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<strong>Summary</strong> guides<br />
<strong>Maths</strong><br />
9<br />
Uncorrected<br />
<strong>Sample</strong> <strong>Pages</strong><br />
Michael Loh
<strong>Summary</strong> <strong>Guides</strong><br />
<strong>Maths</strong> 9<br />
Michael Loh<br />
Uncorrected<br />
<strong>Sample</strong> <strong>Pages</strong>
Table of Contents<br />
Introduction<br />
1. Reviewing numbers and fractions<br />
1.1 Types of numbers<br />
1.2 Arithmetic operations<br />
1.3 Review of fractions<br />
Answers<br />
2. Financial mathematics<br />
2.1 Simple Interest<br />
2.2 Budgeting<br />
Answers<br />
3. Linear relationships<br />
3.1 Distance between two points on the Cartesian plane<br />
3.2 Midpoint between two points on the Cartesian plane<br />
3.3 Gradient of a straight line on the Cartesian plane<br />
3.4 Plotting/sketching a linear graph<br />
3.5 Equation of a straight line<br />
3.6 Horizontal and vertical lines<br />
3.7 Modelling using linear relationships<br />
Uncorrected<br />
3.8 Direct proportion<br />
3.9 Transformation of yy = xx tttt yy = aaaa + bb<br />
Answers<br />
<strong>Sample</strong> <strong>Pages</strong><br />
<strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9<br />
Copyright © Insight Publicaons
4. Algebraic techniques<br />
4.1 Key concepts of algebra<br />
4.2 The distributive law<br />
4.3 Binomial expansion<br />
4.4 Factorising<br />
4.5 Simplifying algebraic fractions<br />
Answers<br />
5. Solving linear equations, simultaneous equations and linear inequalities<br />
5.1 Solving linear equations<br />
5.2 Solving linear equations – Applications<br />
5.3 Solving simultaneous equations<br />
5.4 Solving simultaneous equations – Applications<br />
5.5 Solving linear inequalities<br />
5.6 Solving linear inequalities – Applications<br />
Answers<br />
6. Index laws, scientific notation and surds<br />
6.1 Index laws<br />
6.2 Scientific notation<br />
6.3 Surds<br />
7. Measurement<br />
Answers<br />
Uncorrected<br />
7.1 Units of measurement<br />
7.2 Areas of two-dimensional shapes<br />
7.3 Areas of two-dimensional composite shapes<br />
7.4 Total surface area of a three-dimensional solid<br />
7.5 Volume of a right prism and a cylinder<br />
7.6 Volume of a composite solid<br />
7.7 Volume vs capacity<br />
Answers<br />
<strong>Sample</strong> <strong>Pages</strong><br />
Copyright © Insight Publicaons <strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9
8. Pythagoras’ theorem and trigonometry<br />
8.1 Pythagoras’ theorem<br />
8.2 Applications of Pythagoras’ theorem<br />
8.3 Trigonometry<br />
8.4 Determining the missing length and angle of a right-angled triangle<br />
8.5 Applications of trigonometry<br />
Answers<br />
9. Similarity, enlargement and scale factor<br />
9.1 Similarity<br />
9.2 Scale factor<br />
9.3 Enlargement using scale factor<br />
9.4 Applying the scale factor to area and volume<br />
Answers<br />
10. Geometric reasoning<br />
11. Probability<br />
10.1 Review of angles<br />
10.2 Review of triangles<br />
10.3 Similar triangles<br />
10.4 Applications of similar triangles<br />
Answers<br />
Uncorrected<br />
11.1 Key concepts of probability<br />
11.2 Venn diagrams and set notation<br />
11.3 Two-way tables<br />
11.4 Relative frequency<br />
11.5 Multi-stage experiments<br />
Answers<br />
<strong>Sample</strong> <strong>Pages</strong><br />
<strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9<br />
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12. Statistics<br />
12.1 Data collecting<br />
12.2 Types of data<br />
12.3 Statistical measures<br />
12.4 Displaying data<br />
12.5 Describing and comparing data<br />
12.6 Statistical investigation process<br />
12.7 Interpreting data in real life<br />
Answers<br />
13. Introduction to quadratic equations and graphs<br />
13.1 Introduction<br />
13.2 The equation of the basic parabola and its graph<br />
13.3 General equation of a parabola and its graph<br />
13.3 Identifying a parabola from a table of values<br />
13.4 Solving quadratic equations<br />
13.5 Transformation of a quadratic equation from yy = xx 22<br />
Answers<br />
14. Algorithmic thinking<br />
Uncorrected<br />
<strong>Sample</strong> <strong>Pages</strong><br />
Copyright © Insight Publicaons <strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9
Introduction<br />
The series, <strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong>, has been written by practising teachers with years of<br />
classroom experience, who are passionate about creating user-friendly, accessible guides on<br />
mathematics.<br />
The explanations and exercises in these guides develop core numeracy skills for personal, work<br />
and civic life, and provide the base knowledge for the professional applications of maths as well<br />
as mathematical specialisations. <strong>Maths</strong> is part of your daily life no matter what you choose to do<br />
as an adult – it is important for thinking critically and making sense of the world.<br />
This book summarises key concepts in a clear and comprehensive way. It includes examples<br />
with worked solutions and step-by-step explanations, as well as exercises for you to complete.<br />
The best way to use this book is to make a habit of it – regularly working through the exercises<br />
and examples, and comparing your answers against those provided. Whether you commit to a<br />
daily, weekly or fortnightly routine, this consistent practice is the key to your success.<br />
The content of this book is based on years of experience in the classroom, making maths<br />
comprehensible and achievable for students. Through using this guide, you will be able to<br />
practise your skills and grow in confidence. We hope this resource will help you reap the<br />
rewards of maths.<br />
Michael Loh and Insight Publications<br />
Uncorrected<br />
<strong>Sample</strong> <strong>Pages</strong><br />
<strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9<br />
Copyright © Insight Publicaons
Chapter 1 – Reviewing numbers and fracons<br />
Chapter 1 – Reviewing numbers and fractions<br />
1.1 Types of numbers<br />
Real numbers:<br />
• are the numbers we normally use in calculations<br />
• include all the types of numbers listed below.<br />
Counting<br />
numbers<br />
Name Characteristics Examples<br />
(also known as<br />
natural numbers)<br />
Whole numbers<br />
Integers<br />
Rational numbers<br />
Irrational<br />
numbers<br />
• Any numbers used for counting starting<br />
from 1<br />
• Excludes all negative numbers, 0,<br />
fractions and decimals<br />
• Numbers used for counting starting<br />
from 0<br />
• Excludes all negative numbers, 0,<br />
fractions and decimals<br />
• All numbers with no fraction or decimal<br />
part<br />
• Includes negative numbers<br />
Uncorrected<br />
• A number which can be made by<br />
dividing two integers<br />
• A number which cannot be made by<br />
dividing two integers<br />
1<br />
140<br />
2000<br />
0<br />
340<br />
1500<br />
−300<br />
−5<br />
0<br />
2500<br />
1.25 (= 1 1 4 )<br />
27<br />
5<br />
−2.5<br />
1<br />
3<br />
<strong>Sample</strong> <strong>Pages</strong><br />
√2<br />
ππ (=3,14159…)<br />
3<br />
√5<br />
Copyright © Insight Publicaons <strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9
Chapter 1 – Reviewing numbers and fracons<br />
Exercise 1.1<br />
Identify the following real numbers.<br />
a. 0.75 b. 3 c. −4.5 d. √3 e. −100.52 f.<br />
22<br />
5<br />
1.2 Arithmetic operations<br />
When an expression is to be evaluated, always use the correct order of arithmetic<br />
operations: Brackets, Indices, Multiplication, Division, Addition and Subtraction (BIMDAS).<br />
B<br />
I<br />
Brackets ( ), { }, [ ], etc.<br />
Indices 2 3 , √3<br />
M Multiplication ×<br />
D Division ÷<br />
A Addition +<br />
S<br />
Example<br />
Subtraction −<br />
Evaluate the following:<br />
If multiplication and division appear on the same<br />
line, always do it from left to right.<br />
If addition and subtraction appear on the same<br />
line, always do it from left to right.<br />
a. 4 − 5 × 2 2 b. (5 − 9)/(6 + 2) c. 4 ÷ (2 + 5) d. 6 ÷ 2(2 + 1)<br />
Solution<br />
Uncorrected<br />
Working<br />
a. 4 − 5 × 2 2 = 4 − 5 × 4 = 4 − 20 = −16<br />
b. 5−9<br />
6+2 = (5−9)<br />
(6+2) = − 4 8 = − 1 2<br />
c. 4 ÷ (2 + 5) = 4 ÷ (7) = 4 7<br />
Explanation<br />
Evaluate the index first, then multiply<br />
followed by subtraction.<br />
Rewrite in fraction format (optional),<br />
evaluate the brackets, then simplify the<br />
fraction.<br />
<strong>Sample</strong> <strong>Pages</strong><br />
Evaluate the bracket, divide, then write as a<br />
fraction.<br />
<strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9<br />
Copyright © Insight Publicaons
Chapter 1 – Reviewing numbers and fracons<br />
d. 6 ÷ 2(2 + 1) = 6 ÷ 2 × (2 + 1)<br />
= 6 ÷ 2 × 3<br />
= 9<br />
Rewrite the expression to show the<br />
multiplication sign, evaluate the bracket<br />
first, then evaluate from left to right.<br />
(NOTE: if you input the expression into a<br />
scientific calculator with the multiplication<br />
sign, you will get the incorrect answer).<br />
Exercise 1.2<br />
Evaluate the following:<br />
a. 6 ÷ 2 × 2 + 1 b. (9 2 − 18)/3 c. (4 + 2) × (3 − 1)<br />
d. 6/[2(2 + 1)] e. √36 − 5 × 4 f.<br />
1.3 Review of fractions<br />
Proper<br />
e.g. 2 3 , 4 37 , 22<br />
101<br />
• The numerator is<br />
always smaller than<br />
the denominator.<br />
• Always simplify the<br />
fraction at the end<br />
by cancelling the<br />
common factor.<br />
Fraction = numerator<br />
denominator<br />
Fractions<br />
Improper<br />
e.g. 3 2 , 24<br />
, 122<br />
7 7<br />
• The numerator is<br />
always greater than<br />
the denominator.<br />
• Always simplify the<br />
fraction at the end<br />
by cancelling the<br />
common factor.<br />
• Unless stated, you<br />
do not need to<br />
convert improper<br />
fractions to a mixed<br />
number.<br />
42<br />
7 × 3 − 10/2<br />
Mixed numbers<br />
e.g. 1 2 5 , 12 4 7<br />
Uncorrected<br />
• Made up of an<br />
integer and a proper<br />
fraction.<br />
<strong>Sample</strong> <strong>Pages</strong><br />
Copyright © Insight Publicaons <strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9
Chapter 1 – Reviewing numbers and fracons<br />
Example<br />
Simplify the following fractions:<br />
a. 45<br />
50<br />
b.<br />
26<br />
338<br />
Solution<br />
a.<br />
b.<br />
45<br />
= 45÷5<br />
= 9<br />
50 50÷5 10<br />
26<br />
= 26÷26<br />
= 1<br />
338 338÷26 13<br />
Example<br />
Working<br />
Convert the following improper fractions to mixed numbers.<br />
a. 17<br />
12<br />
Solution<br />
a. 17<br />
b. 70<br />
12 = 1 5 12<br />
= 4<br />
10<br />
= 4 2<br />
15 15 3<br />
Example<br />
Working<br />
Explanation<br />
Determine the highest common factor<br />
(HCF) for both numbers and divide the<br />
numerator and denominator with the HCF.<br />
b. 70<br />
15<br />
Convert the following mixed numbers to improper fractions.<br />
a. 2 5 6<br />
Solution<br />
Working<br />
a. 2 5 = 2×6<br />
+ 5 = 12<br />
+ 5 = 17<br />
6 6 6 6 6<br />
b. 8 2 = 8×3<br />
+ 2 = 26<br />
3 3 3 3<br />
Explanation<br />
Divide the numerator by the denominator.<br />
The resulting quotient is the whole number<br />
and the remainder is the new numerator.<br />
Simply if possible.<br />
Uncorrected<br />
b. 8 2 3<br />
<strong>Sample</strong> <strong>Pages</strong><br />
Explanation<br />
6 Multiply the whole number by the<br />
denominator and add it to the numerator.<br />
<strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9<br />
Copyright © Insight Publicaons
Chapter 1 – Reviewing numbers and fracons<br />
When performing arithmetic operations on fractions, follow these rules:<br />
Addition and<br />
subtraction<br />
Ensure that the denominators are the same for all fractions;<br />
i.e. determine the common denominator.<br />
Example<br />
Multiplication<br />
Division<br />
Evaluate the following:<br />
a. 2 5 + 1 3<br />
Solution<br />
Working<br />
Multiply all the numerators together and then multiply all the<br />
denominators together. Ensure that you simplify the final<br />
fractions. [Hint: cancelling first makes the multiplication<br />
easier.]<br />
Take the reciprocal of the fraction (i.e. reverse the numerator<br />
and denominator) after the division sign. Then multiply the<br />
fraction.<br />
b. 1 8 − 3 4<br />
a. 2 5 + 1 3 = 2 5 × 3 3 + 1 3 × 5 5 = 6 15 + 5 15 = 11<br />
b. 1 8 − 3 4 = 1 8 − 3 4 × 2 2 = 1 8 − 6 8 = − 5 8<br />
c. 2 × 7 = 2×7<br />
= 14<br />
5 11 5×11 55<br />
d. 2 5 ÷ 5 7 = 2 5 × 7 5 = 14<br />
25<br />
15<br />
c. 2 5 × 7 11<br />
d. 2 5 ÷ 5 7<br />
Explanation<br />
Determine the common denominator and<br />
then add the numerators and keep the<br />
denominator. Simplify the fraction where<br />
appropriate.<br />
Determine the common denominator and<br />
then subtract the second numerator from<br />
the first numerator and keep the<br />
denominator.<br />
Uncorrected<br />
Note: the BIMDAS rule must be observed.<br />
Multiply the numerators and then the<br />
denominators.<br />
Convert the division sign to a multiplication<br />
sign and take the reciprocal of the divisor<br />
(in this case, change 5 to 7 ) and then<br />
7 5<br />
multiply.<br />
<strong>Sample</strong> <strong>Pages</strong><br />
Copyright © Insight Publicaons <strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9
Chapter 1 – Reviewing numbers and fracons<br />
Answers<br />
Exercise 1.1<br />
a. rational b. counting or whole<br />
or integer<br />
c. rational<br />
d. irrational e. rational f. rational<br />
Exercise 1.2<br />
a. 7 b. 21 c. 12 d. 1 e. −14 f. 13<br />
Exercise 1.3.1<br />
a.<br />
11<br />
100<br />
Exercise 1.3.2<br />
a. 1 29<br />
32<br />
Exercise 1.3.3<br />
a.<br />
73<br />
12<br />
Exercise 1.3.4<br />
a.<br />
e.<br />
17<br />
35<br />
33<br />
112<br />
b.<br />
3<br />
5<br />
b. 2 5 8<br />
b.<br />
b.<br />
f.<br />
37<br />
18<br />
136<br />
105<br />
14<br />
75<br />
c.<br />
2<br />
9<br />
c. 3 29<br />
c.<br />
c.<br />
37<br />
3<br />
5<br />
42<br />
33<br />
d.<br />
1<br />
2<br />
d. 5 21<br />
d.<br />
67<br />
222<br />
11<br />
d. − 31<br />
Uncorrected<br />
g.<br />
7<br />
9<br />
<strong>Sample</strong> <strong>Pages</strong><br />
h.<br />
35<br />
48<br />
105<br />
<strong>Summary</strong> <strong>Guides</strong> – <strong>Maths</strong> 9<br />
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