Table of Contents
Introduction 4
1 Percentages 6
1.1 Simple interest 7
1.2 Percentage increases and decreases 10
1.3 Calculating the original value 14
1.4 Repeated percentage changes 17
Ready to progress? 20
Review questions 20
Challenge: Exponential growth 22
2 Equations and formulae 24
2.1 Multiplying out brackets 25
2.2 Factorising algebraic expressions 26
2.3 Expressions with several variables 28
2.4 Equations with fractions 31
Ready to progress? 34
Review questions 34
Investigation: Body mass index 36
3 Polygons 38
3.1 Properties of polygons 39
3.2 Interior and exterior angles of regular polygons 44
3.3 Tessellations and regular polygons 46
Ready to progress? 48
Review questions 48
Mathematical reasoning: Semi-regular tessellations 50
4 Using data 52
4.1 Scatter graphs and correlation 53
4.2 Two-way tables 59
4.3 Estimation of a mean from grouped data 62
4.4 Cumulative frequency diagrams 65
4.5 Statistical investigations 70
Ready to progress? 72
Review questions 72
Challenge: Census 74
5 Applications of graphs 76
5.1 Step graphs 77
5.2 Time graphs 82
5.3 Exponential growth graphs 87
Ready to progress? 92
Review questions 92
Problem solving: Mobile phone tariffs 94
6 Pythagoras' theorem 96
6.1 Introducing Pythagoras' theorem 97
6.2 Using Pythagoras' theorem to solve problems 101
6.3 The converse of Pythagoras' theorem 104
Ready to progress? 106
Review questions 106
Activity: Practical Pythagoras 108
7 Fractions 110
7.1 Adding and subtracting fractions 111
7.2 Multiplying fractions and mixed numbers 113
7.3 Dividing fractions and mixed numbers 115
7.4 Algebraic fractions 118
Ready to progress? 120
Review questions 120
Investigation: Fractions, from one to six 122
8 Algebra 124
8.1 Expanding the product of two brackets 125
8.2 Expanding expressions with more than two brackets 128
8.3 Factorising quadratic expressions with positive coefficients 130
8.4 Factorising quadratic expressions with negative coefficients 132
8.5 The difference of two squares 134
Ready to progress? 136
Review questions 136
Challenge: Graphs from expressions 138
9 Decimal numbers 140
9.1 Powers of 10 141
9.2 Standard form 143
9.3 Multiplying with numbers in standard form 147
9.4 Dividing with numbers in standard form 149
9.5 Upper and lower bounds 151
Ready to progress? 154
Review questions 154
Mathematical reasoning: To the stars and back 156
10 Surface area and volume of cylinders 158
10.1 Volume of a cylinder 159
10.2 Surface area of a cylinder 162
10.3 Composite shapes 166
Ready to progress? 170
Review questions 170
Problem solving: Packaging soup 172
11 Solving equations graphically 174
11.1 Graphs from equations in the form ay ± bx = c 175
11.2 Solving simultaneous equations by drawing graphs 177
11.3 Solving quadratic equations by drawing graphs 179
11.4 Solving cubic equations by drawing graphs 181
Ready to progress? 184
Review questions 184
Challenge: Maximum packages 186
12 Compound units 188
12.1 Speed 189
12.2 More compound units 193
12.3 Unit costs 196
Ready to progress? 200
Review questions 200
Challenge: Population density 202
13 Right-angled triangles 204
13.1 Introduction to trigonometric ratios 205
13.2 How to find trigonometric ratios of angles 207
13.3 Using trigonometric ratios to find angles 210
13.4 Using trigonometric ratios to find lengths 215
Ready to progress? 218
Review questions 218
Investigation: Barnes Wallis and the bouncing bomb 220
14 Revision and GCSE preparation 222
14.1 Practice 223
14.2 Revision 232
GCSE-type questions 239
Glossary 242
Index 245