Year 9 Maths PDF PDF
Preview Year 9 Maths PDF
Mathematics 9 Fluency Problem Solving Statistics Measurement Number Probability Geometry Algebra Understanding Reasoning National Curriculum Dr Terry Dwyer Dr Terry Dwyer CertT, BAppSc, BEd, GDipEd, MEd(Hons), PhD Head of Mathematics Mathematics 9 National Curriculum www.drdwyer.com.au The cover: Maths is the pathway to success in our technological economy and society. “In today’s world, economic access and full citizenship depend crucially on math and science literacy.” - Robert P. Moses. “Mathematics makes a significant contribution to our modern society; its basic skills are vital for the life opportunities of the youth. Let us all embrace and encourage the study of this indispensable subject.“ - Alfred L. Teye. Dr Dwyer Pty Ltd ABN 27105593922 10 Moss Court Stanthorpe 4380 www.drdwyer.com.au Copyright © Dr Dwyer Pty Ltd First Published 2011 COPYRIGHT Apart from fair dealing for the purposes of study, research, criticism or review, or as permitted under Part VB of the Copyright Act, no part of this book may be reproduced by any process without permission. National Library of Australia Cataloguing-in-Publication entry Dwyer, T. (Terry), Mathematics 9 : National Curriculum 1st ed. Includes index. For secondary school year age. ISBN: 978-0-646-55957-5 1. Mathematics--Problems, exercises etc. 510 Published in Australia by Dr Dwyer Pty Ltd Printed in Malaysia Contents - Overview Chapter 1 Indices 1 1 Chapter 2 Algebra 1 17 Chapter 3 Area 33 Chapter 4 Linear & Non-linear Graphs 49 Chapter 5 Review 1 65 Chapter 6 Proportion 73 Chapter 7 Pythagoras' Theorem 89 Chapter 8 Geometry 103 Chapter 9 Statistics 119 Chapter 10 Review 2 135 Chapter 11 Indices 2 143 Chapter 12 Trigonometry 1 159 Chapter 13 Volume 175 Chapter 14 Probability 189 Chapter 15 Review 3 205 Chapter 16 Coordinate Geometry 213 Chapter 17 Trigonometry 2 229 Chapter 18 Algebra 2 245 Chapter 19 Data 261 Chapter 20 Review 4 275 Glossary 285 Answers 295 Index 311 Chapter 1 Indices 1 1 A Task 1 History 1 WarmUp 2 Indices 3 Index Law 1 4 Index Law 2 4 Index Law 3 5 Index Law 4 5 Index Law 5 6 Summary 8 Mental Computation 9 NAPLAN Questions 10 Competition Questions 11 Investigations 12 A Couple of Puzzles 13 A Game 13 A Sweet Trick 13 Technology 14 Chapter Review 1 15 Chapter Review 2 16 Chapter 2 Algebra 1 17 A Task 17 History 17 Algebra Warmup 18 Distributive Law 20 Factorisation 23 Mental Computation 25 NAPLAN Questions 26 Competition Questions 27 Investigations 28 A Couple of Puzzles 29 A Game 29 A Sweet Trick 29 Technology 30 Chapter Review 1 31 Chapter Review 2 32 Chapter 3 Area 33 A Task 33 History 33 Area Warmup 34 Composite Shapes 35 Prisms 36 Surface Area 37 Cylinders 38 Circle Review 38 Cylinder Surface Area 39 All Together 40 Mental Computation 41 NAPLAN Questions 42 Competition Questions 43 Investigations 44 Technology 45 A Couple of Puzzles 46 A Game 46 A Sweet Trick 46 Chapter Review 1 47 Chapter Review 2 48 History 49 Chapter 4 Linear & Non-linear 49 A Task 49 Linear Rules 50 Sketching Linear Graphs 53 Linear Graphs 54 Non-Linear Graphs 55 Mental Computation 57 NAPLAN Questions 58 Competition Questions 59 Investigations 60 Technology 61 A Couple of Puzzles 62 A Game 62 A Sweet Trick 62 Chapter Review 1 63 Chapter Review 2 64 Chapter 5 Review 1 65 Review 1 66 Review 2 70 Chapter 6 Proportion 73 A Task 73 History 73 Warmup 74 Proportion 76 Direct Proportion 78 Inverse Proportion 80 Money and Proportion 81 Mental Computation 82 Competition Questions 83 Investigations 84 A Couple of Puzzles 85 A Game 85 A Sweet Trick 85 Technology 86 Chapter Review 1 87 Chapter Review 2 88 History 89 Chapter 7 Pythagoras' Theorem 89 A Task 89 Pythagorean Triads 90 Pythagoras' Theorem 91 Hypotenuse 92 The Shorter Sides 93 Length of a line 94 Our Number System 95 Mental Computation 96 Competition Questions 97 Investigations 98 Technology 99 A Couple of Puzzles 100 A Game 100 A Sweet Trick 100 Chapter Review 1 101 Chapter Review 2 102 Chapter 8 Geometry 103 A Task 103 History 103 Congruent Triangles 104 Tests for Congruent Triangles 105 Similarity Transformation 106 Similar Triangles 108 Tests for Similar Triangles 109 Mental Computation 112 Competition Questions 113 Technology 114 A Couple of Puzzles 116 A Game 116 A Sweet Trick 116 Chapter Review 1 117 Chapter Review 2 118 History 119 Chapter 9 Statistics 119 A Task 119 Descriptive Statistics 120 Stem-and-Leaf Plots 122 Histograms 124 Comparative Analysis 126 Mental Computation 128 Competition Questions 129 Investigations 130 A Couple of Puzzles 131 A Game 131 A Sweet Trick 131 Technology 132 Chapter Review 1 133 Chapter Review 2 134 Chapter 10 Review 2 135 Review 1 136 Review 2 139 Chapter 11 Indices 2 143 A Task 143 History 143 Warmup 144 Index Law 1 145 Index Law 2 145 Index Law 3 146 Index Law 4 146 Index Law 5 147 Scientific Notation 148 Mental Computation 152 Competition Questions 153 Investigations 154 A Couple of Puzzles 155 A Game 155 A Sweet Trick 155 Technology 156 Chapter Review 1 157 Chapter Review 2 158 Chapter 12 Trigonometry 1 159 A Task 159 History 159 Pythagoras' Theorem 160 Naming Sides 162 Trigonometry 163 The Tan Ratio 164 Mental Computation 168 Competition Questions 169 A Couple of Puzzles 170 A Game 170 A Sweet Trick 170 Investigations 171 Technology 172 Chapter Review 1 173 Chapter Review 2 174 Chapter 13 Volume 175 A Task 175 History 175 Area Warmup 176 Composite Shapes 177 Prisms 178 Units of Volume 178 Volume of Prisms 179 Composite Solids 180 Practical Applications 181 Mental Computation 182 Competition Questions 183 Investigations 184 A Couple of Puzzles 185 A Game 185 A Sweet Trick 185 Technology 186 Chapter Review 1 187 Chapter Review 2 188 Chapter 14 Probability 189 A Task 189 History 189 Warm Up 190 Theoretical Probability 191 Experimental Probability 192 Venn Diagrams 197 Mental Computation 198 Competition Questions 199 Investigations 200 Technology 201 A Couple of Puzzles 202 A Game 202 A Sweet Trick 202 Chapter Review 1 203 Chapter Review 2 204 Chapter 15 Review 3 205 Review 1 206 Review 2 209 Chapter 16 Coordinate Geometry 213 A Task 213 History 213 Distance Between 2 Points 214 Midpoint 216 Gradient 218 Formulas 220 Mental Computation 222 Competition Questions 223 Investigations 224 A Couple of Puzzles 225 A Game 225 A Sweet Trick 225 Technology 226 Chapter Review 1 227 Chapter Review 2 228 Chapter 17 Trigonometry 2 229 A Task 229 History 229 Pythagoras' Theorem 230 The Tan Ratio 231 The Sine Ratio 232 The Cos Ratio 233 Trigonometry 234 Mental Computation 238 Competition Questions 239 A Couple of Puzzles 240 A Game 240 A Sweet Trick 240 Investigations 241 Technology 242 Chapter Review 1 243 Chapter Review 2 244 Chapter 18 Algebra 2 245 A Task 245 History 245 Integer Warmup 246 Index Law Warmup 247 Algebra Warmup 248 Distributive Law 250 Factorisation 251 Mental Computation 254 Competition Questions 255 Investigations 256 A Couple of Puzzles 257 A Game 257 A Sweet Trick 257 Technology 258 Chapter Review 1 259 Chapter Review 2 260 Chapter 19 Data 261 A Task 261 History 261 Data 262 Collecting Data 263 Sampling 264 Stratified Sampling 265 Questionnaires 266 Mental Computation 268 Competition Questions 269 Investigations 270 Technology 271 A Couple of Puzzles 272 A Game 272 A Sweet Trick 272 Chapter Review 1 273 Chapter Review 2 274 Paradoxes 283 Isometric Drawing 284 Chapter 20 Review 4 275 Review 1 276 Review 2 279 Glossary 285 Answers 295 Index 311 Preface This text has been written for Year 9 students. The aim of the text is to assist students in investigating and understanding the exciting and very important world of Mathematics and to implement the intent of the Australian Mathematics Curriculum. A literature review of learning from school textbooks was used to enhance the format of this textbook. Each chapter, apart from Review, contains: Numerous worked examples Numerous sets of graded exercises An open-ended rich task Mental computation Technology in mathematics Investigations Puzzles NAPLAN questions Maths competition preparation A mathematics game A mathematics trick A bit of mathematics history Careers using mathematics Chapter review Acknowledgments A heart-felt thank you to my wife Karen for your encouragement, advice, text design, images, illustrations, and above all, your loving support. Public Domain Images: pp. 17,40,48,73,143,175,189,213,229,245,261. Public Domain Images courtesy of nasaimages.org: pp. 151,156. Resources www.drdwyer.com.au 1 A TASK • Everyone should know about infinity. • Every child learns to count to ten, then to twenty, then to one hundred. • Then what? Infinity of course. Research infinity and then give a five minute explanation of infinity to the rest of the class (Try to include one of the interesting stories about infinity). Number and Algebra → Real Numbers Apply index laws to numerical expressions with integer indices. Connect different strategies for simplifying expressions with indices to illustrate the meaning of negative indices. Move fluently between representations of numeric and algebraic terms with negative indices. Apply knowledge of index laws to algebraic terms and simplify algebraic expressions, using both positive and negative integral indices. A 1 m cubic block will have 1 million 1 cm cubes. Infinite, adj. 1. exceedingly great, unlimited, immeasurably large. A LITTLE BIT OF HISTORY • The Romans used "Decies centena milia (ten hundred thousand) as the Roman words for 1 000 000. • The French, in the 13th century, were the first to use the word 1 million. • The Indians were the most advanced with large numbers and had by the 7th century defined infinity as having a denominator of zero. 1 0 1 10 . = 1 0 001 1000 . = 1 0 000001 1000000 . = 1 0 = ∞ 2 WarmUp Exercise 1.1 Calculate each of the following: 1 1×1 2 2×2 3 3×3 4 4×4 5 10×10 6 1×1×1×1 7 2×2×2 8 3×3×3 9 4×4×4 10 5×5×5 11 10×10×10 12 2×2×2×2 13 3×3×3×3 14 10×10×10×10 15 5×5×5×5 16 Copy and complete the following table: 17 10×10 18 10×10×10 19 10×10×10×10 20 10×10×10×10×10 21 10×10×10×10×10×10 22 10×10×10×10×10×10×10 Exercise 1.2 Use a calculator to calculate each of the following: 5×5×5×5×5×5×5x5 On a calculator: 5 ^ 8 = 390625 1 2×2×2×2×2×2 2 5×5×5×5×5×5×5 3 1×1×1×1×1×1×1×1 4 10×10×10×10×10×10 5 2×2×2×2×2×2×2×2 6 22×22×22×22 7 (22)4 8 22 4 9 3×3×3×3×3×3×3×3×3 10 33×33×33 11 (33)3 12 33 3 Square Cube Fourth Fifth 1 2 3 4 5 Cubes happen now and then. (Volume cube = side3) Squares are often used. (Area square = side2) On some calculators, the power button is yx. yx 3 Chapter 1 Indices 1 A convenient way of writing 2×2×2 is Exercise 1.3 Write the following in index form: 2×2×2×2×2 = 25 a×a×a×a = a4 1 4×4×4 2 2×2×2×2 3 a×a×a 4 10×10×10 5 b×b×b×b×b 6 h×h×h 7 m×m×m×m×m 8 9×9×9×9 9 3×3×3×3×3×3 Exercise 1.4 Write the following in factor form: 34 = 3×3×3×3 b3 = b×b×b 1 43 2 b4 3 52 4 27 5 62 6 m5 7 x4 8 p5 9 14 Exercise 1.5 Write the following in index form: 2×2×2×4×4 = 23×42 abbaaab = a4×b3 1 aabbbaa 2 3×3×3×2×2 3 abaaababb 4 2×3×2×3×2×3×2 5 bggggbbbg 6 zzzzzzzzzzz 7 ppqrppqqrrrp 8 2×2×2×3×4×4×4×3 9 4gg4g4g4gg Indices 23 Index Base Biochemists study the chemistry of living things in the fields of medicine, agriculture, the environment, and manufacturing. • Relevant school subjects are English, Mathematics, Chemsitry, Biology. • Courses normally involve a Degree with a major in chemistry/biochemistry. Indices save a lot of effort. 4 Exercise 1.6 Simplify and write the following in index form: 23×22 = 2×2×2 × 2×2 = 25 or 23×22 = 23+2 = 25 a2×a5 = a×a × a×a×a×a×a = a7 or a2×a5 = a2+5 = a7 1 22×23 2 33×32 3 24×22 4 44×43 5 a2×a2 6 b3×b2 7 z5×z3 8 w4×w3 9 23×21 10 t2×t5 11 52×53 12 104×105 13 c4×c3 14 102×104 15 a2×a3 16 b5×b 17 2.45×2.42 18 z3×z2 19 0.55×0.54 20 105×102 21 22×23×22 22 34×33×32 23 a2×a3×a3 24 u2×u2×u3 Exercise 1.7 Simplify and write the following in index form: 23÷22 = 2 2 2 2 2 × × × = 2 or 23÷22 = 23−2 = 2 a6÷a2 = a a a a a a a a × × × × × × = a×a×a×a = a4 or a6÷a2 = a6−2 = a4 1 25÷23 2 45÷42 3 24÷22 4 44÷43 5 a6÷a2 6 b7÷b2 7 105÷103 8 w4÷w3 9 23÷21 10 t7÷t4 11 56÷54 12 105÷102 13 a4÷a3 14 1.24÷1.22 15 29×29÷23 16 p3÷p3 17 m m 4 2 18 e e 5 2 19 c c 7 4 20 10 10 4 2 Index Law 1 Index Law 2 m4÷m2 = m m 4 2 They are the same thing. Multiplying Indices: 24×22 = 2×2×2×2 × 2×2 = 26 Multiplying Indices: 24×22 = 24+2 = 26 Index Law 1 am×an = am+n or Index Law 2 am÷an = am−n Dividing Indices: a3÷a2 = a a a a a × × × = a Dividing Indices: a3÷a2 = a3−2 = a or 5 Chapter 1 Indices 1 Exercise 1.8 Simplify and write the following in index form: (b4)2 = (b×b×b×b)2 = (b×b×b×b)×(b×b×b×b) = b8 or (b4)2 = b4×2 = b8 34×(32)3 = 34×36 = 310 (b4)2b3 = b8×b3 = b11 1 (22)3 2 (23)3 3 (32)3 4 (42)3 5 (a2)2 6 (b3)4 7 (t2)4 8 (n2)5 9 (52)2 10 (s2)5 11 (m2)3 12 (34)2 13 (53)3 14 (102)3 15 (10)3 16 (g2)5 17 (103)2 18 (h4)3 19 (d2)4 20 (25)3 21 22×(22)3 22 22×(23)3 23 (32)3×33 24 44×(42)3 25 d3(d2)2 26 b4(b3)4 27 (t2)4t5 28 n3(n2)5 29 (52)2×5 30 s2(s2)5 31 m(m2)3 32 3×(34)2 Exercise 1.9 Simplify each of the following: 30 = 1 h0 = 1 3×50 = 3×1 = 3 5b0 = 5×1 = 5 1 20 2 50 3 b0 4 k0 5 a0 6 30 7 100 8 d0 9 5×20 10 3a0 11 6×40 12 2×10 13 7w0 14 3e0 15 9×30 16 8×20 17 (50)2×5 18 s2(s0)5 19 m(m2)0 20 3×(30)2 Index Law 3 Index Law 4 Power Indices: (23)2 = (2×2×2)2 = (2×2×2)×(2×2×2) = 26 Power Indices: (23)2 = 23×2 = 26 Index Law 3 (am)n = am×n or Zero Index: p3÷p3 = 1 or p3÷p3 = p3−3 = p0 Which must be = 1 Zero Index: p0 = 1 Zero Index a0 = 1 or Try 50 on your calculator. Is your answer 1? 6 Exercise 1.10 Write each of the following using a negative index: 1 103 = 10−3 1 5 b = b−5 1 x = x−1 1 8 = 1 23 = 2−3 {23=2×2×2=8} 1 1 102 2 1 25 3 1 4 a 4 1 104 5 1 7 x 6 1 x 7 1 106 8 1 34 9 1 4 10 1 9 11 1 27 12 1 16 Find the value of each the following: 2−3 = 1 2 2 2 × × = 1 8 {also = 0.125} 10−4 = 1 10 10 10 10 × × × = 1 10000 {also = 0.0001} 13 2−2 14 4−1 15 2−4 16 10−1 17 10−2 18 10−3 19 10−4 20 10−5 21 5−2 22 2−5 23 0.4−2 24 1.5−2 Copy each of the following tables and use a calculator to complete them: 25 26 27 Index Law 5 Negative Index a−m = 1 am a2÷a5 = a a a a a a a × × × × × = 1 a a a × × a2÷a5 = a2−5 = a−3 or What happens when the index is negative? Power 24 23 22 21 20 2−1 2−2 2−3 Value 4 2 Power 54 53 52 51 50 5−1 5−2 5−3 Value 25 5 Power 104 103 102 101 100 10−1 10−2 10−3 Value 100 10
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