BEAMS KPM

Basic Essential Additional Mathematics Skills

Curriculum Development Division
Ministry of Education Malaysia

Putrajaya
2010

First published 2010

© Curriculum Development Division,
Ministry of Education Malaysia
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 i
ii
Preface iii
Acknowledgement iii
Introduction iii
Objective
Module Layout
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 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

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

Curriculum Development Division

Dr. Lee Boon Hua
Deputy Director (Humanities)
Curriculum Development Division

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

Curriculum Development Division

Editorial Advisor:

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

Editors:

Dr. Rusilawati binti Othman
Assistant Director

(Head of Secondary Mathematics Unit)
Curriculum Development Division

Aszunarni binti Ayob
Assistant Director

Curriculum Development Division

Rosita binti Mat Zain
Assistant Director

Curriculum Development Division

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

Curriculum Development Division
Ministry of Education Malaysia

TABLE OF CONTENTS

Module Overview 1
Part A: Addition and Subtraction of Integers Using Number Lines 2
3
1.0 Representing Integers on a Number Line 3
2.0 Addition and Subtraction of Positive Integers 8
3.0 Addition and Subtraction of Negative Integers 15
Part B: Addition and Subtraction of Integers Using the Sign Model 19
Part C: Further Practice on Addition and Subtraction of Integers 25
Part D: Addition and Subtraction of Integers Including the Use of Brackets 33
Part E: Multiplication of Integers 37
Part F: Multiplication of Integers Using the Accept-Reject Model 40
Part G: Division of Integers 44
Part H: Division of Integers Using the Accept-Reject Model 49
Part I: Combined Operations Involving Integers 52
Answers

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

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.

Curriculum Development Division 1
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

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.

Curriculum Development Division 2
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

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.

–3 –2 –1 0 1 2 3 4 Positive integers
Note: i) –3 is the opposite of +3 may have a plus sign

in front of them,
like +3, or no sign in

front, like 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 3
 When subtracting a positive integer, you move to the left

on a number line.

–3 –2 –1 0 1 2 3 4

Curriculum Development Division
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

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 4
Adding a positive integer:

Start at 2 and move 3 units to the right:
2+3=5

Curriculum Development Division
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

(ii) –2 + 5

Add a
positive 5

–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 5 units to the right:
–2 + 5 = 3

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

Adding a positive integer:
Start at –2 and move 5 units to the right:

–2 + 5 = 3

Curriculum Development Division 5
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

(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:

Make sure you start from
the position of the first

integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Subtracting a positive integer:
Start at 2 and move 5 units to the left:

2 – 5 = –3

Curriculum Development Division 6
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

(iv) –3 – 2 = –5

Subtract a
positive 2

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Subtracting a positive integer:
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:

Make sure you start from
the position of the first

integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Subtracting a positive integer:
Start at –3 and move 2 units to the left:

–3 – 2 = –5

Curriculum Development Division 7
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

3.0 Addition and Subtraction of Negative Integers
Consider the following operations:

4–1=3 –3 –2 –1 0 1 2 3 4 4 + (–1) = 3
4–2=2 –3 –2 –1 0 1 2 3 4 4 + (–2) = 2
4–3=1 –3 –2 –1 0 1 2 3 4 4 + (–3) = 1
4–4=0 –3 –2 –1 0 1 2 3 4 4 + (–4) = 0
4 – 5 = –1 –3 –2 –1 0 1 2 3 4 4 + (–5) = –1
4 – 6 = –2 –3 –2 –1 0 1 2 3 4 4 + (–6) = –2

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.

Curriculum Development Division 8
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

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

Curriculum Development Division 9
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

EXAMPLES

(i) –2 + (–1) = –3 This operation of

Add a –2 + (–1) = –3
negative 1
is the same as

–2 –1 = –3.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Adding a negative integer:
Start by drawing an arrow from 0 to –2, and

then, draw an arrow of 1 unit to the left:
–2 + (–1) = –3

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 10

Adding a negative integer:
Start at –2 and move 1 unit to the left:

–2 + (–1) = –3

Curriculum Development Division
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

(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

Adding a negative integer:
Start by drawing an arrow from 0 to 1, then, draw an arrow of

3 units to the left:
1 + (–3) = –2

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

Adding a negative integer:
Start at 1 and move 3 units to the left:

1 + (–3) = –2

Curriculum Development Division 11
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

(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:

Make sure you start from
the position of the first

integer.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 12

Subtracting a negative integer:
Start at 3 and move 3 units to the right:

3 – (–3) = 6

Curriculum Development Division
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

(iv) –5 – (–8) = 3 This operation of

Subtract a –5 – (–8) = 3
negative 8
is the same as

–5 + 8 = 3

3+3=6

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Subtracting a negative integer:
Start by drawing an arrow from 0 to –5, and
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

Subtracting a negative integer:
Start at –5 and move 8 units to the right:

–5 – (–8) = 3

Curriculum Development Division 13
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

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

Curriculum Development Division 14
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

PART B:

ADDITION AND SUBTRACTION
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.

TEACHING AND LEARNING STRATEGIES
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.

Curriculum Development Division 15
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

PART B:
ADDITION AND SUBTRACTION OF INTEGERS

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 SIGN
What is the value of 3 – 5?
++ +
NUMBER –––––
3
–5 +++
   
WORKINGS
i. Pair up the opposite signs. –2
ii. The number of the unpaired signs is

the answer.

Answer

Curriculum Development Division 16
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

Example 2 SIGN
What is the value of  3 5 ?
___
NUMBER –––––
–3
________
–5
–8
WORKINGS
There is no opposite sign to pair up, so SIGN

just count the number of signs. –––
Answer + + + ++
___
Example 3 + + + ++
What is the value of  3  5 ?
2
NUMBER
–3
+5

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

answer.
Answer

Curriculum Development Division 17
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF B

Solve the following. 2. –8 – 4 3. 12 – 7
1. –4 + 8

4. –5 – 5 5. 5 – 7 – 4 6. –7 + 4 – 3

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

Curriculum Development Division 18
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

PART C:
FURTHER PRACTICE ON
ADDITION AND SUBTRACTION

OF INTEGERS

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

TEACHING AND LEARNING STRATEGIES
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.

Curriculum Development Division 19
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

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.

Curriculum Development Division 20
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

Examples: – Add the numbers and then put the
i) 34 + 37 = positive sign in the answer.

+ We can just write the answer as
71 instead of +71.
34
37 Subtract the smaller number from
the larger number and put the sign
+71
of the larger number in the
ii) 65 – 20 = answer.
+–
65 20 We can just write the answer as
+45 45 instead of +45.

iii) –73 + 22 = – Subtract the smaller number from
73 the larger number and put the sign
+
22 of the larger number in the
answer.
–51
Subtract the smaller number from
iv) 228 – 338 = the larger number and put the sign
+–
228 338 of the larger number in the
–110 answer.

Curriculum Development Division 21
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

v) –428 – 316 = –
+

428
316

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

vi) –863 – 127 + 225 = Add the two numbers in the „–‟
+– column and bring down the number
225 863
127 in the „+‟ column.
225 990
–765 Subtract the smaller number from
the larger number in the third row
vii) 234 – 675 – 567 =
+– and put the sign of the larger
234 675 number in the answer.
567
234 1242 Add the two numbers in the „–‟
–1008 column and bring down the number

in the „+‟ column.

Subtract the smaller number from
the larger number in the third row

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

Curriculum Development Division 22
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

viii) –482 + 236 – 718 = Add the two numbers in the „–‟
+– column and bring down the number
236 482
718 in the „+‟ column.
236 1200
–964 Subtract the smaller number from
the larger number in the third row
ix) –765 – 984 + 432 =
and put the sign of the larger
number in the answer.

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

984 in the „+‟ column.

432 1749 Subtract the smaller number from
–1317 the larger number in the third row

x) –1782 + 436 + 652 = and put the sign of the larger
+– number in the answer.

436 1782 Add the two numbers in the „+‟
column and bring down the number
652
in the „–‟ column.
1088 1782
Subtract the smaller number from
–694 the larger number in the third row

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

Curriculum Development Division 23
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF C

Solve the following. 2. –54 – 48 3. 33 – 125
1. 47 – 89

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

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

Curriculum Development Division 24
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

PART D:

ADDITION AND SUBTRACTION
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.

TEACHING AND LEARNING STRATEGIES
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.

Curriculum Development Division 25
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

PART D:
ADDITION AND SUBTRACTION OF INTEGERS

INCLUDING THE USE OF BRACKETS
LESSON NOTES

The Accept - Reject Model
 „+‟ sign means to accept.
 „–‟ sign means to reject.

+(5) To Accept or To Reject? Answer
–(2) Accept +5 +5
+ (–4) Reject +2 –2
– (–8) Accept –4 –4
Reject –8 +8

Curriculum Development Division 26
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

i) 5 + (–1) = EXAMPLES Answer
+5
Number To Accept or To Reject? –1
5 Accept 5
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 – Subtract the smaller number from
1 the larger number and put the sign
+
5 of the larger number in the
answer.
+4
We can just write the answer as 4
instead of +4.

Curriculum Development Division 27
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

ii) –6 + (–3) =

Number To Accept or To Reject? Answer

–6 Reject 6 –6
+ (–3) Accept –3 –3

–6 + (–3) = ––––––
–––
This operation of
–9
–6 + (–3) = –9

is the same as

–6 –3 = –9

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

–6 + (–3) = –6 – 3 =

+– Add the numbers and then put the
negative sign in the answer.
6
3

–9

Curriculum Development Division 28
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

iii) –7 – (–4) = To Accept or To Reject? Answer
Number
–7 Reject 7 –7
– (–4) Reject –4 +4

–7 – (–4) = –––––––
+ ++ +

–3

This operation of

–7 – (–4) = –3

is the same as

–7 + 4 = –3

We can also solve this question by using the Table Model as follows:
–7 – (–4) = –7 + 4 =

+– Subtract the smaller number from
47 the larger number and put the sign

–3 of the larger number in the
answer.

Curriculum Development Division 29
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

iv) –5 – (3) =

Number To Accept or To Reject? Answer

–5 Reject 5 –5
– (3) Reject 3 –3

– 5 – (3) = –––––
–––
This operation of
–8
–5 – (3) = –8

is the same as

–5 – 3 = –8

We can also solve this question by using the Table Model as follows:
–5 – (3) = –5 – 3 =

+– Add the numbers and then put the
negative sign in the answer.
5
3

–8

Curriculum Development Division 30
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

v) –35 + (–57) = –35 – 57 = This operation of
–35 + (–57)

is the same as
–35 – 57

Using the Table Model:

+– Add the numbers and then put the
negative sign in the answer.
35
57

–92

vi) –123 – (–62) = –123 + 62 = This operation of
Using the Table Model: –123 – (–62)

is the same as
–123 + 62

+– Subtract the smaller number from
62 123 the larger number and put the sign
of the larger number in the answer.
–61

Curriculum Development Division 31
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF D

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

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)

Curriculum Development Division 32
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

PART E:

MULTIPLICATION OF
INTEGERS

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

TEACHING AND LEARNING STRATEGIES
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.

Curriculum Development Division 33
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

PART E:
MULTIPLICATION OF INTEGERS

LESSON NOTES

Consider the following pattern:
3×3=9

32  6

31 3 positive × positive = positive
(+) × (+) = (+)
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 The result is increased by 3 in negative × positive = negative
(3)  2  6 every step. (–) × (+) = (–)
(3) 1  3
(3)  0  0 negative × negative = positive
(3)  (1)  3 (–) × (–) = (+)
(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.

Curriculum Development Division 34
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

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

Curriculum Development Division 35
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF E

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

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

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

Curriculum Development Division 36
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

PART F:

MULTIPLICATION OF INTEGERS
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.

TEACHING AND LEARNING STRATEGIES
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.

Curriculum Development Division 37
Ministry of Education Malaysia