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o Mastering
Mathematics
Second Edition
Geoff Buckwell
~
MACMlUAN
© Geoff Buckwell 1991, 1997
All rights reserved. No reproduction, copy or transmission of
this publication may be made without written permission.
No paragraph of this publication may be reproduced, copied or
transmitted save with written permission or in accordance with
the provisions of the Copyright, Designs and Patents Act 1988,
or under the terms of any licence permitting limited copying
issued by the Copyright Licensing Agency, 90 Tottenham Court
Road, London WIP 9HE.
Any person who does any unauthorised act in relation to
this publication may be liable to criminal prosecution and
civil claims for damages.
The author has asserted his right to be identified as the author
of this work in accordance with the Copyright, Designs and
Patents Act 1988.
First edition 1991
Reprinted four times
Second edition 1997
Published by
MACMILLAN PRESS LTD
Houndmills, Basingstoke, Hampshire RG21 6XS
and London
Companies and representatives
throughout the world
ISBN 978-0-333-66508-4 ISBN 978-1-349-14131-9 (eBook)
DOI 10.1007/978-1-349-14131-9
A catalogue record for this book is available
from the British Library.
This book is printed on paper suitable for recycling and
made from fully managed and sustained forest sources.
10 9 8 7 6 5 432 1
06 05 04 03 02 01 00 99 98 97
Copy-edited and typeset by Povey-Edmondson
Tavistock and Rochdale, England
Ocontents
Introduction IX
Acknowledgements X
1 Working with whole numbers 1
1.1 Number systems 1
1.2 The four ru1es of working with numbers 3
1.3 The use of brackets 8
1.4 Multiples and factors 10
1.5 Negative numbers 14
1.6 Other number bases 20
2 Working with fractions and decimals 27
2.1 U nderstanding fractions 27
2.2 Decima1 fractions 39
2.3 Decimal places and significant figures 41
2.4 Standard index or scientific notation 42
2.5 Working with decima1s 45
2.6 Squares and square roots 48
2.7 Approximation and the calculator 50
2.8 The metric system 51
2.9 Speed, distance, time 54
3 Introducing algebra 60
3.1 Basic definitions 60
3.2 Formulae and substitution 60
3.3 Forming expressions 62
3.4 Simplifying expressions 64
3.5 Equations (linear) 67
3.6 Problem solving with equations 72
3.7 Rearranging formulae (changing the subject) 73
4 Ratios and percentages 76
4.1 Percentages 76
4.2 Interest 83
4.3 Ratio and proportion 86
4.4 Changing units and currency conversion 90
5 Angle and shape 97
5.1 Definitions 97
5.2 Polygons 103
5.3 Symmetry of polygons 106
CONTENTS v
5.4 Properties of three and four sided shapes 108
5.5 Similarity and congruence 113
5.6 Tessellations 117
6 Length, area and volume 122
6.1 Perimeter of plane figures 122
6.2 Area 125
6.3 Nets and surface area 133
6.4 Volume 137
6.5 Similar shapes 142
7 Trigonometry 146
7.1 Tangent ratio 146
7.2 Sine ratio 150
7.3 Cosine ratio 154
7.4 Angle of elevation (or depression), bearings 158
7.5 Pythagoras' theorem 163
7.6 Three-dimensional problems 170
8 Geometry and construction 175
8.1 Geometrical constructions 175
8.2 Drawing three-dimensional shapes 182
8.3 Plans and elevations 183
8.4 Latitude and longitude 186
8.5 Locus 191
9 Statistics 194
9.1 Statistical data 194
9.2 Statistical diagrams 195
9.3 Statistical averages 203
9.4 Cumulative frequency 208
9.5 Scatter dia grams 214
10 More algebra 219
10.1 Fractional and negative indices 219
10.2 Algebraic fractions 222
10.3 Quadratic expressions 223
10.4 Number patterns 225
10.5 Factorisation 228
10.6 Quadratic equations 232
10.7 Trial and improvement 235
10.8 Problem solving 236
10.9 Further rearrangement of formulae 240
10.10 Simultaneous equations (linear) 244
10.11 Variation and proportion 246
10.12 Venn diagrams 249
11 Probability 254
11.1 Definition 254
11.2 Probability from graphs 257
vi CONTENTS
11.3 Using Venn diagrarns 258
11.4 Surn and produet rules 261
11.5 Complement 262
11.6 Tree diagrams 265
12 Coordinates and straight line graphs 272
12.1 Using straight line graphs 272
12.2 Coordinates and the equation of a straight line 273
12.3 y = mx + c 276
12.4 Simultaneous equations 278
12.5 Trave1 graphs 282
12.6 Experimentallaws 288
12.7 Inequalities 292
12.8 Errors 294
12.9 Regions and linear programming 295
13 Curves, functions and equations 300
13.1 Plotting eurves 300
13.2 Solving equations 302
13.3 The trapezium rule 305
13.4 Funetion notation 311
13.5 Flow charts 316
13.6 Iterative methods 318
14 The circle and further trigonometry 322
14.1 Geometry of the eircle 322
14.2 The sine rule 334
14.3 The eosine rule 339
14.4 Angles greater than 90° 343
14.5 Trigonometrie equations 344
15 Matrices 348
15.1 Basic definitions 348
15.2 Route matriees 351
15.3 Algebra of matriees 356
16 Vectors and transformations 363
16.1 Column vectors 363
16.2 Geometrieal proofs 368
16.3 Transformations 372
17 The calculus 382
17.1 Basic ideas of differentiation 382
17.2 Maxima and minima 388
17.3 Integration 389
17.4 Area and volume 391
Hints and solutions to investigations 395
Answers to exercises 399
Index 423
CONTENTS vii
o
Introduction
This book covers basic mathematical principles in a clear, straightforward style.
The purpose of this book is to help those of you who are finding some (or all!)
aspect of mathematics difficult. Very often, it is a small point that needs
explanation, and many books do not cover everything in sufficient detail.
Mastering Mathematics is designed to overcome these problems. It includes pencil
and paper techniques as weH as the correct use of a calculator. It also covers a
wide range of topics so that if you are following a GCSE, BTEC, college course,
or just 'doing mathematics for fun', you will almost certainly find the topics you
need.
Scattered throughout the book are a number of Investigations which are
designed to help you explore topics in a more practical way. Also as you progress
through the book, there are Examples and Exercises to see if you have indeed
mastered those subjects covered in that chapter. None of these investigations
requires any specialist equipment. It is important to remember when studying
mathematics at any time that methods sometimes need to be read several times.
Perseverance will bring its reward, particularly if the exercises are tried and
checked against the answers at the back. If you get anything wrong, go back to the
text and read the relevant section.
Geoff Buckwell
INTRODUCTION ix
o
Acknowledgements
I am indebted to Colin Prior for his illustrations which I hope give some light
relief, to Josie Buckwell for her work in preparing and checking the manuscript,
and Matthew Buckwell for his work in producing the diagrams.
The author and publishers wish to thank the following for permission to use
copyright material:
Express Newspapers pIe for articles 'Quiet for God's Sake' and 'No Camping',
Daily Star, 4 March 1991. The London East Anglian Group: Midland Examining
Group; Northern Examining Association comprising Associated Lancashire
Schools Examining Board, Joint Matriculation Board, North Regional
Examinations Board, North West Regional Examinations Board and Yorkshire
& Humberside Regional Examinations Board; Southern Examining Group,
University of Cambridge Local Examinations Syndicate, and the Welsh Joint
Education Committee for questions from past examination papers. Times
Newspapers Ltd for 'Britain will honour troops with Gulf victory parade' by
Nicholas Wood and Ruth Gledill, The Times, 4 March 1991.
Every effort has been made to trace all the copyright-holders, but if any have been
inadvertently overlooked the publishers will be pleased to make the necessary
arrangement at the first opportunity.
Geoff Buckwell
x ACKNOWLEDGEMENTS
