Beams Latest 24 Mac 2010 Edited Version

Basic Essential Additional Mathematics Skills

Curriculum Development Division
Ministry of Education Malaysia
Putrajaya

2010

First published 2010

© Curriculum Development Division,
Aras 4-8, Blok E9
Pusat Pentadbiran Kerajaan Persekutuan
62604 Putrajaya
Tel.: 03-88842000 Fax.: 03-88889917
Website: http://www.moe.gov.my/bpk

Copyright reserved. Except for use in a review, the reproduction or utilization of this
work in any form or by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, and recording is forbidden without prior
written permission from the Director of the Curriculum Development Division, Ministry
of Education Malaysia.

TABLE OF CONTENTS

Preface i

Acknowledgement ii

Introduction iii

Objective iii

Module Layout iii

BEAMS Module:

Unit 1: Negative Numbers

Unit 2: Fractions

Unit 3: Algebraic Expressions and Algebraic Formulae

Unit 4: Linear Equations

Unit 5: Indices

Unit 6: Coordinates and Graphs of Functions

Unit 7: Linear Inequalities

Unit 8: Trigonometry

Panel of Contributors

ACKNOWLEDGEMENT

The Curriculum Development Division,

Ministry of Education wishes to express our

deepest gratitude and appreciation to all

panel of contributors for their expert

views and opinions, dedication,

and continuous support in

the development of

this module.

ii

INTRODUCTION
Additional Mathematics is an elective subject taught at the upper secondary level. This
subject demands a higher level of mathematical thinking and skills compared to that required
by the more general Mathematics KBSM. A sound foundation in mathematics is deemed
crucial for pupils not only to be able to grasp important concepts taught in Additional
Mathematics classes, but also in preparing them for tertiary education and life in general.

This Basic Essential Additional Mathematics Skills (BEAMS) Module is one of the
continuous efforts initiated by the Curriculum Development Division, Ministry of Education,
to ensure optimal development of mathematical skills amongst pupils at large. By the
acronym BEAMS itself, it is hoped that this module will serve as a concrete essential
support that will fruitfully diminish mathematics anxiety amongst pupils. Having gone
through the BEAMS Module, it is hoped that fears induced by inadequate basic
mathematical skills will vanish, and pupils will learn mathematics with the due excitement
and enjoyment.

OBJECTIVE
The main objective of this module is to help pupils develop a solid essential mathematics
foundation and hence, be able to apply confidently their mathematical skills, specifically
in school and more significantly in real-life situations.

MODULE LAYOUT
This module encompasses all mathematical skills and knowledge
taught in the lower secondary level and is divided into eight units as
follows:

Unit 2: Fractions
Unit 3: Algebraic Expressions and Algebraic Formulae
Unit 4: Linear Equations
Unit 5: Indices
Unit 6: Coordinates and Graphs of Functions
Unit 7: Linear Inequalities
Unit 8: Trigonometry

iii

Each unit stands alone and can be used as a comprehensive revision of a particular topic.
Most of the units follow as much as possible the following layout:
Module Overview
Objectives
Teaching and Learning Strategies
Lesson Notes
Examples
Test Yourself
Answers

The “Lesson Notes”, “Examples” and “Test Yourself” in each unit can be used as
supplementary or reinforcement handouts to help pupils recall and understand the basic
concepts and skills needed in each topic.

Teachers are advised to study the whole unit prior to classroom teaching so as to familiarize
with its content. By completely examining the unit, teachers should be able to select any part
in the unit that best fit the needs of their pupils. It is reminded that each unit in this module is
by no means a complete lesson, rather as a supporting material that should be ingeniously
integrated into the Additional Mathematics teaching and learning processes.

At the outset, this module is aimed at furnishing pupils with the basic mathematics
foundation prior to the learning of Additional Mathematics, however the usage could be
broadened. This module can also be benefited by all pupils, especially those who are
preparing for the Penilaian Menengah Rendah (PMR) Examination.

iv

PANEL OF CONTRIBUTORS

Advisors:

Haji Ali bin Ab. Ghani AMN
Director

Dr. Lee Boon Hua
Deputy Director (Humanities)

Mohd. Zanal bin Dirin
Deputy Director (Science and Technology)

Editorial Advisor:

Aziz bin Saad
Principal Assistant Director
(Head of Science and Mathematics Sector)

Editors:

Dr. Rusilawati binti Othman
Assistant Director
(Head of Secondary Mathematics Unit)

Aszunarni binti Ayob
Assistant Director

Rosita binti Mat Zain
Assistant Director

Writers:

Abdul Rahim bin Bujang Hon May Wan
SM Tun Fatimah, Johor SMK Tasek Damai, Ipoh, Perak

Ali Akbar bin Asri Horsiah binti Ahmad
SM Sains, Labuan SMK Tun Perak, Jasin, Melaka

Amrah bin Bahari Kalaimathi a/p Rajagopal
SMK Dato’ Sheikh Ahmad, Arau, Perlis SMK Sungai Layar, Sungai Petani, Kedah

Aziyah binti Paimin Kho Choong Quan
SMK Kompleks KLIA, , Negeri Sembilan SMK Ulu Kinta, Ipoh, Perak

Bashirah binti Seleman Lau Choi Fong
SMK Sultan Abdul Halim, Jitra, Kedah SMK Hulu Klang, Selangor

Bibi Kismete binti Kabul Khan Loh Peh Choo
SMK Jelapang Jaya, Ipoh, Perak SMK Bandar Baru Sungai Buloh, Selangor

Che Rokiah binti Md. Isa Mohd. Misbah bin Ramli
SMK Dato’ Wan Mohd. Saman, Kedah SMK Tunku Sulong, Gurun, Kedah

Cheong Nyok Tai Noor Aida binti Mohd. Zin
SMK Perempuan, Kota Kinabalu, Sabah SMK Tinggi Kajang, Kajang, Selangor

Ding Hong Eng Noor Ishak bin Mohd. Salleh
SM Sains Alam Shah, Kuala Lumpur SMK Laksamana, Kota Tinggi, Johor

Esah binti Daud Noorliah binti Ahmat
SMK Seri Budiman, Kuala Terengganu SM Teknik, Kuala Lumpur

Haspiah binti Basiran Nor A’idah binti Johari
SMK Tun Perak, Jasin, Melaka SMK Teknik Setapak, Selangor

Noorliah binti Ahmat
SM Teknik, Kuala Lumpur

Ali Akbar bin Asri Nor A’idah binti Johari

SM Sains, Labuan SMK Teknik Setapak, Selangor

Amrah bin Bahari Nor Dalina binti Idris

SMK Dato’ Sheikh Ahmad, Arau, Perlis SMK Syed Alwi, Kangar, Perlis

Writers:

Nor Dalina binti Idris Suhaimi bin Mohd. Tabiee
SMK Syed Alwi, Kangar, Perlis SMK Datuk Haji Abdul Kadir, Pulau Pinang

Norizatun binti Abdul Samid Suraiya binti Abdul Halim
SMK Sultan Badlishah, Kulim, Kedah SMK Pokok Sena, Pulau Pinang

Pahimi bin Wan Salleh Tan Lee Fang
Maktab Sultan Ismail, Kelantan SMK Perlis, Perlis

Rauziah binti Mohd. Ayob Tempawan binti Abdul Aziz
SMK Bandar Baru Salak Tinggi, Selangor SMK Mahsuri, Langkawi, Kedah

Rohaya binti Shaari Turasima binti Marjuki
SMK Tinggi Bukit Merajam, Pulau Pinang SMKA Simpang Lima, Selangor

Roziah binti Hj. Zakaria Wan Azlilah binti Wan Nawi
SMK Taman Inderawasih, Pulau Pinang SMK Putrajaya Presint 9(1), WP Putrajaya

Shakiroh binti Awang Zainah binti Kebi
SM Teknik Tuanku Jaafar, Negeri Sembilan SMK Pandan, Kuantan, Pahang

Sharina binti Mohd. Zulkifli Zaleha binti Tomijan
SMK Agama, Arau, Perlis SMK Ayer Puteh Dalam, Pendang, Kedah

Sim Kwang Yaw Zariah binti Hassan
SMK Petra, Kuching, Sarawak SMK Dato’ Onn, Butterworth, Pulau Pinang

Layout and Illustration:

Aszunarni binti Ayob Mohd. Lufti bin Mahpudz
Assistant Director Assistant Director
Curriculum Development Division Curriculum Development Division

Basic Essential

Additional Mathematics Skills

UNIT 1
NEGATIVE NUMBERS

Unit 1:
Negative Numbers


TABLE OF CONTENTS

Module Overview 1

Part A: Addition and Subtraction of Integers Using Number Lines 2

1.0 Representing Integers on a Number Line 3

2.0 Addition and Subtraction of Positive Integers 3

3.0 Addition and Subtraction of Negative Integers 8

Part B: Addition and Subtraction of Integers Using the Sign Model 15

Part C: Further Practice on Addition and Subtraction of Integers 19

Part D: Addition and Subtraction of Integers Including the Use of Brackets 25

Part E: Multiplication of Integers 33

Part F: Multiplication of Integers Using the Accept-Reject Model 37

Part G: Division of Integers 40

Part H: Division of Integers Using the Accept-Reject Model 44

Part I: Combined Operations Involving Integers 49

Answers 52

Basic Essential Additional Mathematics Skills (BEAMS) Module

MODULE OVERVIEW

1. Negative Numbers is the very basic topic which must be mastered by every
pupil.

2. The concept of negative numbers is widely used in many Additional
Mathematics topics, for example:
(a) Functions (b) Quadratic Equations
(c) Quadratic Functions (d) Coordinate Geometry
(e) Differentiation (f) Trigonometry
Thus, pupils must master negative numbers in order to cope with topics in
Additional Mathematics.

3. The aim of this module is to reinforce pupils‟ understanding on the concept of
negative numbers.

4. This module is designed to enhance the pupils‟ skills in

 using the concept of number line;
 using the arithmetic operations involving negative numbers;
 solving problems involving addition, subtraction, multiplication and
division of negative numbers; and
 applying the order of operations to solve problems.

5. It is hoped that this module will enhance pupils‟ understanding on negative
numbers using the Sign Model and the Accept-Reject Model.

6. This module consists of nine parts and each part consists of learning objectives
which can be taught separately. Teachers may use any parts of the module as
and when it is required.

1


PART A:
ADDITION AND SUBTRACTION
OF INTEGERS USING
NUMBER LINES

LEARNING OBJECTIVE

Upon completion of Part A, pupils will be able to perform computations
involving combined operations of addition and subtraction of integers using a
number lines.

TEACHING AND LEARNING STRATEGIES

The concept of negative numbers can be confusing and difficult for pupils to
grasp. Pupils face difficulty when dealing with operations involving positive and
negative integers.

Strategy:

Teacher should ensure that pupils understand the concept of positive and negative
integers using number lines. Pupils are also expected to be able to perform
computations involving addition and subtraction of integers with the use of the
number line.

2


PART A:
ADDITION AND SUBTRACTION OF INTEGERS
USING NUMBER LINES

LESSON NOTES

1.0 Representing Integers on a Number Line

 Positive whole numbers, negative numbers and zero are all integers.

 Integers can be represented on a number line.
Positive integers
may have a plus sign
–3 –2 –1 0 1 2 3 4 in front of them,
like +3, or no sign in
front, like 3.
Note: i) –3 is the opposite of +3

ii) – (–2) becomes the opposite of negative 2, that is, positive 2.

2.0 Addition and Subtraction of Positive Integers

Rules for Adding and Subtracting Positive Integers

 When adding a positive integer, you move to the right on a
number line.

–3 –2 –1 0 1 2 3 4

 When subtracting a positive integer, you move to the left
on a number line.

–3 –2 –1 0 1 2 3 4

3


EXAMPLES

(i) 2 + 3
Start Add a
with 2 positive 3

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Adding a positive integer:

Start by drawing an arrow from 0 to 2, and then,
draw an arrow of 3 units to the right:

2+3=5

Alternative Method:

Make sure you start from
the position of the first
integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start at 2 and move 3 units to the right:

2+3=5

4


(ii) –2 + 5

Add a
positive 5

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start by drawing an arrow from 0 to –2, and then,
draw an arrow of 5 units to the right:

–2 + 5 = 3

Alternative Method:

integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start at –2 and move 5 units to the right:

–2 + 5 = 3

5


(iii) 2 – 5 = –3
Subtract a
positive 5

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Subtracting a positive integer:

Start by drawing an arrow from 0 to 2, and then,
draw an arrow of 5 units to the left:

2 – 5 = –3

Alternative Method:

integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start at 2 and move 5 units to the left:

2 – 5 = –3

6


(iv) –3 – 2 = –5

Subtract a
positive 2

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start by drawing an arrow from 0 to –3, and
then, draw an arrow of 2 units to the left:

–3 – 2 = –5

Alternative Method:

integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start at –3 and move 2 units to the left:

–3 – 2 = –5

7


3.0 Addition and Subtraction of Negative Integers

Consider the following operations:

4 + (–1) = 3
4–1=3
–3 –2 –1 0 1 2 3 4

4–2=2 4 + (–2) = 2
–3 –2 –1 0 1 2 3 4

4–3=1 4 + (–3) = 1
–3 –2 –1 0 1 2 3 4

4–4=0 4 + (–4) = 0
–3 –2 –1 0 1 2 3 4

4 + (–5) = –1
4 – 5 = –1
–3 –2 –1 0 1 2 3 4

4 – 6 = –2 4 + (–6) = –2
–3 –2 –1 0 1 2 3 4

Note that subtracting an integer gives the same result as adding its opposite. Adding or
subtracting a negative integer goes in the opposite direction to adding or subtracting a positive
integer.

8


Rules for Adding and Subtracting Negative Integers

 When adding a negative integer, you move to the left on a
number line.

–3 –2 –1 0 1 2 3 4

 When subtracting a negative integer, you move to the right
on a number line.

–3 –2 –1 0 1 2 3 4

9


EXAMPLES

(i) –2 + (–1) = –3
This operation of
–2 + (–1) = –3
is the same as
Add a
negative 1 –2 –1 = –3.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Adding a negative integer:

then, draw an arrow of 1 unit to the left:

–2 + (–1) = –3

Alternative Method: Make sure you start from
integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start at –2 and move 1 unit to the left:

–2 + (–1) = –3

10


(ii) 1 + (–3) = –2
This operation of
1 + (–3) = –2
is the same as
1 – 3 = –2
Add a
negative 3

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Start by drawing an arrow from 0 to 1, then, draw an arrow of
3 units to the left:
1 + (–3) = –2

Alternative Method:
integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start at 1 and move 3 units to the left:

1 + (–3) = –2

11


(iii) 3 – (–3) = 6

This operation of
3 – (–3) = 6
is the same as
3+3=6
Subtract a
negative 3

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Subtracting a negative integer:

Start by drawing an arrow from 0 to 3, and
then, draw an arrow of 3 units to the right:

3 – (–3) = 6

Alternative Method:

integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start at 3 and move 3 units to the right:

3 – (–3) = 6

12


(iv) –5 – (–8) = 3 This operation of
–5 – (–8) = 3
is the same as
–5 + 8 = 3

Subtract a 3+3=6
negative 8

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


then, draw an arrow of 8 units to the right:

–5 – (–8) = 3

Alternative Method:

–5 –4 –3 –2 –1 0 1 2 3 4 5 6


Start at –5 and move 8 units to the right:

–5 – (–8) = 3

13


TEST YOURSELF A

Solve the following.

1. –2 + 4

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

2. 3 + (–6)

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

3. 2 – (–4)

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

4. 3 – 5 + (–2)

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

5. –5 + 8 + (–5)

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

14


PART B:
OF INTEGERS USING
THE SIGN MODEL

LEARNING OBJECTIVE

Upon completion of Part B, pupils will be able to perform computations
involving combined operations of addition and subtraction of integers using
the Sign Model.


This part emphasises the first alternative method which include activities and
mathematical games that can help pupils understand further and master the
operations of positive and negative integers.

Strategy:

Teacher should ensure that pupils are able to perform computations involving
addition and subtraction of integers using the Sign Model.

15


PART B:

USING THE SIGN MODEL

LESSON NOTES

In order to help pupils have a better understanding of positive and negative integers, we have
designed the Sign Model.

The Sign Model

 This model uses the „+‟ and „–‟ signs.
 A positive number is represented by „+‟ sign.
 A negative number is represented by „–‟ sign.

EXAMPLES

Example 1

What is the value of 3 – 5?

NUMBER SIGN

3 + + +
–5 – – – – –

WORKINGS
+ + +
i. Pair up the opposite signs.
    
ii. The number of the unpaired signs is
the answer.

Answer –2

16


Example 2

What is the value of  3  5 ?

NUMBER SIGN

–3 _ _ _

–5 – – – – –

WORKINGS

There is no opposite sign to pair up, so _ _ _ _ _ _ _ _
just count the number of signs.

Answer –8

Example 3

What is the value of  3  5 ?

NUMBER SIGN

–3 – – –
+5 + + + + +

WORKINGS _ _ _
i. Pair up the opposite signs. + + + + +
ii. The number of unpaired signs is the
answer.
Answer 2

17


TEST YOURSELF B


1. –4 + 8 2. –8 – 4 3. 12 – 7

4. –5 – 5 5. 5–7–4 6. –7 + 4 – 3

7. 4+3–7 8. 6–2 +8 9. –3 + 4 + 6

18


PART C:
FURTHER PRACTICE ON
OF INTEGERS

LEARNING OBJECTIVE

Upon completion of Part C, pupils will be able to perform computations
involving addition and subtraction of large integers.


This part emphasises addition and subtraction of large positive and negative integers.

Strategy:

Teacher should ensure the pupils are able to perform computation involving addition
and subtraction of large integers.

19


PART C:
FURTHER PRACTICE ON ADDITION AND SUBTRACTION OF INTEGERS

LESSON NOTES

In Part A and Part B, the method of counting off the answer on a number line and the Sign
Model were used to perform computations involving addition and subtraction of small integers.
However, these methods are not suitable if we are dealing with large integers. We can use the
following Table Model in order to perform computations involving addition and subtraction
of large integers.

Steps for Adding and Subtracting
Integers

1. Draw a table that has a column for + and a column
for –.

2. Write down all the numbers accordingly in the
column.

3. If the operation involves numbers with the same
signs, simply add the numbers and then put the
respective sign in the answer. (Note that we
normally do not put positive sign in front of a
positive number)

4. If the operation involves numbers with different
signs, always subtract the smaller number from
the larger number and then put the sign of the
larger number in the answer.

20


Examples:

i) 34 + 37 =

+ – Add the numbers and then put the
positive sign in the answer.
34
37 We can just write the answer as
71 instead of +71.

+71

ii) 65 – 20 =
Subtract the smaller number from
+ – the larger number and put the sign
of the larger number in the
65 20 answer.

+45 We can just write the answer as
45 instead of +45.

iii) –73 + 22 =

+ – Subtract the smaller number from
the larger number and put the sign
22 73
answer.
–51

iv) 228 – 338 =

+ –
228 338 the larger number and put the sign
–110 answer.

21


v) –428 – 316 =

+ –

428
316
Add the numbers and then put the
negative sign in the answer.
–744

vi) –863 – 127 + 225 =

+ –

225 863 Add the two numbers in the „–‟
column and bring down the number
127 in the „+‟ column.

225 990 Subtract the smaller number from
the larger number in the third row
–765 and put the sign of the larger
number in the answer.

vii) 234 – 675 – 567 =

+ –

567
in the „+‟ column.


22


viii) –482 + 236 – 718 =

+ –

718 in the „+‟ column.


ix) –765 – 984 + 432 =

+ –

432 765
Add the two numbers in the „–‟
984 column and bring down the number
in the „+‟ column.

432 1749
–1317 the larger number in the third row
and put the sign of the larger
x) –1782 + 436 + 652 =

+ –

436 1782 Add the two numbers in the „+‟
652 in the „–‟ column.

1782
1088 Subtract the smaller number from

23


TEST YOURSELF C


1. 47 – 89 2. –54 – 48 3. 33 – 125

4. –352 – 556 5. 345 – 437 – 456 6. –237 + 564 – 318

7. –431 + 366 – 778 8. –652 – 517 + 887 9. –233 + 408 – 689

24


PART D:
OF INTEGERS INCLUDING THE
USE OF BRACKETS

LEARNING OBJECTIVE

Upon completion of Part D, pupils will be able to perform computations
involving combined operations of addition and subtraction of integers, including
the use of brackets, using the Accept-Reject Model.


This part emphasises the second alternative method which include activities to
enhance pupils‟ understanding and mastery of the addition and subtraction of
integers, including the use of brackets.

Strategy:

Teacher should ensure that pupils understand the concept of addition and subtraction
of integers, including the use of brackets, using the Accept-Reject Model.

25


PART D:
INCLUDING THE USE OF BRACKETS

LESSON NOTES

The Accept - Reject Model

 „+‟ sign means to accept.
 „–‟ sign means to reject.

To Accept or To Reject? Answer

+(5) Accept +5 +5

–(2) Reject +2 –2

+ (–4) Accept –4 –4

– (–8) Reject –8 +8

26


EXAMPLES

i) 5 + (–1) =

Number To Accept or To Reject? Answer
5 Accept 5 +5
+ (–1) Accept –1 –1

+ + + + +
–

5 + (–1) = 4

This operation of
5 + (–1) = 4
is the same as
5–1=4

We can also solve this question by using the Table Model as follows:

5 + (–1) = 5 – 1

+ – the larger number and put the sign
5 1
answer.
+4 We can just write the answer as 4
instead of +4.

27


ii) –6 + (–3) =


–6 Reject 6 –6
+ (–3) Accept –3 –3

– – – – – –
– – –
–6 + (–3) = –9

This operation of
–6 + (–3) = –9
is the same as
–6 –3 = –9


–6 + (–3) = –6 – 3 =

+ –

6
3 Add the numbers and then put the

–9

28


iii) –7 – (–4) =


–7 Reject 7 –7
– (–4) Reject –4 +4

– – – – – – –
+ + + +

–7 – (–4) = –3

This operation of
–7 – (–4) = –3
is the same as
–7 + 4 = –3


–7 – (–4) = –7 + 4 =

+ –
4 7 the larger number and put the sign
–3 answer.

29


iv) –5 – (3) =


–5 Reject 5 –5
– (3) Reject 3 –3

– – – – –
– – –
– 5 – (3) = –8

This operation of
–5 – (3) = –8
is the same as
–5 – 3 = –8


–5 – (3) = –5 – 3 =

+ –

5

–8

30


v) –35 + (–57) = –35 – 57 = This operation of
–35 + (–57)
is the same as
–35 – 57

Using the Table Model:

+ –

35
–92

vi) –123 – (–62) = –123 + 62 =
This operation of
–123 – (–62)
is the same as
–123 + 62

Using the Table Model:

+ –

the larger number and put the sign
of the larger number in the answer.
–61

31


TEST YOURSELF D


1. –4 + (–8) 2. 8 – (–4) 3. –12 + (–7)

4. –5 + (–5) 5. 5 – (–7) + (–4) 6. 7 + (–4) – (3)

7. 4 + (–3) – (–7) 8. –6 – (2) + (8) 9. –3 + (–4) + (6)

10. –44 + (–81) 11. 118 – (–43) 12. –125 + (–77)

13. –125 + (–239) 14. 125 – (–347) + (–234) 15. 237 + (–465) – (378)

16. 412 + (–334) – (–712) 17. –612 – (245) + (876) 18. –319 + (–412) + (606)

32


PART E:
MULTIPLICATION OF
INTEGERS

LEARNING OBJECTIVE

Upon completion of Part E, pupils will be able to perform computations
involving multiplication of integers.


This part emphasises the multiplication rules of integers.

Strategy:

Teacher should ensure that pupils understand the multiplication rules to perform
computations involving multiplication of integers.

33


PART E:
MULTIPLICATION OF INTEGERS

LESSON NOTES

Consider the following pattern:
3×3=9

3 2  6
positive × positive = positive
3 1  3 (+) × (+) = (+)

3 0  0 The result is reduced by 3 in
positive × negative = negative
3  (1)  3 every step. (+) × (–) = (–)

3  (2)  6

3  (3)  9

(3)  3  9

(3)  2  6
negative × positive = negative
(3)  1  3 (–) × (+) = (–)

(3)  0  0 The result is increased by 3 in
negative × negative = positive
(3)  (1)  3 every step. (–) × (–) = (+)

(3)  (2)  6

(3)  (3)  9

Multiplication Rules of Integers
1. When multiplying two integers of the same signs, the answer is positive integer.
2. When multiplying two integers of different signs, the answer is negative integer.
3. When any integer is multiplied by zero, the answer is always zero.

34


EXAMPLES

1. When multiplying two integers of the same signs, the answer is positive integer.

(a) 4 × 3 = 12

(b) –8 × –6 = 48

2. When multiplying two integers of the different signs, the answer is negative integer.

(a) –4 × (3) = –12

(b) 8 × (–6) = –48

3. When any integer is multiplied by zero, the answer is always zero.

(a) (4) × 0 = 0

(b) (–8) × 0 = 0

(c) 0 × (5) = 0

(d) 0 × (–7) = 0

35


TEST YOURSELF E


1. –4 × (–8) 2. 8 × (–4) 3. –12 × (–7)

4. –5 × (–5) 5. 5 × (–7) × (–4) 6. 7 × (–4) × (3)

7. 4 × (–3) × (–7) 8. (–6) × (2) × (8) 9. (–3) × (–4) × (6)

36


PART F:
USING
THE ACCEPT-REJECT MODEL

LEARNING OBJECTIVE

Upon completion of Part F, pupils will be able to perform computations
involving multiplication of integers using the Accept-Reject Model.


This part emphasises the second alternative method which include activities to
enhance the pupils‟ understanding and mastery of the multiplication of integers.

Strategy:

Teacher should ensure that pupils understand the multiplication rules of integers
using the Accept-Reject Model. Pupils can then perform computations involving
multiplication of integers.

37


PART F:
USING THE ACCEPT-REJECT MODEL

LESSON NOTES

The Accept-Reject Model

 In order to help pupils have a better understanding of multiplication of integers, we have
designed the Accept-Reject Model.

 Notes: (+) × (+) : The first sign in the operation will determine whether to accept
or to reject the second sign.

Multiplication Rules:

Sign To Accept or To Reject Answer

(+) × (+) Accept + 
(–) × (–) Reject – 
(+) × (–) Accept – –
(–) × (+) Reject + –

EXAMPLES

To Accept or to Reject Answer
(2) × (3) Accept + 6

(–2) × (–3) Reject – 6

(2) × (–3) Accept – –6
(–2) × (3) Reject + –6

38


TEST YOURSELF F


1. 3 × (–5) = 2. –4 × (–8) = 3. 6 × (5) =

4. 8 × (–6) = 5. – (–5) × 7 = 6. (–30) × (–4) =

7. 4 × 9 × (–6) = 8. (–3) × 5 × (–6) = 9. (–2) × ( –9) × (–6) =

10. –5× (–3) × (+4) = 11. 7 × (–2) × (+3) = 12. 5 × 8 × (–2) =

39


PART G:
DIVISION OF INTEGERS

LEARNING OBJECTIVE

Upon completion of Part G, pupils will be able to perform computations
involving division of integers.


This part emphasises the division rules of integers.

Strategy:

Teacher should ensure that pupils understand the division rules of integers to
perform computation involving division of integers.

40


PART G:

LESSON NOTES

Consider the following pattern:
3 × 2 = 6, then 6÷2=3 and 6÷3=2

3 × (–2) = –6, then (–6) ÷ 3 = –2 and (–6) ÷ (–2) = 3

(–3) × 2 = –6, then (–6) ÷ 2 = –3 and (–6) ÷ (–3) = 2

(–3) × (–2) = 6, then 6 ÷ (–3) = –2 and 6 ÷ (–2) = –3

Rules of Division

1. Division of two integers of the same signs results in a positive integer.

i.e. positive ÷ positive = positive
(+) ÷ (+) = (+)

negative ÷ negative = positive
(–) ÷ (–) = (+)

2. Division of two integers of different signs results in a negative integer.

i.e. positive ÷ negative = negative
(+) ÷ (–) = (–)

Undefined means “this
negative ÷ positive = negative operation does not have a
(–) ÷ (+) = (–) meaning and is thus not
assigned an interpretation!”

Source:
3. Division of any number by zero is undefined. http://www.sn0wb0ard.com

41


EXAMPLES

1. Division of two integers of the same signs results in a positive integer.

(a) (12) ÷ (3) = 4

(b) (–8) ÷ (–2) = 4

2. Division of two integers of different signs results in a negative integer.

(a) (–12) ÷ (3) = –4

(b) (+8) ÷ (–2) = –4

3. Division of zero by any number will always give zero as an answer.

(a) 0 ÷ (5) = 0

(b) 0 ÷ (–7) = 0

42


TEST YOURSELF G


1. (–24) ÷ (–8) 2. 8 ÷ (–4) 3. (–21) ÷ (–7)

4. (–5) ÷ (–5) 5. 60 ÷ (–5) ÷ (–4) 6. 36 ÷ (–4) ÷ (3)

7. 42 ÷ (–3) ÷ (–7) 8. (–16) ÷ (2) ÷ (8) 9. (–48) ÷ (–4) ÷ (6)

43


PART H:
USING
THE ACCEPT-REJECT MODEL

LEARNING OBJECTIVE

Upon completion of Part H, pupils will be able to perform computations
involving division of integers using the Accept-Reject Model.


This part emphasises the alternative method that include activities to help pupils
further understand and master division of integers.

Strategy:

Teacher should make sure that pupils understand the division rules of integers using
the Accept-Reject Model. Pupils can then perform division of integers, including
the use of brackets.

44


PART H:
DIVISION OF INTEGERS USING THE ACCEPT-REJECT MODEL

LESSON NOTES

 In order to help pupils have a better understanding of division of integers, we have designed
the Accept-Reject Model.

 Notes: (+) ÷ (+) : The first sign in the operation will determine whether to accept
or to reject the second sign.

() : The sign of the numerator will determine whether to accept or
() to reject the sign of the denominator.

Division Rules:


(+) ÷ (+) Accept + +

(–) ÷ (–) Reject – +

(+) ÷ (–) Accept – –

(–) ÷ (+) Reject + –

45


EXAMPLES

To Accept or To Reject Answer

(6) ÷ (3) Accept + 2

(–6) ÷ (–3) Reject – 2

(+6) ÷ (–3) Accept – –2

(–6) ÷ (3) Reject + –2

Division [Fraction Form]:


() Accept + +
()

()
Reject – +
()

()
Accept – –
()

()
Reject + –
()

46


EXAMPLES

To Accept or To Reject Answer

(  8)
Accept + 4
(  2)

(  8)
Reject – 4
(  2)

(  8)
Accept – –4
( 2)

(  8)
Reject + –4
( 2)

47


TEST YOURSELF H


1. 18 ÷ (–6) 12 24
2. 3.
2 8

 25 6 6. – (–35) ÷ 7
4. 5.
5 3

7. (–32) ÷ (–4) 8. (–45) ÷ 9 ÷ (–5) (30 )
9.
(6)

80 11. 12 ÷ (–3) ÷ (–2) 12. – (–6) ÷ (3)
10.
(5)

48


PART I:
COMBINED OPERATIONS
INVOLVING INTEGERS

LEARNING OBJECTIVES

Upon completion of Part I, pupils will be able to:

1. perform computations involving combined operations of addition,
subtraction, multiplication and division of integers to solve problems; and

2. apply the order of operations to solve the given problems.


This part emphasises the order of operations when solving combined operations
involving integers.

Strategy:

Teacher should make sure that pupils are able to understand the order of operations
or also known as the BODMAS rule. Pupils can then perform combined operations
involving integers.

49


PART I:
COMBINED OPERATIONS INVOLVING INTEGERS

LESSON NOTES

 A standard order of operations for calculations involving +, –, ×, ÷ and
brackets:

Step 1: First, perform all calculations inside the brackets.

Step 2: Next, perform all multiplications and divisions,
working from left to right.

Step 3: Lastly, perform all additions and subtractions, working
from left to right.

 The above order of operations is also known as the BODMAS Rule
and can be summarized as:

Brackets
power of
Division
Multiplication
Addition
Subtraction

EXAMPLES

1. 10 – (–4) × 3 2. (–4) × (–8 – 3 ) 3. (–6) + (–3 + 8 ) ÷5
= (–4) × (–11 ) = (–6 )+ (5) ÷5
=10 – (–12)
= 44 = (–6 )+ 1
= 10 + 12 = –5
= 22

50


TEST YOURSELF I


1. 12 + (8 ÷ 2) 2. (–3 – 5) × 2 3. 4 – (16 ÷ 2) × 2

4. (– 4) × 2 + 6 × 3 5. ( –25) ÷ (35 ÷ 7) 6. (–20) – (3 + 4) × 2

7. (–12) + (–4 × –6) ÷ 3 8. 16 ÷ 4 + (–2) 9. (–18 ÷ 2) + 5 – (–4)

51


ANSWERS

TEST YOURSELF A:

1. 2

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

2. –3

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

3. 6

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

4. –4

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

5. –2

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

52


TEST YOURSELF B:

1) 4 2) –12 3) 5
4) –10 5) –6 6) –6
7) 0 8) 12 9) 7

TEST YOURSELF C:

1) –42 2) –102 3) –92
4) –908 5) –548 6) 9
7) –843 8) –282 9) –514

TEST YOURSELF D:

1) –12 2) 12 3) –19
4) –10 5) 8 6) 0
7) 8 8) 0 9) –1
10) –125 11) 161 12) –202
13) –364 14) 238 15) –606
16) 790 17) 19 18) –125

TEST YOURSELF E:

1) 32 2) –32 3) 84
4) 25 5) 140 6) –84
7) 84 8) –96 9) 72

53


TEST YOURSELF F:

1) –15 2) 32 3) 30
4) –48 5) 35 6) 120
7) –216 8) 90 9) –108
10) 60 11) –42 12) –80

TEST YOURSELF G:

1) 3 2) –2 3) 3
4) 1 5) 3 6) –3
7) 2 8) –1 9) 2

TEST YOURSELF H:

1. –3 2. –6 3. 3
4. 5 5. –2 6. 5
7. 8 8. 1 9. 5
10. –16 11. 2 12. 2

TEST YOURSELF I:

1. 16 2. –16 3. –12
4. 10 5. –5 6. –34
7. –4 8. 2 9. 0

54

Basic Essential

Additional Mathematics Skills

UNIT 2

FRACTIONS

Unit 1:
Negative Numbers


TABLE OF CONTENTS

Module Overview 1

Part A: Addition and Subtraction of Fractions 2
1.0 Addition and Subtraction of Fractions with the Same Denominator 5
1.1 Addition of Fractions with the Same Denominators 5
1.2 Subtraction of Fractions with The Same Denominators 6
1.3 Addition and Subtraction Involving Whole Numbers and Fractions 7
1.4 Addition or Subtraction Involving Mixed Numbers and Fractions 9
2.0 Addition and Subtraction of Fractions with Different Denominator 10
2.1 Addition and Subtraction of Fractions When the Denominator
of One Fraction is A Multiple of That of the Other Fraction 11
2.2 Addition and Subtraction of Fractions When the Denominators
Are Not Multiple of One Another 13
2.3 Addition or Subtraction of Mixed Numbers with Different
Denominators 16
2.4 Addition or Subtraction of Algebraic Expression with Different
Denominators 17

Part B: Multiplication and Division of Fractions 22
1.0 Multiplication of Fractions 24
1.1 Multiplication of Simple Fractions 28
1.2 Multiplication of Fractions with Common Factors 29
1.3 Multiplication of a Whole Number and a Fraction 29
1.4 Multiplication of Algebraic Fractions 31
2.0 Division of Fractions 33
2.1 Division of Simple Fractions 36
2.2 Division of Fractions with Common Factors 37

Answers 42

UNIT 2: Fractions

MODULE OVERVIEW

1. The aim of this module is to reinforce pupils’ understanding of the concept
of fractions.

2. It serves as a guide for teachers in helping pupils to master the basic
computation skills (addition, subtraction, multiplication and division)
involving integers and fractions.

3. This module consists of two parts, and each part consists of learning

PART 1
objectives which can be taught separately. Teachers may use any parts of the
module as and when it is required.

Ministry of Education Malaysia 1

UNIT 2: Fractions

PART A:
OF FRACTIONS

LEARNING OBJECTIVES

Upon completion of Part A, pupils will be able to:

1. perform computations involving combination of two or more operations
on integers and fractions;

2. pose and solve problems involving integers and fractions;

3. add or subtract two algebraic fractions with the same denominators;

4. add or subtract two algebraic fractions with one denominator as a
multiple of the other denominator; and

5. add or subtract two algebraic fractions with denominators:

(i) not having any common factor;
(ii) having a common factor.


UNIT 2: Fractions


Pupils have difficulties in adding and subtracting fractions with different
denominators.

Strategy:

Teachers should emphasise that pupils have to find the equivalent form of
the fractions with common denominators by finding the lowest common
multiple (LCM) of the denominators.


UNIT 2: Fractions

LESSON NOTES

Fraction is written in the form of:

a numerator
b denominator

Examples:
2 4
,
3 3
Proper Fraction Improper Fraction Mixed Numbers

The numerator is smaller The numerator is larger A whole number and
than the denominator. than or equal to the denominator. a fraction combined.

Examples: Examples: Examples:

2 9 15 108 2 1 ,85
, , 7 6
3 20 4 12

Rules for Adding or Subtracting Fractions

1. When the denominators are the same, add or subtract only the numerators and
keep the denominator the same in the answer.

2. When the denominators are different, find the equivalent fractions that have the
same denominator.

Note: Emphasise that mixed numbers and whole numbers must be converted to improper
fractions before adding or subtracting fractions.


UNIT 2: Fractions

EXAMPLES

1.0 Addition And Subtraction of Fractions with the Same Denominator

1.1 Addition of Fractions with the Same Denominators

Add only the numerators and keep the
1 4 5
i)   denominator same.
8 8 8

1  4  5
8 8 8

1 3 4 denominator the same.
ii)  
8 8 8
1 Write the fraction in its simplest form.

2

1 5 6
iii)   denominator the same.
f f f


UNIT 2: Fractions

1.2 Subtraction of Fractions with The Same Denominators

Subtract only the numerators and keep
5 1 4
i)   the denominator the same.
8 8 8
1
 Write the fraction in its simplest form.
2

4 1
5
 1 
8

2
8 8

1 5 4
ii)   the denominator the same.
7 7 7

3 1 2
iii)   the denominator the same.
n n n


UNIT 2: Fractions

1.3 Addition and Subtraction Involving Whole Numbers and Fractions

1
i) Calculate 1  .
8

1
1 +
8

9
8 1 
 + 8
8 8
1
 1
8

 First, convert the whole number to an improper fraction with the
same denominator as that of the other fraction.
 Then, add or subtract only the numerators and keep the denominator
the same.

1 28 1 2 20 2 1 12 1
4    4    4 y  y
7 7 7 5 5 5 3 3 3

29 18 12  y
  
7 5 3

1 3
 4  3
7 5


UNIT 2: Fractions

 First, convert the whole number to an improper fraction with
the same denominator as that of the other fraction.
 Then, add or subtract only the numerators and keep the
denominator the same.

5 2n 5 2 2 3k
2     3 
n n n k k k

2n  5 2  3k
 
n k


UNIT 2: Fractions

1.4 Addition or Subtraction Involving Mixed Numbers and Fractions
1 4
i) Calculate 1  .
8 8

1 + 4
1
8 8

 9 + 4  13 5
 1
8 8 8 8

 First, convert the mixed number to improper fraction.
 Then, add or subtract only the numerators and keep the
denominator the same.

1 5 15 5 2 4 29 4 3 x 11 x
2    3    1   
7 7 7 7 9 9 9 9 8 8 8 8

20 6 25 7 11  x
= = 2 = = 2 =
7 7 9 9 8


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