)

Haese & Harris Publications
3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA
Telephone: +61 8 8355 9444, Fax: + 61 8 8355 9471
Email: [email protected]
Web: www.haeseandharris.com.au

National Library of Australia Card Number & ISBN 978-1-876543-10-5

© Haese & Harris Publications 2008

Published by Raksar Nominees Pty Ltd
3 Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA

First Edition 2008

Cartoon artwork by John Martin. Artwork by Piotr Poturaj and David Purton.
Cover design by Piotr Poturaj.
Computer software by David Purton and Thomas Jansson.

Typeset in Australia by Susan Haese (Raksar Nominees). Typeset in Times Roman 10\Qw_ /11\_Qw

The textbook and its accompanying CD have been developed independently of the International
Baccalaureate Organization (IBO). The textbook and CD are in no way connected with, or endorsed by,
the IBO.

This book is copyright. Except as permitted by the Copyright Act (any fair dealing for the purposes of
private study, research, criticism or review), no part of this publication may be reproduced, stored in a
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 &
Harris Publications.

Copying for educational purposes: Where copies of part or the whole of the book are made under Part
VB of the Copyright Act, the law requires that the educational institution or the body that administers it
has given a remuneration notice to Copyright Agency Limited (CAL). For information, contact the
Copyright Agency Limited.

Acknowledgements: The publishers acknowledge the cooperation of Oxford University Press, Australia,
for the reproduction of material originally published in textbooks produced in association with
Haese & Harris Publications.

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 pleased
to come to a suitable agreement with the rightful owner.

Disclaimer: All the internet addresses (URL’s) given in this book were valid at the time of printing.
While the authors and publisher regret any inconvenience that changes of address may cause readers, no
responsibility for any such changes can be accepted by either the authors or the publisher.

FOREWORD

Pre-Diploma Studies SL (MYP 5) is an attempt to cover, in one volume, the Presumed Knowledge
required for the IB Diploma course “Mathematical Studies SL” as well as including some extension topics.
It may also be used as a general textbook at about Grade 10 level in classes where students might be
expected to embark on an “Applications” type of Mathematics course in their final two years of high
school.

In terms of the IB Middle Years Programme (MYP), this book does not pretend to be a definitive course. In
response to requests from teachers who use “Mathematics for the International Student” at Diploma level,
we have endeavoured to interpret their requirements, as expressed to us, for a book that would prepare
students for Mathematical Studies SL at Diploma level. We have developed the book independently of the
International Baccalaureate Organization (IBO) in consultation with experienced teachers of IB
Mathematics. The text is not endorsed by the IBO.

It is not our intention that each chapter be worked through in full. Time constraints may 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 content as possible.

To avoid producing a book that would be too bulky for students, we have presented these chapters on the
CD as printable pages:

Chapter 25: Transformation geometry

Chapter 26: Sine and cosine rules

The above were selected because the content could be regarded as extension beyond what might be
regarded as an essential prerequisite for Diploma.

This package is language rich and technology rich. We hope the combination of textbook and interactive
Student CD will foster the mathematical development of students in a stimulating way. 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 intended to help students
who have been absent from classes or who experience difficulty understanding the material.

The book contains many problems from the basic to the advanced, to cater for a range of student abilities
and interests. While some of the exercises are simply designed to build skills, every effort has been made
to contextualise problems, so that students can see everyday uses and practical applications of the
mathematics they are studying, and appreciate the universality of mathematics. We understand the
emphasis that the IB MYP places on the five Areas of Interaction and in response there are links on the CD
to printable pages which offer ideas for projects and investigations to help busy teachers (see p. 8).

The interactive CD also allows immediate access to our own specially designed geometry packages,
graphing packages and more.

In this changing world of mathematics education, we believe that the contextual approach shown in this
book, with the associated use of technology, will enhance the students’ understanding, knowledge and
appreciation of mathematics, and its universal application.

We welcome your feedback. Email: [email protected]

Web: www.haeseandharris.com.au

KB, PV, PMH, RCH, SHH, MH

Acknowledgements

The authors and publishers would like to thank all those teachers who have read proofs and offered advice
and encouragement.

Among those who submitted courses of study for Middle Years 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, Alessandra Pecoraro, Jeanne-Mari Neefs, Ray Wiens, John Bush, Jane Forrest, Dr Andrzej
Cichy, 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, Vivienne
Verschuren, Mark Willis, Curtis Wood, Ufuk Genc, Fran O'Connor. Special thanks to Heather Farish. To
anyone we may have missed, we offer our apologies.

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.

USING THE INTERACTIVE CD INTERACTIVE
LINK
The interactive CD is ideal for independent study.

Students can revisit concepts taught in class and undertake their own
revision and practice. The CD also has the text of the book, allowing
students to leave the textbook at school and keep the CD at home.

By clicking on the relevant icon, a range of new interactive features
can be accessed:

t Self Tutor
t Areas of Interaction links to printable pages
t Printable Chapters
t Interactive Links – to spreadsheets, video clips, graphing and

geometry software, computer demonstrations and simulations

SELF TUTOR is a new exciting feature of this book. NEW!

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 move-
ment and colour on the screen.

Ideal for students who have missed lessons or need extra help.

Example 8 Self Tutor

Solve for x: 4x + 3 = ¡2
5

4x + 3 = ¡2
5

) 5 £ (4x + 3) = ¡2 £ 5 fmultiply both sides by 5g
5
fsimplifyg
) 4x + 3 = ¡10 fsubtract 3 from both sidesg
fsimplifyg
) 4x + 3 ¡ 3 = ¡10 ¡ 3
fdivide both sides by 4g
) 4x = ¡13
fsimplifyg
) 4x = ¡ 13
4 4

) x = ¡3 1
4

AREAS OF INTERACTION

The International Baccalaureate Middle Years Programme focuses teaching and learning
through fiveAreas of Interaction:

t Approaches to learning t Environments
t Community and service t Health and social education
t Human ingenuity

The Areas of Interaction are intended as a focus for developing connections between different
subject areas in the curriculum and to promote an understanding of the interrelatedness of
different branches of knowledge and the coherence of knowledge as a whole.

In an effort to assist busy teachers, we offer the following printable pages of ideas for projects and
investigations:

Click on the heading to
access a printable ‘pop-up’

version of the link.

LINKS ERRORS IN MEASUREMENT

click here Areas of interaction:
Approaches to learning

Links to printable pages of ideas for projects and investigations

Chapter 1: Measurement and units ERRORS IN MEASUREMENT
p. 39 Approaches to learning

Chapter 4: Rounding and estimating AREA AND VOLUME ERRORS
p. 88 Approaches to learning

Chapter 5: The Rule of Pythagoras PYTHAGORAS
p. 99 Human ingenuity

Chapter 7: Length and area ESTIMATING AREAS
p. 137 Approaches to learning/Human ingenuity

Chapter 10: Statistics HOW MANY TROUT ARE IN THE LAKE?
Environments/Human ingenuity
p. 217
ALL THAT GLITTERS IS NOT GOLD
Chapter 12: Ratios and rates Human ingenuity
p. 252
HOW WIDE IS THE CANAL?
Chapter 14: Congruence and Human ingenuity

similarity p. 294 HOW FAR AWAY IS THE MOON AND HOW LARGE IS IT?
Human ingenuity
Chapter 16: Trigonometry
p. 326 WHAT ARE YOUR SURVIVAL PROSPECTS?
Environments/Health and social education
Chapter 19: Probability
WHAT REGION CAN BE EATEN BY A GOAT?
p. 392 Approaches to learning/Environments

Chapter 21: Geometry HOW MUCH CAN I SAVE BY NOT SMOKING?
Environments/Health and social education
p. 434
WHAT IS THE STRONGEST ARCH?
Chapter 23: Finance Approaches to learning/Environments/Human ingenuity

p. 472

Chapter 24: Quadratic functions
p. 485

6 TABLE OF CONTENTS

TABLE OF CONTENTS

GRAPHICS CALCULATOR 5 THE RULE OF PYTHAGORAS 97

INSTRUCTIONS 9 A The Rule of Pythagoras (Review) 99

A Basic calculations 10 B Further problem solving 103
B Basic functions 12 C Testing for right angles 106
C Secondary function and alpha keys 15 D Navigation 107
D Memory 15 Review set 5A 109
E Lists 18 Review set 5B 110

F Statistical graphs 20 111
G Working with functions 21 6 ALGEBRA 112

A Changing words into symbols

1 MEASUREMENT AND UNITS 25 B Generalising arithmetic 113

A Standard units 26 C Converting into algebraic form 115
B Converting units 29 D Formula construction 116
C Area units 32 E Number patterns and rules 118
D Volume units 33 F The value of an expression 120
E Capacity 34 Review set 6A 124
F Mass 35 Review set 6B 125

G Time 36 127
H 24-hour time 39 7 LENGTH AND AREA
40 A Perimeter and length 128
Review set 1A 41 B Area 133
Review set 1B 139
C Surface area

2 NUMBER OPERATIONS 43 D Problem solving 145

A Operations with integers 44 Review set 7A 147
B Operations with fractions 49 Review set 7B 148

C Index notation 53
D Laws of indices 57 8 DECIMALS AND PERCENTAGE 149

Review set 2A 59 A Decimal numbers 150
Review set 2B
60 B Percentage 152

C Working with percentages 154

3 SETS, SEQUENCES AND LOGIC 61 D Unitary method in percentage 156

A Set notation 62 E Percentage increase and decrease 157
B Important number sets 63 F Scientific notation (Standard form) 159
C Constructing sets (Interval notation) 66 Review set 8A 163
D Venn diagrams 67 Review set 8B 164

E Union and intersection 69
F Simple set problems 72 9 ALGEBRAIC SIMPLIFICATION
G Number sequences
73 AND EXPANSION 167

H Introduction to logic 75 A Collecting like terms 168

Review set 3A 79 B Product notation 170

Review set 3B 80 C The distributive law 172

4 ROUNDING AND ESTIMATION 81 D The expansion of¡ (a+b)(c+d) 177
E The expansion rules
A Rounding numbers 82 F Perimeters and areas 179
183
B Rounding money 83 Review set 9A 185
185
C One figure approximations 86 Review set 9B

D Rounding decimal numbers 88

E Using a calculator to round off 90 10 STATISTICS 187
F Significant figure rounding 92 A Terminology for the study of statistics 189
G Rounding time 94 B Quantitative (numerical) data
95 C Grouped discrete data 194
Review set 4A
Review set 4B 96 197