Table Of ContentMathematics 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.
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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
