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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 />

Copyright © Insight Publicaons

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