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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
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
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.
INTRODUCTION
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.
iii
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
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
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
PANEL OF CONTRIBUTORS
Abdul Rahim bin Bujang
SM Tun Fatimah, Johor
Ali Akbar bin Asri SM Sains, Labuan
Amrah bin Bahari
SMK Dato’ Sheikh Ahmad, Arau, Perlis
Aziyah binti Paimin SMK Kompleks KLIA, , Negeri Sembilan
Bashirah binti Seleman
SMK Sultan Abdul Halim, Jitra, Kedah
Bibi Kismete binti Kabul Khan SMK Jelapang Jaya, Ipoh, Perak
Che Rokiah binti Md. Isa
SMK Dato’ Wan Mohd. Saman, Kedah
Cheong Nyok Tai SMK Perempuan, Kota Kinabalu, Sabah
Ding Hong Eng
SM Sains Alam Shah, Kuala Lumpur
Esah binti Daud SMK Seri Budiman, Kuala Terengganu
Haspiah binti Basiran
SMK Tun Perak, Jasin, Melaka
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
Hon May Wan
SMK Tasek Damai, Ipoh, Perak
Horsiah binti Ahmad SMK Tun Perak, Jasin, Melaka
Kalaimathi a/p Rajagopal
SMK Sungai Layar, Sungai Petani, Kedah
Kho Choong Quan SMK Ulu Kinta, Ipoh, Perak
Lau Choi Fong
SMK Hulu Klang, Selangor
Loh Peh Choo SMK Bandar Baru Sungai Buloh, Selangor
Mohd. Misbah bin Ramli
SMK Tunku Sulong, Gurun, Kedah
Noor Aida binti Mohd. Zin SMK Tinggi Kajang, Kajang, Selangor
Noor Ishak bin Mohd. Salleh
SMK Laksamana, Kota Tinggi, Johor
Noorliah binti Ahmat SM Teknik, Kuala Lumpur
Nor A’idah binti Johari
SMK Teknik Setapak, Selangor
Writers:
Layout and Illustration:
Aszunarni binti Ayob Mohd. Lufti bin Mahpudz
Assistant Director Assistant Director
Curriculum Development Division Curriculum Development Division
Writers:
Nor Dalina binti Idris
SMK Syed Alwi, Kangar, Perlis
Norizatun binti Abdul Samid
SMK Sultan Badlishah, Kulim, Kedah
Pahimi bin Wan Salleh Maktab Sultan Ismail, Kelantan
Rauziah binti Mohd. Ayob
SMK Bandar Baru Salak Tinggi, Selangor
Rohaya binti Shaari SMK Tinggi Bukit Merajam, Pulau Pinang
Roziah binti Hj. Zakaria
SMK Taman Inderawasih, Pulau Pinang
Shakiroh binti Awang SM Teknik Tuanku Jaafar, Negeri Sembilan
Sharina binti Mohd. Zulkifli
SMK Agama, Arau, Perlis
Sim Kwang Yaw SMK Petra, Kuching, Sarawak
Suhaimi bin Mohd. Tabiee
SMK Datuk Haji Abdul Kadir, Pulau Pinang
Suraiya binti Abdul Halim
SMK Pokok Sena, Pulau Pinang
Tan Lee Fang SMK Perlis, Perlis
Tempawan binti Abdul Aziz
SMK Mahsuri, Langkawi, Kedah
Turasima binti Marjuki SMKA Simpang Lima, Selangor
Wan Azlilah binti Wan Nawi
SMK Putrajaya Presint 9(1), WP Putrajaya
Zainah binti Kebi SMK Pandan, Kuantan, Pahang
Zaleha binti Tomijan
SMK Ayer Puteh Dalam, Pendang, Kedah
Zariah binti Hassan SMK Dato’ Onn, Butterworth, Pulau Pinang
Unit 1:
Negative Numbers
UNIT 1
NEGATIVE NUMBERS
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
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
Unit 1: Negative Numbers
1
Curriculum Development Division
Ministry of Education Malaysia
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
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.
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
PART A:
ADDITION AND SUBTRACTION OF INTEGERS
USING NUMBER LINES
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.
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
–3 –2 –1 0 1 2 3 4
LESSON NOTES
Rules for Adding and Subtracting Positive Integers
When adding a positive integer, you move to the right on a
number line.
When subtracting a positive integer, you move to the left
on a number line.
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
Positive integers
may have a plus sign
in front of them,
like +3, or no sign in
front, like 3.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
(i) 2 + 3
Alternative Method:
EXAMPLES
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
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Start
with 2
Add a
positive 3
Adding a positive integer:
Start at 2 and move 3 units to the right:
2 + 3 = 5
Make sure you start from
the position of the first
integer.
–5 –4
–3 –2 –1 0 1 2 3 4 5 6
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
(ii) –2 + 5
Alternative Method:
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
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Add a
positive 5
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
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
(iii) 2 – 5 = –3
Alternative Method:
–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
Subtract a
positive 5
Subtracting a positive integer:
Start at 2 and move 5 units to the left:
2 – 5 = –3
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Make sure you start from
the position of the first
integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
(iv) –3 – 2 = –5
Alternative Method:
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
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtract a
positive 2
–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
Make sure you start from
the position of the first
integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
3.0 Addition and Subtraction of Negative Integers
Consider the following operations:
4 – 1 = 3
4 – 2 = 2
4 – 3 = 1
4 – 4 = 0
4 – 5 = –1
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.
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
4 + (–5) = –1
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
4 + (–6) = –2
4 + (–1) = 3
4 + (–2) = 2
4 + (–3) = 1
4 + (–4) = 0
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
Rules for Adding and Subtracting Negative Integers
When adding a negative integer, you move to the left on a
number line.
When subtracting a negative integer, you move to the right
on a number line.
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
(i) –2 + (–1) = –3
Alternative Method:
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Adding a negative integer:
Start at –2 and move 1 unit to the left:
–2 + (–1) = –3
EXAMPLES
–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
Add a
negative 1
Make sure you start from
the position of the first
integer.
This operation of
–2 + (–1) = –3
is the same as
–2 –1 = –3.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
(ii) 1 + (–3) = –2
Alternative Method:
–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
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
Make sure you start from
the position of the first
integer.
This operation of
1 + (–3) = –2
is the same as
1 – 3 = –2
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
(iii) 3 – (–3) = 6
Alternative Method:
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtracting a negative integer:
Start at 3 and move 3 units to the right:
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 3, and
then, draw an arrow of 3 units to the right:
3 – (–3) = 6
Subtract a
negative 3
This operation of
3 – (–3) = 6
is the same as
3 + 3 = 6
Make sure you start from
the position of the first
integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
(iv) –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
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtract a
negative 8
This operation of
–5 – (–8) = 3
is the same as
–5 + 8 = 3
3 + 3 = 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
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
Solve the following.
1. –2 + 4
2. 3 + (–6)
3. 2 – (–4)
4. 3 – 5 + (–2)
5. –5 + 8 + (–5)
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
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.
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
PART B:
ADDITION AND SUBTRACTION OF INTEGERS
USING THE SIGN MODEL
In order to help pupils have a better understanding of positive and negative integers, we have
designed the Sign Model.
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
+
+
+
LESSON NOTES
EXAMPLES
The Sign Model
This model uses the „+‟ and „–‟ signs.
A positive number is represented by „+‟ sign.
A negative number is represented by „–‟ sign.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
Example 2
What is the value of 53 ?
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 53 ?
NUMBER SIGN
–3 – – –
+5 + + + + +
WORKINGS
i. Pair up the opposite signs.
ii. The number of unpaired signs is the
answer.
Answer 2
_
+ + +
_
+
_
+
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
Solve the following.
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
TEST YOURSELF B
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
PART C:
FURTHER PRACTICE ON
ADDITION AND SUBTRACTION
OF 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.
LEARNING OBJECTIVE
Upon completion of Part C, pupils will be able to perform computations
involving addition and subtraction of large integers.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
PART C:
FURTHER PRACTICE ON ADDITION AND SUBTRACTION OF INTEGERS
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.
LESSON NOTES
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
Examples:
i) 34 + 37 =
+ –
34
37
+71
ii) 65 – 20 =
+ –
65 20
+45
iii) –73 + 22 =
+ –
22 73
–51
iv) 228 – 338 =
+ –
228 338
–110
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
We can just write the answer as
45 instead of +45.
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
Add the numbers and then put the
positive sign in the answer.
We can just write the answer as
71 instead of +71.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
v) –428 – 316 =
+ –
428
316
–744
vi) –863 – 127 + 225 =
+ –
225
863
127
225 990
–765
vii) 234 – 675 – 567 =
+ –
234
675
567
234 1242
–1008
Add the numbers and then put the
negative sign in the answer.
Add the two numbers in the „–‟
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.
Add the two numbers in the „–‟
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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viii) –482 + 236 – 718 =
+ –
236
482
718
236 1200
–964
ix) –765 – 984 + 432 =
+ –
432
765
984
432
1749
–1317
x) –1782 + 436 + 652 =
+ –
436
652
1782
1088 1782
–694
Add the two numbers in the „–‟
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.
Add the two numbers in the „–‟
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.
Add the two numbers in the „+‟
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
Solve the following.
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
TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
Ministry of Education Malaysia
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.
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
26
Curriculum Development Division
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PART D:
ADDITION AND SUBTRACTION OF INTEGERS
INCLUDING THE USE OF BRACKETS
To Accept or To Reject? Answer
+ ( 5 ) Accept +5 +5
– ( 2 ) Reject +2 –2
+ (–4) Accept –4 –4
– (–8) Reject –8 +8
LESSON NOTES
The Accept - Reject Model
„+‟ sign means to accept.
„–‟ sign means to reject.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
27
Curriculum Development Division
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i) 5 + (–1) =
Number To Accept or To Reject? Answer
5
+ (–1)
Accept 5
Accept –1
+5
–1
+ + + + +
–
5 + (–1) = 4
We can also solve this question by using the Table Model as follows:
5 + (–1) = 5 – 1
+ –
5 1
+4
EXAMPLES
This operation of
5 + (–1) = 4
is the same as
5 – 1 = 4
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
We can just write the answer as 4
instead of +4.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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ii) –6 + (–3) =
Number To Accept or To Reject? Answer
–6
+ (–3)
Reject 6
Accept –3
–6
–3
– – – – – –
– – –
–6 + (–3) = –9
We can also solve this question by using the Table Model as follows:
–6 + (–3) = –6 – 3 =
+ –
6
3
–9
This operation of
–6 + (–3) = –9
is the same as
–6 –3 = –9
Add the numbers and then put the
negative sign in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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iii) –7 – (–4) =
Number To Accept or To Reject? Answer
–7
– (–4)
Reject 7
Reject –4
–7
+4
– – – – – – –
+ + + +
–7 – (–4) = –3
We can also solve this question by using the Table Model as follows:
–7 – (–4) = –7 + 4 =
+ –
4
7
–3
This operation of
–7 – (–4) = –3
is the same as
–7 + 4 = –3
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
30
Curriculum Development Division
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iv) –5 – (3) =
Number To Accept or To Reject? Answer
–5
– (3)
Reject 5
Reject 3
–5
–3
– – – – –
– – –
– 5 – (3) = –8
We can also solve this question by using the Table Model as follows:
–5 – (3) = –5 – 3 =
+ –
5
3
–8
This operation of
–5 – (3) = –8
is the same as
–5 – 3 = –8
Add the numbers and then put the
negative sign in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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v) –35 + (–57) = –35 – 57 =
Using the Table Model:
+ –
35
57
–92
vi) –123 – (–62) = –123 + 62 =
Using the Table Model:
+ –
62
123
–61
This operation of
–35 + (–57)
is the same as
–35 – 57
Add the numbers and then put the
negative sign in the answer.
Subtract the smaller number from
the larger number and put the sign
of the larger number in the answer.
This operation of
–123 – (–62)
is the same as
–123 + 62
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
32
Curriculum Development Division
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Solve the following.
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)
TEST YOURSELF D
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Unit 1: Negative Numbers
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PART E:
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.
LEARNING OBJECTIVE
Upon completion of Part E, pupils will be able to perform computations
involving multiplication of integers.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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PART E:
MULTIPLICATION OF INTEGERS
Consider the following pattern:
3 × 3 = 9
623
313
003 The result is reduced by 3 in
3)1(3 every step.
6)2(3
9)3(3
93)3(
62)3(
31)3(
00)3( The result is increased by 3 in
3)1()3( every step.
6)2()3(
9)3()3(
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.
positive × positive = positive
(+) × (+) = (+)
positive × negative = negative
(+) × (–) = (–)
negative × positive = negative
(–) × (+) = (–)
negative × negative = positive
(–) × (–) = (+)
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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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
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
36
Curriculum Development Division
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Solve the following.
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)
TEST YOURSELF E
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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PART F:
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.
LEARNING OBJECTIVE
Upon completion of Part F, pupils will be able to perform computations
involving multiplication of integers using the Accept-Reject Model.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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PART F:
MULTIPLICATION OF INTEGERS
USING THE ACCEPT-REJECT MODEL
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:
To Accept or to Reject Answer
(2) × (3) Accept + 6
(–2) × (–3) Reject – 6
(2) × (–3) Accept – –6
(–2) × (3) Reject + –6
Sign To Accept or To Reject Answer
( + ) × ( + ) Accept +
( – ) × ( – ) Reject –
( + ) × ( – ) Accept – –
( – ) × ( + ) Reject + –
LESSON NOTES
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
39
Curriculum Development Division
Ministry of Education Malaysia
Solve the following.
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) =
TEST YOURSELF F
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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TEACHING AND LEARNING STRATEGIES
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.
PART G:
DIVISION OF INTEGERS
LEARNING OBJECTIVE
Upon completion of Part G, pupils will be able to perform computations
involving division of integers.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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PART G:
DIVISION OF INTEGERS
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
(+) ÷ (–) = (–)
negative ÷ positive = negative
(–) ÷ (+) = (–)
3. Division of any number by zero is undefined.
LESSON NOTES
Undefined means “this
operation does not have a
meaning and is thus not
assigned an interpretation!”
Source:
http://www.sn0wb0ard.com
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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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
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
43
Curriculum Development Division
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Solve the following.
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)
TEST YOURSELF G
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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PART H:
DIVISION OF INTEGERS
USING
THE ACCEPT-REJECT MODEL
TEACHING AND LEARNING STRATEGIES
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.
LEARNING OBJECTIVE
Upon completion of Part H, pupils will be able to perform computations
involving division of integers using the Accept-Reject Model.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
45
Curriculum Development Division
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PART H:
DIVISION OF INTEGERS USING THE ACCEPT-REJECT MODEL
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:
Sign To Accept or To Reject Answer
( + ) ÷ ( + )
Accept +
+
( – ) ÷ ( – )
Reject – +
( + ) ÷ ( – ) Accept – –
( – ) ÷ ( + ) Reject + –
)(
)(
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
46
Curriculum Development Division
Ministry of Education Malaysia
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]:
Sign To Accept or To Reject Answer
)(
)(
Accept +
+
)(
)(
Reject – +
)(
)(
Accept – –
)(
)(
Reject + –
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
47
Curriculum Development Division
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To Accept or To Reject Answer
)2(
)8(
Accept + 4
)2(
)8(
Reject – 4
)2(
)8(
Accept – – 4
)2(
)8(
Reject + – 4
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
48
Curriculum Development Division
Ministry of Education Malaysia
Solve the following.
1. 18 ÷ (–6)
2. 2
12
3.
8
24
4. 5
25
5. 3
6
6. – (–35) ÷ 7
7. (–32) ÷ (–4)
8. (–45) ÷ 9 ÷ (–5) 9.
)6(
)30(
10. )5(
80
11. 12 ÷ (–3) ÷ (–2) 12. – (–6) ÷ (3)
TEST YOURSELF H
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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TEACHING AND LEARNING STRATEGIES
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.
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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PART I:
COMBINED OPERATIONS INVOLVING INTEGERS
1. 10 – (–4) × 3
=10 – (–12)
= 10 + 12
= 22
2. (–4) × (–8 – 3 )
= (–4) × (–11 )
= 44
3. (–6) + (–3 + 8 ) ÷5
= (–6 )+ (5) ÷5
= (–6 )+ 1
= –5
LESSON NOTES
EXAMPLES
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
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
51
Curriculum Development Division
Ministry of Education Malaysia
Solve the following.
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)
TEST YOURSELF I
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
52
Curriculum Development Division
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TEST YOURSELF A:
1. 2
2. –3
3. 6
4. –4
5. –2
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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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
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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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
Unit 1:
Negative Numbers
UNIT 2
FRACTIONS
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
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
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
1
Curriculum Development Division
Ministry of Education Malaysia
PART 1
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
objectives which can be taught separately. Teachers may use any parts of the
module as and when it is required.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
2
Curriculum Development Division
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PART A:
ADDITION AND SUBTRACTION
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
3
Curriculum Development Division
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TEACHING AND LEARNING STRATEGIES
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
4
Curriculum Development Division
Ministry of Education Malaysia
numerator
denominator
Fraction is written in the form of:
b
a
Examples:
3
4 ,
3
2
Proper Fraction Improper Fraction Mixed Numbers
The numerator is smaller
than the denominator.
Examples:
20
9 ,
3
2
The numerator is larger
than or equal to the denominator.
Examples:
12
108 ,
4
15
A whole number and
a fraction combined.
Examples:
65
71 8 ,2
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.
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
5
Curriculum Development Division
Ministry of Education Malaysia
1.0 Addition And Subtraction of Fractions with the Same Denominator
1.1 Addition of Fractions with the Same Denominators
8
5
8
4
8
1 i)
2
1
8
4
8
3
8
1 ii)
fff
651 iii)
EXAMPLES
Add only the numerators and keep the
denominator same.
Write the fraction in its simplest form.
Add only the numerators and keep the
denominator the same.
Add only the numerators and keep the
denominator the same.
8
1
8
4
8
5
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
6
Curriculum Development Division
Ministry of Education Malaysia
1.2 Subtraction of Fractions with The Same Denominators
2
1
8
4
8
1
8
5 i)
7
4
7
5
7
1 ii)
nnn
213 iii)
Write the fraction in its simplest form.
Subtract only the numerators and keep
the denominator the same.
Subtract only the numerators and keep
the denominator the same.
Subtract only the numerators and keep
the denominator the same.
8
5
8
1
2
1
8
4
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
7
Curriculum Development Division
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1.3 Addition and Subtraction Involving Whole Numbers and Fractions
.8
11 Calculate i)
7
29
7
1
7
28
7
14
7
14
5
18
5
2
5
20
5
24
5
33
3
12
3
1
3
12
3
14
y
yy
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 8
1
8
11
8
9
+
8
8
+
8
1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
8
Curriculum Development Division
Ministry of Education Malaysia
n
n
nn
n
n
52
5252
k
k
k
k
kk
32
323
2
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.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
9
Curriculum Development Division
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1.4 Addition or Subtraction Involving Mixed Numbers and Fractions
.8
4
8
11 Calculate i)
7
5
7
15
7
5
7
12
= 7
20 =
7
62
9
4
9
29
9
4
9
23
= 9
25 =
9
72
88
11
88
31
xx
= 8
11 x
First, convert the mixed number to improper fraction.
Then, add or subtract only the numerators and keep the denominator the same.
8
11
8
4
8
51
8
13
+
8
9
+
8
4
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UNIT 2: Fractions
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2.0 Addition and Subtraction of Fractions with Different Denominators
.2
1
8
1 Calculate i)
To make the denominators the same, multiply both the numerator and the denominator of
the second fraction by 4:
Now, the question can be visualized like this:
?
The denominators are not the same.
See how the slices are different in
sizes? Before we can add the
fractions, we need to make them the
same, because we can't add them
together like this!
8
1
8
4
+
8
5
8
4
2
1
4
4
Now, the denominators
are the same. Therefore,
we can add the fractions
together!
8
1
2
1
+
?
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UNIT 2: Fractions
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Hint: Before adding or subtracting fractions with different denominators, we must
convert each fraction to an equivalent fraction with the same denominator.
2.1 Addition and Subtraction of Fractions When the Denominator of One Fraction is
A Multiple of That of the Other Fraction
Multiply both the numerator and the denominator with an integer that makes the
denominators the same.
(i) 6
5
3
1
6
5
6
2
6
7
= 6
11
(ii) 4
3
12
7
12
9
12
7
12
2
6
1
Change the first fraction to an equivalent
fraction with denominator 6.
(Multiply both the numerator and the denominator of the first fraction by 2):
6
2
3
1
2
2
Add only the numerators and keep the
denominator the same.
Change the second fraction to an equivalent fraction with denominator 12.
(Multiply both the numerator and the
denominator of the second fraction by 3):
12
9
4
3
3
3
Subtract only the numerators and keep the
denominator the same.
Write the fraction in its simplest form.
Convert the fraction to a mixed number.
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UNIT 2: Fractions
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(iii) vv 5
91
vv 5
9
5
5
v5
14
Change the first fraction to an equivalent
fraction with denominator 5v.
(Multiply both the numerator and the denominator of the first fraction by 5):
vv 5
51
5
5
Add only the numerators and keep the
denominator the same.
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UNIT 2: Fractions
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2.2 Addition and Subtraction of Fractions When the Denominators Are Not Multiple of
One Another
Method I
4
3
6
1
(i) Find the Least Common Multiple (LCM)
of the denominators.
2) 4 , 6
2) 2 , 3
3) 1 , 3
- , 1
LCM = 2 2 3 = 12
The LCM of 4 and 6 is 12.
(ii) Change each fraction to an equivalent
fraction using the LCM as the
denominator.
(Multiply both the numerator and the
denominator of each fraction by a whole
number that will make their
denominators the same as the LCM
value).
= 4
3
6
1
= 12
9
12
2
= 12
11
Method II
4
3
6
1
(i) Multiply the numerator and the
denominator of the first fraction with
the denominator of the second fraction
and vice versa.
= 4
3
6
1
= 24
18
24
4
= 24
22
= 12
11
Write the fraction in its
simplest form.
This method is preferred but you
must remember to give the
answer in its simplest form. 3
3 2
2
4
4 6
6
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UNIT 2: Fractions
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Multiply the first fraction with the second denominator and
multiply the second fraction with the first denominator.
1. 5
1
3
2
= 5
5
3
2
+
3
3
5
1
15
3
15
10
= 15
13
2. 8
3
6
5
=
8
8
6
5
–
6
6
8
3
= 48
18
48
40
= 48
22
= 24
11
Write the fraction in its simplest form.
EXAMPLES
Multiply the first fraction by the
denominator of the second fraction and multiply the second fraction by the
denominator of the first fraction.
Multiply the first fraction by the
denominator of the second fraction and
multiply the second fraction by the denominator of the first fraction.
Add only the numerators and keep the
denominator the same.
Subtract only the numerators and keep
the denominator the same.
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UNIT 2: Fractions
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3. 7
1
3
2g
= 3
3
7
7
7
1
3
2
g
= 21
3
21
14
g
= 21
314 g
4. 53
2 hg
3
3
55
5
3
2
hg
15
3
15
10 hg
15
310 hg
5. dc
46
= c
c
d
d
dc
46
cd
c
cd
d 46
= cd
cd 46
Multiply the first fraction by the denominator of the second fraction and
multiply the second fraction by the
denominator of the first fraction.
Write as a single fraction.
Write as a single fraction.
Write as a single fraction.
Multiply the first fraction by the
denominator of the second fraction and
multiply the second fraction by the denominator of the first fraction.
Multiply the first fraction by the
denominator of the second fraction and
multiply the second fraction by the denominator of the first fraction.
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UNIT 2: Fractions
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Convert the mixed numbers to improper fractions.
Convert the mixed numbers to improper fractions.
2.3 Addition or Subtraction of Mixed Numbers with Different Denominators
1. 4
32
2
12
= 4
11
2
5
= 4
11
2
5
2
2
= 4
11
4
10
= 4
21
4
15
2. 4
31
6
53
= 4
7
6
23
= 6
6
4
4
4
7
6
23
= 24
42
24
92
= 24
50
= 12
25
= 12
12
Change the first fraction to an equivalent fraction
with denominator 4. (Multiply both the numerator and the denominator
of the first fraction by 2)
The denominators are not multiples of one another:
Multiply the first fraction by the denominator
of the second fraction.
Multiply the second fraction by the
denominator of the first fraction.
Convert the mixed numbers to improper fractions.
Convert the mixed numbers to improper fractions.
Add only the numerators and keep the
denominator the same.
Change the fraction back to a mixed number.
Add only the numerators and keep the
denominator the same.
Change the fraction back to a mixed number.
Write the fraction in its simplest form.
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UNIT 2: Fractions
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The denominators are not multiples of one another Multiply the first fraction with the second denominator
Multiply the second fraction with the first denominator
The denominators are not multiples of one another Multiply the first fraction with the second denominator
Multiply the second fraction with the first denominator
2.4 Addition or Subtraction of Algebraic Expression with Different Denominators
1. 22
m
m
m
= )2(
)2(
2
2
22
m
mm
m
m
=
22
2
22
2
m
mm
m
m
= )2(2
)2(2
m
mmm
= )2(2
22 2
m
mmm
= )2(2
2
m
m
2. y
y
y
y 1
1
= )1(
)1(1
1
y
y
y
y
y
y
y
y
= )1(
)1)(1(2
yy
yyy
= )1(
)1( 22
yy
yy
= )1(
122
yy
yy
= )1(
1
yy
Remember to use brackets
Write the above fractions as a single fraction.
The denominators are not multiples of one another:
Multiply the first fraction by the denominator
of the second fraction.
Multiply the second fraction by the
denominator of the first fraction.
Expand:
m (m – 2) = m2 – 2m
Expand:
(y – 1) (y + 1) = y2 + y – y – 1
2
= y2 – 1
Expand:
– (y2 – 1) = –y
2 + 1
Write the fractions as a single fraction.
The denominators are not multiples of one another:
Multiply the first fraction by the denominator
of the second fraction.
Multiply the second fraction by the
denominator of the first fraction.
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UNIT 2: Fractions
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The denominators are not multiples of one another Multiply the first fraction with the second denominator
Multiply the second fraction with the first denominator
3. 24
5
8
3
n
n
n
= n
n
n
n
n
n
n 8
8
24
4
4
5
8
3
2
2
= )4(8
)5(8
)4(8
1222
2
nn
nn
nn
n
= )4(8
)5(812
2
2
nn
nnn
= )4(8
84012
2
22
nn
nnn
= )4(8
404
2
2
nn
nn
= )8(4
)10(42nn
nn
= 28
10
n
n
Factorise and simplify the fraction by canceling
out the common factors.
Expand:
– 8n (5 + n) = –40n – 8n2
Subtract the like terms.
Write as a single fraction.
The denominators are not multiples of one another:
Multiply the first fraction by the denominator
of the second fraction.
Multiply the second fraction by the
denominator of the first fraction.
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UNIT 2: Fractions
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Calculate each of the following.
1. 7
1
7
2
2. 12
5
12
11
3. 14
1
7
2
4. 12
5
3
2
5. 5
4
7
2
6. 7
5
2
1
7. 313
22
8. 9
72
5
24
9. ss
12
10. ww
511
TEST YOURSELF A
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UNIT 2: Fractions
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Curriculum Development Division
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11. aa 2
12
12. ff 3
52
13. ba
42
14. qp
51
15. nmnm5
3
7
2
5
2
7
5
16.
)2(2
1p
p
17.
5
3
2
32 yxyx
18.
xx
x 5
2
412
19.
x
x
x
x 1
1
20.
2
4
2 x
x
x
x
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UNIT 2: Fractions
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21.
4
84
2
36 yxyx
22.
29
4
3
2
n
n
n
23.
r
rr
15
25
5
2
24.
p
p
p
p
2
232
25.
n
n
n
n
10
34
5
322
26.
n
n
mn
nm 33
27.
mn
nm
m
m
5
5
28.
mn
mn
m
m
3
3
29.
24
5
8
3
n
n
n
30.
m
p
m
p 1
3
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UNIT 2: Fractions
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PART B:
MULTIPLICATION AND DIVISION
OF FRACTIONS
LEARNING OBJECTIVES
Upon completion of Part B, pupils will be able to:
1. multiply:
(i) a whole number by a fraction or mixed number;
(ii) a fraction by a whole number (include mixed numbers); and
(iii) a fraction by a fraction.
2. divide:
(i) a fraction by a whole number;
(ii) a fraction by a fraction;
(iii) a whole number by a fraction; and
(iv) a mixed number by a mixed number.
3. solve problems involving combined operations of addition, subtraction,
multiplication and division of fractions, including the use of brackets.
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UNIT 2: Fractions
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TEACHING AND LEARNING STRATEGIES
Pupils face problems in multiplication and division of fractions.
Strategy:
Teacher should emphasise on how to divide fractions correctly. Teacher should
also highlight the changes in the positive (+) and negative (–) signs as follows:
Multiplication Division
(+) (+) = + (+) (+) = +
(+) (–) = – (+) (–) = –
(–) (+) = – (–) (+) = –
(–) (–) = + (–) (–) = +
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UNIT 2: Fractions
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1.0 Multiplication of Fractions
Recall that multiplication is just repeated addition.
Consider the following:
32
First, let’s assume this box as 1 whole unit.
Therefore, the above multiplication 32 can be represented visually as follows:
This means that 3 units are being repeated twice, or mathematically can be written as:
6
33 32
Now, let’s calculate 2 x 2. This multiplication can be represented visually as:
This means that 2 units are being repeated twice, or mathematically can be written as:
4
22 22
LESSON NOTES
3 + 3 = 6
2 + 2 = 4
2 groups of 3 units
2 groups of 2 units
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UNIT 2: Fractions
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Now, let’s calculate 2 x 1. This multiplication can be represented visually as:
This means that 1 unit is being repeated twice, or mathematically can be written as:
211 12
It looks simple when we multiply a whole number by a whole number. What if we
have a multiplication of a fraction by a whole number? Can we represent it visually?
Let’s consider .2
12
Since represents 1 whole unit, therefore 2
1unit can be represented by the
following shaded area:
Then, we can represent visually the multiplication of 2
12 as follows:
This means that 2
1unit is being repeated twice, or mathematically can be written as:
1
2
2
2
1
2
1
2
12
1 + 1 = 2
2
1 +
2
1 = 1
2
2
2 groups of 1 unit
2 groups of 2
1 unit
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UNIT 2: Fractions
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Let’s consider again .22
1 What does it mean? It means ‘
2
1 out of 2 units’ and the
visualization will be like this:
Notice that the multiplications2
12 and 2
2
1 will give the same answer, that is, 1.
How about ?23
1
Since represents 1 whole unit, therefore 3
1unit can be represented by the
following shaded area:
Then, we can represent visually the multiplication 23
1 as follows:
This means that 3
1unit is being repeated twice, or mathematically can be written as:
3
2
3
1
3
1 2
3
1
3
1 +
3
1 =
3
2
The shaded area is 3
1unit.
2
1 out of 2 units 12
2
1
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UNIT 2: Fractions
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Let’s consider 23
1 . What does it mean? It means ‘
3
1out of 2 units’ and the visualization
will be like this:
Notice that the multiplications3
12 and 2
3
1 will give the same answer, that is,
3
2.
Consider now the multiplication of a fraction by a fraction, like this:
2
1
3
1
This means ‘3
1 out of
2
1 units’ and the visualization will be like this:
Consider now this multiplication:
2
1
3
2
This means ‘3
2 out of
2
1 units’ and the visualization will be like this:
2
1unit
3
1 out of 2 units
3
22
3
1
3
1 out of
2
1 units
6
1
2
1
3
1
2
1unit
3
2 out of
2
1 units
6
2
2
1
3
2
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UNIT 2: Fractions
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What do you notice so far?
The answer to the above multiplication of a fraction by a fraction can be obtained by
just multiplying both the numerator together and the denominator together:
6
1
2
1
3
1
9
2
3
1
3
2
So, what do you think the answer for 3
1
4
1 ? Do you get
12
1 as the answer?
The steps to multiply a fraction by a fraction can therefore be summarized as follows:
1.1 Multiplication of Simple Fractions
Examples:
a) 35
6
7
3
5
2
b) 35
6
5
3
7
2
c) 35
12
5
2
7
6
d) 35
12
5
2
7
6
Steps to Multiply Fractions:
1) Multiply the numerators together and
multiply the denominators together.
2) Simplify the fraction (if needed).
Remember!!!
(+) (+) = +
(+) (–) = –
(–) (+) = –
(–) (–) = +
Multiply the two numerators together and the two denominators together.
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UNIT 2: Fractions
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1.2 Multiplication of Fractions with Common Factors
6
5
7
12 or
6
5
7
12
1.3 Multiplication of a Whole Number and a Fraction
6
152
=
6
31
1
2
=
6
31
1
2
= 3
31
= 3
110
Second Method:
(i) Simplify the fraction by canceling
out the common factors.
6
5
7
12
(i) Then, multiply the two
numerators together and the two
denominators together, and
convert to a mixed number, if
needed.
6
5
7
12
7
31
7
10
2
1
Convert the mixed number to improper
fraction.
Simplify by canceling out the common
factors.
Remember
2 = 1
2
First Method:
(ii) Multiply the two numerators
together and the two
denominators together:
6
5
7
12 =
42
60
(ii) Then, simplify.
7
31
7
10
42
60
10
7
3 Multiply the two numerators together and
the two denominators together.
Remember: (+) (–) = (–)
Change the fraction back to a mixed number.
1
1
2
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UNIT 2: Fractions
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1. Find 10
15
12
5
Solution: 10
15
12
5
= 8
5
2. Find 5
2
6
21
Solution : 5
2
6
21
= 5
2
6
21
5
7
= 5
21
Simplify by canceling out the common
factors.
Note that 3
21 can be further simplified.
Simplify further by canceling out the
common factors.
3
1
Simplify by canceling out the common factors.
EXAMPLES
Multiply the two numerators together and the
two denominators together.
Remember: (+) (–) = (–)
Multiply the two numerators together and
the two denominators together.
Remember: (+) (–) = (–)
3
1
1
7
Change the fraction back to a mixed
number.
2
1
4
5
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UNIT 2: Fractions
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1.4 Multiplication of Algebraic Fractions
1. Simplify 4
52 x
x
Solution : 4
52 x
x
= 2
5
= 2
12
2. Simplify
m
n
n4
9
2
Solution:
m
n
n4
9
2
=
1
4
2
9
2
mn
n
n
= 1
)2(
2
9 mn
= nm22
9
1 2
1 1 Simplify the fraction by canceling out the x’s.
Multiply the two numerators together and
the two denominators together.
Simplify the fraction by canceling the
common factor and the n.
Multiply the two numerators together
and the two denominators together.
Write the fraction in its simplest form.
Change the fraction back to a mixed
number.
2
1
1
1
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UNIT 2: Fractions
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1. Calculate 27
25
5
9
2. Calculate – 20
14
7
3
12
45
3. Calculate
4
112
4. Calculate
5
14
3
1
5. Simplify
k
m3
6. Simplify )5(2
mn
7. Simplify
14
3
6
11
x
8. Simplify )32(2
dan
9. Simplify
yx
10
95
3
2
10. Simplify
x
x 120
4
TEST YOURSELF B1
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UNIT 2: Fractions
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2.0 Division of Fractions
Consider the following:
36
First, let’s assume this circle as 1 whole unit.
Therefore, the above division can be represented visually as follows:
This means that 6 units are being divided into a group of 3 units, or mathematically
can be written as:
2 36
The above division can also be interpreted as ‘how many 3’s can fit into 6’. The answer is
‘2 groups of 3 units can fit into 6 units’.
Consider now a division of a fraction by a fraction like this:
.8
1
2
1
LESSON NOTES
How many 8
1 is in
?2
1
6 units are being divided into a group of 3
units:
2 36
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UNIT 2: Fractions
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This means ‘How many is in ?
8
1
2
1
The answer is 4:
Consider now this division:
.4
1
4
3
This means ‘How many is in ?
4
1
4
3
The answer is 3:
But, how do you
calculate the answer?
How many 4
1 is in ?
4
3
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UNIT 2: Fractions
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Consider again .236
Actually, the above division can be written as follows:
3
16
3
636
Notice that we can write the division in the multiplication form. But here, we have to
change the second number to its reciprocal.
Therefore, if we have a division of fraction by a fraction, we can do the same, that is,
we have to change the second fraction to its reciprocal and then multiply the
fractions.
Therefore, in our earlier examples, we can have:
4
2
8
1
8
2
1
8
1
2
1 (i)
The reciprocal of a
fraction is found by
inverting the
fraction.
Change the second fraction to its
reciprocal and change the sign to .
The reciprocal
of 8
1 is .
1
8
These operations are the same!
The reciprocal
of 3 is .3
1
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UNIT 2: Fractions
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3
1
4
4
3
4
1
4
3 (ii)
The steps to divide fractions can therefore be summarized as follows:
2.1 Division of Simple Fractions
Example:
7
3
5
2
= 3
7
5
2
= 15
14
Change the second fraction to its reciprocal
and change the sign to .
Multiply the two numerators together and
the two denominators together.
Steps to Divide Fractions:
1. Change the second fraction to its
reciprocal and change the sign to .
2. Multiply the numerators together and
multiply the denominators together.
3. Simplify the fraction (if needed).
Tips:
(+) (+) = +
(+) (–) = –
(–) (+) = –
(–) (–) = +
Change the second fraction to its
reciprocal and change the sign to .
The reciprocal
of 4
1 is .
1
4
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UNIT 2: Fractions
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2.2 Division of Fractions With Common Factors
Examples:
9
2
21
10
= 2
9
21
10
= 2
9
21
10
= 7
15
= 7
12
7
6
5
3
6
7
5
3
10
7
7
65
3
1
5 3
7
1
2
Express the fraction in division form.
Change the second fraction to its reciprocal and
change the sign to .
Simplify by canceling out the common factors.
Change the fraction back to a mixed number.
Change the second fraction to its reciprocal
and change the sign to .
Then, simplify by canceling out the common
factors.
Multiply the two numerators together and the
two denominators together.
Remember: (+) (–) = (–)
Multiply the two numerators together and the
two denominators together.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
38
Curriculum Development Division
Ministry of Education Malaysia
1. Find 6
25
12
35
Solution : 6
25
12
35
= 25
6
12
35
= 10
7
2. Simplify –4
52 x
x
Solution : –xx 5
42
= –25
8
x
3. Simplify 2
x
y
Solution :
2x
y
2
1
x
y
x
y
2
5
7
Change the second fraction to its reciprocal
and change the sign to . Then, simplify by canceling out the common
factors.
Method I
EXAMPLES
Change the second fraction to its reciprocal
and change the sign to .
Multiply the two numerators together and the two
denominators together.
Express the fraction in division form.
Change the second fraction to its reciprocal
and change to .
Multiply the two numerators together and the two
denominators together.
Remember: (+) (–) = (–)
Multiply the two numerators together and the
two denominators together.
2
1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
39
Curriculum Development Division
Ministry of Education Malaysia
Multiply the numerator and the denominator of
the given fraction with x
2
x
y
= 2
x
y
x
x
= x
xx
y
2
= x
y
2
4. Simplify 5
)1( 1r
Solution:
5
)1( 1r
= 5
)1
1(r
r
r
= r
r
5
1
The given fraction.
r is the denominator of r
1.
Multiply the given fraction with r
r.
Note that:
1)1
1( rrr
Method II
The numerator is also
a fraction with
denominator x
Multiply the numerator and the denominator of the
given fraction by x.
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
40
Curriculum Development Division
Ministry of Education Malaysia
1. Calculate 2
21
7
3
2. Calculate 16
5
8
7
9
5
3. Simplify 3
48 y
y
4. Simplify
k
2
16
5. Simplify
3
5
2
x
6. Simplify n
m
n
m
3
24 2
7. Simplify 8
1
4
y
8. Simplify
x
x
11
TEST YOURSELF B2
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
41
Curriculum Development Division
Ministry of Education Malaysia
9. Calculate 5
)1(341
10. Simplify y
x15
11. Simplify
32
941 x
12. Simplify
15
1
1
p
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
42
Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF A:
1. 7
3
2. 2
1
3. 14
5
4. 4
1
5. 35
38 or
35
31
6. 14
3
7. 13
67 or
13
25
8. 45
73or
45
281
9. s
3
10. w
6
11. a2
5
12. f3
1
13. ab
ab 42
14. pq
pq 5
15. nm
16. 2
33 p
17. 10
1716 yx
18. x
x 12
19. )1(
1
xx
20. 2
21. 2
8 yx
22. 29
47
n
n
23. r
r
3
12
24. 2
2
2
6
p
p
25. 2
2
10
647
n
nn
26. m
m1
27. n
n
5
5
28. n
n
3
3
29. 28
10
n
n
30. m
p
3
34
ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions
43
Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF B1:
1. 3
21
3
5or 2.
8
11
8
9 or 3.
2
15
2
11or
4. 5
21
5
7 or 5.
k
m3 6.
2
5mn
7. 4
x 8. ndna
2
3 9. yx
5
3
3
10
10. 4
15 x
TEST YOURSELF B2:
1. 49
2 2.
9
51
9
14 or 3.
2
6
y
4. 8k
5. x5
6 6.
m
6
7. )1(2
1
y 8.
1
2
x
x
9. 20
9
10. xy
x 15 11.
6
13x 12.
p4
5
Unit 1:
Negative Numbers
UNIT 3
ALGEBRAIC EXPRESSIONS
AND
ALGEBRAIC FORMULAE
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Performing Operations on Algebraic Expressions 2
Part B: Expansion of Algebraic Expressions 10
Part C: Factorisation of Algebraic Expressions and Quadratic Expressions 15
Part D: Changing the Subject of a Formula 23
Activities
Crossword Puzzle 31
Riddles 33
Further Exploration 37
Answers 38
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
1 Curriculum Development Division
Ministry of Education Malaysia
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding of the concepts and skills
in Algebraic Expressions, Quadratic Expressions and Algebraic Formulae.
2. The concepts and skills in Algebraic Expressions, Quadratic Expressions and
Algebraic Formulae are required in almost every topic in Additional Mathematics,
especially when dealing with solving simultaneous equations, simplifying
expressions, factorising and changing the subject of a formula.
3. It is hoped that this module will provide a solid foundation for studies of Additional
Mathematics topics such as:
Functions
Quadratic Equations and Quadratic Functions
Simultaneous Equations
Indices and Logarithms
Progressions
Differentiation
Integration
4. This module consists of four parts and each part deals with specific skills. This format
provides the teacher with the freedom to choose any parts that is relevant to the skills
to be reinforced.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
2 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
Pupils who face problem in performing operations on algebraic expressions might have
difficulties learning the following topics:
Simultaneous Equations - Pupils need to be skilful in simplifying the algebraic
expressions in order to solve two simultaneous equations.
Functions - Simplifying algebraic expressions is essential in finding composite
functions.
Coordinate Geometry - When finding the equation of locus which involves
distance formula, the techniques of simplifying algebraic expressions are required.
Differentiation - While performing differentiation of polynomial functions, skills
in simplifying algebraic expressions are needed.
Strategy:
1. Teacher reinforces the related terminologies such as: unknowns, algebraic terms,
like terms, unlike terms, algebraic expressions, etc.
2. Teacher explains and shows examples of algebraic expressions such as:
8k, 3p + 2, 4x – (2y + 3xy)
3. Referring to the “Lesson Notes” and “Examples” given, teacher explains how to
perform addition, subtraction, multiplication and division on algebraic expressions.
4. Teacher emphasises on the rules of simplifying algebraic expressions.
PART A:
PERFORMING OPERATIONS ON
ALGEBRAIC EXPRESSIONS
LEARNING OBJECTIVES
Upon completion of Part A, pupils will be able to perform operations on algebraic
expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
3 Curriculum Development Division
Ministry of Education Malaysia
PART A:
PERFORMING BASIC ARITHMETIC OPERATIONS ON ALGEBRAIC EXPRESSIONS
1. An algebraic expression is a mathematical term or a sum or difference of mathematical
terms that may use numbers, unknowns, or both.
Examples of algebraic expressions: 2r, 3x + 2y, 6x2 +7x + 10, 8c + 3a – n
2,
g
3
2. An unknown is a symbol that represents a number. We normally use letters such as n, t, or
x for unknowns.
3. The basic unit of an algebraic expression is a term. In general, a term is either a number
or a product of a number and one or more unknowns. The numerical part of the term, is
known as the coefficient.
Examples: Algebraic expression with one term: 2r, g
3
Algebraic expression with two terms: 3x + 2y, 6s – 7t
Algebraic expression with three terms: 6x2 +7x + 10, 8c + 3a – n
2
4. Like terms are terms with the same unknowns and the same powers.
Examples: 3ab, –5ab are like terms.
3x2,
2
5
2x are like terms.
5. Unlike terms are terms with different unknowns or different powers.
Examples: 1.5m, 9k, 3xy, 2x2y are all unlike terms.
LESSON NOTES
6 xy Coefficient Unknowns
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
4 Curriculum Development Division
Ministry of Education Malaysia
6. An algebraic expression with like terms can be simplified by adding or subtracting the
coefficients of the unknown in algebraic terms.
7. To simplify an algebraic expression with like terms and unlike terms, group the like terms
first, and then simplify them.
8. An algebraic expression with unlike terms cannot be simplified.
9. Algebraic fractions are fractions involving algebraic terms or expressions.
Examples: .2
,2
4,
6
2,
15
322
22
2
2
yxyx
yx
grg
gr
h
m
10. To simplify an algebraic fraction, identify the common factor of both the numerator and the
denominator. Then, simplify it by elimination.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
5 Curriculum Development Division
Ministry of Education Malaysia
Simplify the following algebraic expressions and algebraic fractions:
(a) 5x – (3x – 4x) 64
)e(ts
(b) –3r –9s + 6r + 7s z
yx
2
3
6
5)f(
(c) 2
2
2
4
grg
gr
g
f
e2)g(
qp
43)d(
(h) x
x
3
2
13
Solutions:
(a) 5x – (3x – 4x)
= 5x – (– x)
= 5x + x
= 6x
(b) –3r –9s + 6r + 7s
= –3r + 6r –9s + 7s
= 3r – 2s
2
2
2
4)c(
grg
gr
gr
r
grg
gr
2
4
)2(
4
2
2
Perform the operation in the bracket.
Arrange the algebraic terms according to the like terms.
.
Unlike terms cannot be simplified.
Leave the answer in the simplest form as shown.
Algebraic expression with like terms can be simplified by
adding or subtracting the coefficients of the unknown.
Simplify by canceling out the common factor and the
same unknowns in both the numerator and the
denominator.
1
1
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
6 Curriculum Development Division
Ministry of Education Malaysia
pq
pq
pq
p
pq
q
qp
43
43
43)d(
12
23
26
2
34
3
64)e(
ts
ts
ts
z
xy
z
yx
z
yx
4
5
22
5
2
3
6
5)f(
fg
e
gf
eg
f
e
2
2
12)g(
x
x
x
x
x
x
x
x
x
x
6
16
3
1
2
16
3
2
16
3
2
1
2
)2(3
3
2
13
)h(
The LCM of p and q is pq.
The LCM of 4 and 6 is 12.
Simplify by canceling out the common
factor, then multiply the numerators
together and followed by the
denominators.
Change division to multiplication of the
reciprocal of 2g.
Equate the denominator.
2
1
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
7 Curriculum Development Division
Ministry of Education Malaysia
ALTERNATIVE METHOD
Simplify the following algebraic fractions:
(a) x
x
3
2
13
= x
x
3
2
13
2
2
= )2(3
)2(2
1)2(3
x
x
= x
x
6
16
(b) 5
23
x = 5
23
x
x
x
x
x
x
xxx
5
23
)(5
)(2)(3
x
x
x
xx
x
x
xxx
4
316
)2(2
)2(2
3)2(8
2
2
2
2
38
2
2
38
)c(
The denominator of x2
3 is 2x. Therefore,
multiply the algebraic fraction byx
x
2
2.
Each of the terms in the numerator and
denominator is multiplied by 2x.
.
The denominator of 2is2
1. Therefore,
multiply the algebraic fraction by2
2.
Each of the terms in the numerator and
denominator of the algebraic fraction is
multiplied by 2.
The denominator of x
3 is x. Therefore,
multiply the algebraic fraction byx
x.
Each of the terms in the numerator and
denominator is multiplied by x.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
8 Curriculum Development Division
Ministry of Education Malaysia
x
x
x
xx
36
21
288
21
)7(4)7(7
8
)7(3
7
7
47
8
3
47
8
3)d(
The denominator of 7
8 x is 7.
Therefore, multiply the algebraic
fraction by7
7.
Each of the terms in the numerator
and denominator is multiplied by 7.
Simplify the denominator.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
9 Curriculum Development Division
Ministry of Education Malaysia
Simplify the following algebraic expressions:
1. 2a –3b + 7a – 2b
2. − 4m + 5n + 2m – 9n
3. 8k – ( 4k – 2k )
4. 6p – ( 8p – 4p )
xy 5
13.5
5
2
3
4.6
kh
c
ba
2
3
7
4.7
dc
dc
3
8
2
4.8
yzz
xy.9
w
uv
vw
u
2.10
65
2.11
x
54
24
.12
x
x
TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
10 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
Pupils who face problem in expanding algebraic expressions might have
difficulties in learning of the following topics:
Simultaneous Equations – pupils need to be skilful in expanding the
algebraic expressions in order to solve two simultaneous equations.
Functions – Expanding algebraic expressions is essential when finding
composite function.
Coordinate Geometry – when finding the equation of locus which
involves distance formula, the techniques of expansion are applied.
Strategy:
Pupils must revise the basic skills involving expanding algebraic expressions.
PART B:
EXPANSION OF ALGEBRAIC
EXPRESSIONS
LEARNING OBJECTIVE
Upon completion of Part B, pupils will be able to expand algebraic
expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
11 Curriculum Development Division
Ministry of Education Malaysia
PART B:
EXPANSION OF ALGEBRAIC EXPRESSIONS
1. Expansion is the result of multiplying an algebraic expression by a term or another
algebraic expression.
2. An algebraic expression in a single bracket is expanded by multiplying each term in the
bracket with another term outside the bracket.
3(2b – 6c – 3) = 6b – 18c – 9
3. Algebraic expressions involving two brackets can be expanded by multiplying each term of
algebraic expression in the first bracket with every term in the second bracket.
(2a + 3b)(6a – 5b) = 12a2 – 10ab + 18ab – 15b
2
= 12a
2 + 8ab – 15b
2
4. Useful expansion tips:
(i) (a + b)2 = a
2 + 2ab + b
2
(ii) (a – b)2 = a
2 – 2ab + b
2
(iii) (a – b)(a + b) = (a + b)(a – b)
= a2 – b
2
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
12 Curriculum Development Division
Ministry of Education Malaysia
Expand each of the following algebraic expressions:
(a) 2(x + 3y)
(b) – 3a (6b + 5 – 4c)
Solutions:
(a) 2 (x + 3y)
= 2x + 6y
(b) –3a (6b + 5 – 4c)
= –18ab – 15a + 12ac
1293
2)c( y
= 123
29
3
2 y
= 6y + 8
= (a + 3) (a + 3)
= a2 + 3a + 3a + 9
= a2 + 6a + 9
When expanding two brackets, each term
within the first bracket is multiplied by
every term within the second bracket.
1293
2)c( y
2523)e( k
2)3()d( a
)5)(2()f( pp
2)3()d( a
When expanding a bracket, each term
within the bracket is multiplied by the term
outside the bracket.
When expanding a bracket, each term
within the bracket is multiplied by the term
outside the bracket.
1
3
1
4
EXAMPLES
Simplify by canceling out the common
factor, then multiply the numerators
together and followed by the denominators.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
13 Curriculum Development Division
Ministry of Education Malaysia
(c) (4x – 3y)(6x – 5y)
– 18 xy
– 20 xy
– 38 xy
= 24x2 – 38 xy + 15y
2
2523)e( k
= –3(2k + 5) (2k + 5)
= –3(4k2 + 20k + 25)
= –12k2 – 60k – 75
)5( )2( )f( qp
= pq – 5p + 2q – 10
ALTERNATIVE METHOD
Expanding two brackets
(a) (a + 3) (a + 3)
= a2 + 3a + 3a + 9
= a2 + 6a + 9
(b) (2p + 3q) (6p – 5q)
= 12p2 – 10 pq + 18 pq – 15q
2
= 12p2 + 8 pq – 15q
2
When expanding two brackets, each term
within the first bracket is multiplied by
every term within the second bracket.
When expanding two
brackets, write down the
product of expansion and
then, simplify the like
terms.
When expanding two brackets, each term
within the first bracket is multiplied by
every term within the second bracket.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
14 Curriculum Development Division
Ministry of Education Malaysia
Simplify the following expressions and give your answers in the simplest form.
4
324.1 n
162
1.2 q
yxx 326.3
)(22.4 baba
)6()3(2.5 pp
3
26
3
1.6
yxyx
121.72
ee
nmmnm 2.82
gfggfgf 2.9
ihiihih 32.10
TEST YOURSELF B
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
15 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
Some pupils may face problem in factorising the algebraic expressions. For
example, in the Differentiation topic which involves differentiation using the
combination of Product Rule and Chain Rule or the combination of Quotient
Rule and Chain Rule, pupils need to simplify the answers using factorisation.
Examples:
2
2
2
32
3
32
2433
43
)27(
)154()3(
)27(
)2()3(])3(3)[27(
27
)3(.2
)1549()57(2
)6()57(])57(28[2
)57(2.1
x
xx
x
xxx
dx
dy
x
xy
xxx
xxxxdx
dy
xxy
Strategy
1. Pupils revise the techniques of factorisation.
PART C:
FACTORISATION OF
ALGEBRAIC EXPRESSIONS AND
QUADRATIC EXPRESSIONS
LEARNING OBJECTIVE
Upon completion of Part C, pupils will be able to factorise algebraic expressions
and quadratic expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
16 Curriculum Development Division
Ministry of Education Malaysia
PART C:
FACTORISATION OF
ALGEBRAIC EXPRESSIONS AND QUADRATIC EXPRESSIONS
1. Factorisation is the process of finding the factors of the terms in an algebraic expression. It
is the reverse process of expansion.
2. Here are the methods used to factorise algebraic expressions:
(i) Express an algebraic expression as a product of the Highest Common Factor (HCF) of
its terms and another algebraic expression.
ab – bc = b(a – c)
(ii) Express an algebraic expression with three algebraic terms as a complete square of two
algebraic terms.
a2 + 2ab + b
2 = (a + b)
2
a2 – 2ab + b
2 = (a – b)
2
(iii) Express an algebraic expression with four algebraic terms as a product of two algebraic
expressions.
ab + ac + bd + cd = a(b + c) + d(b + c)
= (a + d)(b + c)
(iv) Express an algebraic expression in the form of difference of two squares as a product of
two algebraic expressions.
a2 – b
2 = (a + b)(a – b)
3. Quadratic expressions are expressions which fulfill the following characteristics:
(i) have only one unknown; and
(ii) the highest power of the unknown is 2.
4. Quadratic expressions can be factorised using the methods in 2(i) and 2(ii).
5. The Cross Method can be used to factorise algebraic expression in the general form of
ax2 + bx + c, where a, b, c are constants and a ≠ 0, b ≠ 0, c ≠ 0.
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
17 Curriculum Development Division
Ministry of Education Malaysia
(a) Factorising the Common Factors
i) mn + m = m (n +1)
ii) 3mp + pq = p (3m + q)
iii) 2mn – 6n = 2n (m – 3)
(b) Factorising Algebraic Expressions with Four Terms
i) vy + wy + vz + wz
= y (v + w) + z (v + w)
= (v + w)(y + z)
ii) 21bm – 7bs + 6cm – 2cs
= 7b(3m – s) + 2c(3m – s)
= (3m – s)(7b + 2c)
Factorise the first and the second terms
with the common factor y, then factorise
the third and fourth terms with the
common factor z.
.
(v + w) is the common factor.
Factorise the first and the second terms with
common factor 7b, then factorise the third
and fourth terms with common factor 2c.
(3m – s) is the common factor.
EXAMPLES
Factorise the common factor m.
.
Factorise the common factor p.
.
Factorise the common factor 2n.
.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
18 Curriculum Development Division
Ministry of Education Malaysia
(c) Factorising the Algebraic Expressions by Using Difference of Two Squares
i) x2 – 16 = x
2 – 4
2
= (x + 4)(x – 4)
ii) 4x2
– 25 = (2x)2 – 5
2
= (2x + 5)(2x – 5)
(d) Factorising the Expressions by Using the Cross Method
i) x2
– 5x + 6
xxx
x
x
523
2
3
x2
– 5x + 6 = (x – 3) (x – 2)
ii) 3x2
+ 4x – 4
xxx
x
x
462
2
23
3x2 + 4x – 4 = (3x – 2) (x + 2)
The summation of the cross
multiplication products should
equal to the middle term of the
quadratic expression in the
general form.
The summation of the cross
multiplication products should
equal to the middle term of the
quadratic expression in the
general form.
a2 – b
2 = (a + b)(a – b)
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
19 Curriculum Development Division
Ministry of Education Malaysia
ALTERNATIVE METHOD
Factorise the following quadratic expressions:
i) x 2 – 5x + 6
ac b
+ 6 – 5
–2 –3
(x – 2) (x – 3)
)3)(2(65 2 xxxx
ii) x 2 – 5x – 6
ac b
– 6 – 5
+1 – 6
(x + 1) (x– 6)
)6)(1(65 2 xxxx
+1 (–6) = –6
+1 (–6) = –6
+1 – 6 = –5
a=+1 b= –5 c = –6
REMEMBER!!!
An algebraic expression can
be represented in the general
form of ax2 + bx + c, where
a, b, c are constants and
a ≠ 0, b ≠ 0, c ≠ 0.
+1 (+ 6) = + 6 –2 (–3) = +6
–2 + (–3) = –5
a=+1 b= –5 c =+6
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
20 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF C
(iii) 2x2 – 11x + 5
ac b
+ 10 –11
–1 – 10
2
10
2
1
52
1
(2x – 1) (x – 5)
)5)(12(5112 2 xxxx
(iv) 3x2 + 4x – 4
ac b
– 12 + 4
– 2 +6
23
2
3
6
3
2
The coefficient of x2 is 2,
divide each number by 2.
(+2) (+5) = +10
–1 (–10) = +10
–1 + (–10) = –11
–2 + 6 = 4
The coefficient of x2 is 3, divide each
number by 3.
3 (– 4) = –12
a=+2 b = –11 c =+5
a =+ 3 b=+ 4 c = –4
(3x – 2) (x + 2)
The coefficient of x2 is 2,
multiply by 2:
5)(12
52
5
21
21
xx
xx
xx
The coefficient of x2 is 3, multiply by 3:
2)(23
23
2
32
32
xx
xx
xx
)2)(23(443 2 xxxx
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
21 Curriculum Development Division
Ministry of Education Malaysia
Factorise the following quadratic expressions completely.
1. 3p 2 – 15
2. 2x 2 – 6
3. x 2 – 4x
4. 5m 2 + 12m
5. pq – 2p
6. 7m + 14mn
7. k2 –144
8. 4p 2 – 1
9. 2x 2 – 18
10. 9m2 – 169
TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
22 Curriculum Development Division
Ministry of Education Malaysia
11. 2x 2 + x – 10
12. 3x 2 + 2x – 8
13. 3p 2 – 5p – 12
14. 4p2 – 3p – 1
15. 2x2
– 3x – 5
16. 4x 2 – 12x + 5
17. 5p 2 + p – 6
18. 2x2
– 11x + 12
19. 3p + k + 9pr + 3kr
20. 4c2 – 2ct – 6cw + 3tw
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
23 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
If pupils have difficulties in changing the subject of a formula, they probably
face problems in the following topics:
Functions – Changing the subject of the formula is essential in finding
the inverse function.
Circular Measure – Changing the subject of the formula is needed to
find the r or from the formulae s = r or 2
2
1rA .
Simultaneous Equations – Changing the subject of the formula is the
first step of solving simultaneous equations.
Strategy:
1. Teacher gives examples of formulae and asks pupils to indicate the subject
of each of the formula.
Examples: y = x – 2
hrV
bhA
2
2
1
y, A and V are the
subjects of the
formulae.
PART D:
CHANGING THE SUBJECT
OF A FORMULA
LEARNING OBJECTIVE
Upon completion of this module, pupils will be able to change the subject of
a formula.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
24 Curriculum Development Division
Ministry of Education Malaysia
PART D:
CHANGING THE SUBJECT OF A FORMULA
1. An algebraic formula is an equation which connects a few unknowns with an equal
sign.
Examples:
hrV
bhA
2
2
1
2. The subject of a formula is a single unknown with a power of one and a coefficient
of one, expressed in terms of other unknowns.
Examples: bhA2
1
a2 = b
2 + c
2
hTrT 2
2
1
3. A formula can be rearranged to change the subject of the formula. Here are the
suggested steps that can be used to change the subject of the formula:
(i) Fraction : Get rid of fraction by multiplying each term in the formula with
the denominator of the fraction.
(ii) Brackets : Expand the terms in the bracket.
(iii) Group : Group all the like terms on the left or right side of the formula.
(iv) Factorise : Factorise the terms with common factor.
(v) Solve : Make the coefficient and the power of the subject equal to one.
LESSON NOTES
A is the subject of the formula because it is
expressed in terms of other unknowns.
a
2 is not the subject of the formula
because the power ≠ 1
T is not the subject of the formula
because it is found on both sides of the
equation.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
25 Curriculum Development Division
Ministry of Education Malaysia
1. Given that 2x + y = 2, express x in terms of y.
Solution:
2x + y = 2
2x = 2 – y
x = 2
2 y
2. Given that yyx
52
3
, express x in terms of y.
Solution:
yyx
52
3
3x + y = 10y
3x = 10y – y
3x = 9y
x = 3
9y
x = 3y
No fraction and brackets.
Group:
Retain the x term on the left hand side of the
equation by grouping all the y term to the
right hand side of the equation.
Fraction:
Multiply both sides of the equation by 2.
Group:
Retain the x term on the left hand side of the
equation by grouping all the y term to the
right hand side of the equation.
Solve:
Divide both sides of the equation by 2 to
make the coefficient of x equal to 1.
Solve:
Divide both sides of the equation by 3 to
make the coefficient of x equal to 1.
EXAMPLES
Steps to Change the Subject of a Formula
(i) Fraction
(ii) Brackets
(iii) Group
(iv) Factorise
(v) Solve
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
26 Curriculum Development Division
Ministry of Education Malaysia
3. Given that yx 2 , express x in terms of y.
Solution:
yx 2
x = (2y)2
x = 4y2
4. Given that px
3, express x in terms of p.
Solution:
px
3
2
2
9
)3(
3
px
px
px
5. Given that yxx 23 , express x in terms of y.
Solution:
2
2
2
2
2
22
23
23
yx
yx
yx
yxx
yxx
Solve:
Square both sides of the equation to make the
power of x equal to 1.
Fraction:
Multiply both sides of the equation by 3.
Solve:
Square both sides of the equation to make
the power of x equal to1.
Group:
Group the like terms
Solve:
Divide both sides of the equation by 2 to
make the coefficient of x equal to 1.
Solve:
Square both sides of equation to make the
power of x equal to 1.
Simplify the terms.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
27 Curriculum Development Division
Ministry of Education Malaysia
6. Given that 4
11x – 2(1 – y) = xp2 , express x in terms of y and p.
Solution:
4
11x – 2 (1 – y) = xp2
11x – 8(1 – y) = xp8
11x – 8 + 8y = 8xp
11x – 8xp = 8 – 8y
x(11 – 8p) = 8 – 8y
x = p
y
811
88
7. Given that n
xp
5
32 = 1 – p , express p in terms of x and n.
Solution:
n
xp
5
32 = 1 – p
2p – 3x = 5n – 5pn
2p + 5pn = 5n + 3x
p(2 + 5n) = 5n + 3x
p = n
xn
52
35
Fraction:
Multiply both sides of the equation
by 4.
Bracket:
Expand the bracket.
Group:
Group the like terms.
Factorise:
Factorise the x term.
Solve:
Divide both sides by (11 – 8p) to
make the coefficient of x equal to 1.
Fraction:
Multiply both sides of the equation by
5n.
Solve:
Divide both sides of the equation by
(2 + 5n) to make the coefficient of p
equal to 1.
Group:
Group the like p terms.
Factorise:
Factorise the p terms.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
28 Curriculum Development Division
Ministry of Education Malaysia
1. Express x in terms of y.
a) 02 yx
b) 032 yx
c) 12 xy
d) 22
1 yx
e) 53 yx
f) 43 xy
TEST YOURSELF D
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
29 Curriculum Development Division
Ministry of Education Malaysia
2. Express x in terms of y.
a) xy
b) xy 2
c) 3
2x
y
d) xy 31
e) 13 xyx
f) yx 1
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
30 Curriculum Development Division
Ministry of Education Malaysia
3. Change the subject of the following formulae:
a) Given that 2
ax
ax, express x in terms
of a .
b) Given that x
xy
1
1, express x in terms
of y .
c) Given that vuf
111 , express u in
terms of v and f .
d) Given that 4
3
2
2
qp
qp, express p in
terms of .q
e) Given that mnmp 23 , express m in
terms of n and p .
f) Given that
C
CBA
1, express C in
terms of A and B .
g) Given that yx
xy2
2
, express y in
terms of x.
h) Given that g
lT 2 , express g in
terms of T and l.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
31 Curriculum Development Division
Ministry of Education Malaysia
CROSSWORD PUZZLE
HORIZONTAL
1) – 4p, 10q and 7r are called algebraic .
3) An algebraic term is the of unknowns and numbers.
4) 4m and 8m are called terms.
5) hrV 2 , then V is the of the formula.
7) An can be represented by a letter.
10) 21232 xxxx .
ACTIVITIES
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
32 Curriculum Development Division
Ministry of Education Malaysia
VERTICAL
2) An algebraic consists of two or more algebraic terms combined by
addition or subtraction or both.
6) 252212 2 xxxx .
8) terms are terms with different unknowns.
9) The number attached in front of an unknown is called .
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
33 Curriculum Development Division
Ministry of Education Malaysia
RIDDLES
RIDDLE 1
1. You are given 9 multiple-choice questions.
2. For each of the questions, choose the correct answer and fill the alphabet in the box
below.
3. Rearrange the alphabets to form a word.
4. What is the word?
1
2 3 4 5 6 7 8 9
1. Calculate
.3
5
12
D) 5
1 O) 1
W) 3
11 N)
15
11
2. Simplify yxyx 7693 .
F) yx 23 W) yx 169
E) yx 23 X) yx 29
3. Simplify 23
qp .
L) 6
32 qp A)
6
32 qp
N) 6
23 pq R)
6
23 qp
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
34 Curriculum Development Division
Ministry of Education Malaysia
4. Expand )7()4(2 xx .
A) 1x D) 15x
U) 13 x C) 153 x
5. Expand )52(3 cba .
S ) acab 156 C) acab 156
T) acab 156 R) acab 156
6. Factorise 252 x .
E) )5)(5( xx T) )5)(5( xx
I) )5)(5( xx C) )25)(25( xx
7. Factorise qpq 4 .
D) )41( qpq E) )4( pq
T) )4( qp S) )4( pq
8. Factorise 1282 xx .
I ) )6)(2( xx W) )6)(2( xx
F) )3)(4( xx C) )3)(4( xx
9. Given that 42
3
x
yx, express x in terms of y.
L) 5
yx C)
5
yx
T) 11
yx N)
3
8 yx
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
35 Curriculum Development Division
Ministry of Education Malaysia
RIDDLE 2
1. You are given 9 multiple-choice questions.
2. For each of the questions, choose the correct answer and fill the alphabet in the box
below.
3. Rearrange the alphabets to form a word.
4. What is the word?
1
2 3 4 5 6 7 8 9
1. Calculate
.3
15
x
A) 3
5 x O)
x
x
3
5
I ) 5
3
x
x N)
5
3
x
2. Simplify r
qp
54
3 .
F) q
pr
4
15 R)
pr
q
15
4
W) r
pq
20
3 B)
r
pq
5
3
3. Simplifyz
xy
yz
x
2 .
N)2
2
y D)
2
2
2z
x
L) 22z
x I)
2
2
z
x
4. Solve ).3(2
yxxyx
E) xyyx 222 D) xyyx 222
I ) xyxyx 222 3 N) xyyx 222
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
36 Curriculum Development Division
Ministry of Education Malaysia
5. Expand 25p .
I) 252 p N) 252 p
D) 25102 pp L) 25102 pp
6. Factorise 1572 2 yy .
F) )5)(32( yy D) )5)(32( yy
W) )5)(32( yy L) )52)(3( yy
7. Factorise 5112 2 pp .
R) )5)(12( pp B) )5)(12( pp
F) )5)(1( pp W) )52)(1( pp
8. Given that ACC
B )1( , express C in terms of A and B.
L) AB
BC
R)
ABC
1
C) AB
ABC
N)
AB
ABC
9. Given that 25 xyx , express x in terms of y.
O) 16
42
yx B)
24
42
yx
I )
2
2
1
yx U)
2
4
2
yx
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
37 Curriculum Development Division
Ministry of Education Malaysia
SUGGESTED WEBSITES:
1. http://www.themathpage.com/alg/algebraic-expressions.htm
2. http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut11_si
mp.htm
3. http://www.helpalgebra.com/onlinebook/simplifyingalgebraicexpressions.htm
4. http://www.tutor.com.my/tutor/daily/eharian_06.asp?h=60104&e=PMR&S=MAT&ft=F
TN
FURTHER
EXPLORATION
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
38 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF A:
1. 9a – 5b
2. – 2m – 4n
3. 6k
4. 2p
5. xy
yx
5
15
6.
15
620 kh
7. c
ab
7
6
8. dc
dc
3
)4(4
9. 2z
x
10. 2
2
v
11. x
x
65
2
12. x
x
54
24
TEST YOURSELF B:
1. – 8n + 3 6. x + y
2. 3q + 2
1
7. 2e
3. – 12x2 + 18xy 8. mnmn 22
4. – 3b 9. fgf 22
5. p 10. 22 52 iihh
ANSWERS
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
39 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF C:
1. 3(p 2 – 5)
2. 2(x 2 – 3)
3. x(x – 4)
4. m(5m + 12)
5. p(q – 2)
6. 7m (1 + 2n)
7. (k + 12)(k – 12)
8. (2p – 1)(2p + 1)
9. 2(x – 3)(x + 3)
10. (3m + 13)(3m – 13)
11. (2x + 5)(x – 2)
12. (3x – 4)(x + 2)
13. (3p + 4)(p – 3)
14. (4p + 1)(p – 1)
15. (2x – 5)(x +1)
16. (2x – 5)(2x – 1)
17. (5p + 6)(p – 1)
18. (2x – 3)(x – 4)
19. (1 + 3r)(3p + k) 20. (2c – t)(2c – 3w)
TEST YOURSELF D:
1. (a) x = 2 – y (b)
2
3 yx
(c) x = 2y – 1
(d) x = 4 – y (e) 3
5 yx
(f) x = 3y – 4
2. (a) x = y2
(b) 24yx
(c) 236 yx
(d)
2
3
1
yx
2
2
1)e(
yx (f) 12 yx
3. (a) ax 3
(b) 1
1
y
yx
(c) fv
fvu
(d) 2
7qp
(e) 32
n
pm
(f) AB
BC
(g)
)1(2
x
xy (h)
2
24
T
lg
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
40 Curriculum Development Division
Ministry of Education Malaysia
ACTIVITIES
CROSSWORD PUZZLE
RIDDLES
RIDDLE 1
2 F
3
A
1
N
5
T
4
A
7
S
6
T
8
I
9
C
RIDDLE 2
2
W 1
O
3
N
5
D
4
E
7
R
6
F
9
U
8
L
Unit 1:
Negative Numbers
UNIT 4
LINEAR EQUATIONS
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Linear Equations 2
Part B: Solving Linear Equations in the Forms of x + a = b and x – a = b 6
Part C: Solving Linear Equations in the Forms of ax = b and a
x= b 9
Part D: Solving Linear Equations in the Form of ax + b = c 12
Part E: Solving Linear Equations in the Form of a
x+ b = c 15
Part F: Further Practice on Solving Linear Equations 18
Answers 23
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
1 Curriculum Development Division
Ministry of Education Malaysia
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding on the concept involved in
solving linear equations.
2. The module is written as a guide for teachers to help pupils master the basic skills
required to solve linear equations.
3. This module consists of six parts and each part deals with a few specific skills.
Teachers may use any parts of the module as and when it is required.
4. Overall lesson notes are given in Part A, to stress on the important facts and concepts
required for this topic.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
2 Curriculum Development Division
Ministry of Education Malaysia
PART A:
LINEAR EQUATIONS
LEARNING OBJECTIVES
Upon completion of Part A, pupils will be able to:
1. understand and use the concept of equality;
2. understand and use the concept of linear equations in one unknown; and
3. understand the concept of solutions of linear equations in one unknown
by determining if a numerical value is a solution of a given linear
equation in one unknown.
a. determine if a numerical value is a solution of a given linear equation
in one unknown;
TEACHING AND LEARNING STRATEGIES
The concepts of can be confusing and difficult for pupils to grasp. Pupils might
face difficulty when dealing with problems involving linear equations.
Strategy:
Teacher should emphasise the importance of checking the solutions obtained.
Teacher should also ensure that pupils understand the concept of equality and
linear equations by emphasising the properties of equality.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
3 Curriculum Development Division
Ministry of Education Malaysia
GUIDELINES:
1. The solution to an equation is the value that makes the equation ‘true’. Therefore,
solutions obtained can be checked by substituting them back into the original
equation, and make sure that you get a true statement.
2. Take note of the following properties of equality:
(a) Subtraction
(b) Addition
(c) Division
(d) Multiplication
Arithmetic
8 = (4) (2)
8 – 3 = (4) (2) – 3
Algebra
a = b
a – c = b – c
;
Arithmetic
8 = (4) (2)
8 + 3 = (4) (2) + 3
Algebra
a = b
a + c = b + c
Arithmetic
8 = 6 + 2
8 6 2
3 3
Algebra
a = b
a b
c c c ≠ 0
Arithmetic
8 = (6 +2)
(8)(3) = (6+2) (3)
Algebra
a = b
ac = bc
OVERALL LESSON NOTES
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
4 Curriculum Development Division
Ministry of Education Malaysia
PART A:
LINEAR EQUATIONS
1. An equation shows the equality of two expressions and is joined by an equal sign.
Example: 2 4 = 7 + 1
2. An equation can also contain an unknown, which can take the place of a number.
Example: x + 1 = 3, where x is an unknown
A linear equation in one unknown is an equation that consists of only one unknown.
3. To solve an equation is to find the value of the unknown in the linear equation.
4. When solving equations,
(i) always write each step on a new line;
(ii) keep the left hand side (LHS) and the right hand side (RHS) balanced by:
adding the same number or term to both sides of the equation;
subtracting the same number or term from both sides of the equations;
multiplying both sides of the equation by the same number or term;
dividing both sides of the equation by the same number or term; and
(iii) simplify (whenever possible).
5. When pupils have mastered the skills and concepts involved in solving linear equations,
they can solve the questions by using alternative method.
What is solving
an equation?
LESSON NOTES
Solving an equation is like solving a puzzle to find the value of the unknown.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
5 Curriculum Development Division
Ministry of Education Malaysia
The puzzle can be visualised by using real life and concrete examples.
1. The equality in an equation can be visualised as the state of equilibrium of a balance.
2.
2. The equality in an equation can also be explained by using tiles (preferably coloured tiles).
x x
x + 2 – 2 = 5 – 2
x = 3
x + 2 = 5
(a) x + 2 = 5
x = ?
x x
x + 2 = 5 x + 2 – 2 = 5 – 2
x = 3
x = 3
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
6 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
Some pupils might face difficulty when solving linear equations in one
unknown by solving equations in the form of:
(i) x + a = b
(ii) x – a = b
where a, b, c are integers and x is an unknown.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
PART B:
SOLVING LINEAR EQUATIONS IN
THE FORMS OF
x + a = b AND x – a = b
LEARNING OBJECTIVES
Upon completion of Part B, pupils will be able to understand the concept of
solutions of linear equations in one unknown by solving equations in the
form of:
(i) x + a = b
(ii) x – a = b
where a, b, c are integers and x is an unknown.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
7 Curriculum Development Division
Ministry of Education Malaysia
PART B:
SOLVING LINEAR EQUATIONS IN THE FORM OF
x + a = b OR x – a = b
Solve the following equations.
(i) 52 x (ii) 3 5x
Solutions:
(ii) 3 5x
x – 3 + 3 = 5 + 3
x = 5 + 3
x = 8
(i) 52 x
x + 2 – 2 = 5 – 2
x = 5 – 2
x = 3
Subtract 2 from both
sides of the equation.
Simplify the LHS.
Add 3 to both sides of
the equation.
Alternative Method:
3
25
52
x
x
x
Alternative Method:
8
35
53
x
x
x
Simplify the LHS.
Simplify the RHS.
Simplify the RHS.
EXAMPLES
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
8 Curriculum Development Division
Ministry of Education Malaysia
Solve the following equations.
1. x + 1 = 6
2. x – 2 = 4 3. x – 7 = 2
4. 7 + x = 5
5. 5 + x = – 2
6. – 9 + x = – 12
7. –12 + x = 36
8. x – 9 = –54
9. – 28 + x = –78
10. x + 9 = –102
11. –19 + x = 38
12. x – 5 = –92
13. –13 + x = –120
14. –35 + x = 212
15. –82 + x = –197
TEST YOURSELF B
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
9 Curriculum Development Division
Ministry of Education Malaysia
PART C:
SOLVING LINEAR EQUATIONS IN
THE FORMS OF
ax = b AND ba
x
LEARNING OBJECTIVES
Upon completion of Part C, pupils will be able to understand the concept of
solutions of linear equations in one unknown by solving equations in the
form of:
(a) ax = b
b
a
xb )(
where a, b, c are integers and x is an unknown.
TEACHING AND LEARNING STRATEGIES
Pupils face difficulty when solving linear equations in one unknown by solving
equations in the form of:
(a) ax = b
b
a
xb )(
where a, b, c are integers and x is an unknown.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
10 Curriculum Development Division
Ministry of Education Malaysia
PART C:
SOLVING LINEAR EQUATION
ax = b AND ba
x
Solve the following equations.
(i) 3m = 12 (ii)
43
m
Solutions:
(i) 3m = 12
3 12
3 3
m
3
12m
m = 4
(ii) 43
m
3433
m
m = 4 3
m = 12
Divide both sides of
the equation by 3.
Multiply both sides of
the equation by 3.
Simplify the LHS.
Simplify the LHS.
Simplify the RHS.
Alternative Method:
4
3
12
123
m
m
m
Alternative Method:
12
43
43
m
m
m
Simplify the RHS.
EXAMPLES
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
11 Curriculum Development Division
Ministry of Education Malaysia
Solve the following equations.
1. 2p = 6
2. 5k = – 20
3. – 4h = 24
4. 567 l
5. 728 j
6. 605 n
7. 726 v
8. 427 y
9. 9612 z
10. 42
m
11. 4
r = 5
12. 8
w= –7
13. 88
t
14. 912
s
15. 65
u
TEST YOURSELF C
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
12 Curriculum Development Division
Ministry of Education Malaysia
LEARNING OBJECTIVE
Upon completion of Part D, pupils will be able to understand the concept of
solutions of linear equations in one unknown by solving equations in the
form of ax + b = c where a, b, c are integers and x is an unknown.
PART D:
SOLVING LINEAR EQUATIONS IN
THE FORM OF
ax + b = c
TEACHING AND LEARNING STRATEGIES
Some pupils might face difficulty when solving linear equations in one
unknown by solving equations in the form of ax + b = c where a, b, c are
integers and x is an unknown.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
13 Curriculum Development Division
Ministry of Education Malaysia
PART D:
SOLVING LINEAR EQUATIONS IN THE FORM OF ax + b = c
Solve the equation 2x – 3 = 11.
Solution:
Method 1
2x – 3 = 11
2x – 3 + 3 = 11 + 3
2x = 14
22
142
x
2
14x
x = 7
Method 2
1132 x
222
1132
x
2
11
2
3x
2
3
2
3
2
11
2
3x
2
14x
7x
Add 3 to both sides of
the equation.
Simplify both sides of
the equation.
Divide both sides of
the equation by 2.
Simplify the LHS.
Divide both sides of
the equation by 2.
Simplify the LHS.
Add 2
3 to both sides
of the equation.
Simplify both sides of
the equation.
Alternative Method:
2
2
14
142
3112
1132
x
x
x
x
x
Alternative Method:
7
2
14
2
3
2
11
2
11
2
3
2
2
1132
x
x
x
x
x
Simplify the RHS.
EXAMPLES
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
14 Curriculum Development Division
Ministry of Education Malaysia
Solve the following equations.
1. 2m + 3 = 7
2. 3p – 1 = 11 3. 3k + 4 = 10
4. 4m – 3 = 9
5. 4y + 3 = 9
6. 4p + 8 = 11
7. 2 + 3p = 8
8. 4 + 3k = 10
9. 5 + 4x = 1
10. 4 – 3p = 7
11. 10 – 2p = 4 12. 8 – 2m = 6
TEST YOURSELF D
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
15 Curriculum Development Division
Ministry of Education Malaysia
PART E
SOLVING LINEAR EQUATIONS IN
THE FORM OF
cba
x
LEARNING OBJECTIVES
Upon completion of Part E, pupils will be able to understand the concept of
solutions of linear equations in one unknown by solving equations in the form
of ba
x where a, b, c are integers and x is an unknown.
TEACHING AND LEARNING STRATEGIES
Pupils face difficulty when solving linear equations in one unknown by solving
equations in the form of ba
x where a, b, c are integers and x is an unknown.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
16 Curriculum Development Division
Ministry of Education Malaysia
PART E:
SOLVING LINEAR EQUATIONS IN THE FORM OF cba
x
Solve the equation 143
x
.
Solution:
Method 1
143
x
4 43
x = 1 + 4
53
x
33 53
x
35x
x = 15
Method 2
33
14
3
x
313433
x
312x
x – 12 + 12 = 3 + 12
123x
15x
Add 4 to both sides of
the equation.
Simplify both sides of
the equation.
Multiply both sides of
the equation by 3.
Simplify both sides of the
equation.
Multiply both sides of
the equation by 3.
Expand the LHS.
Simplify both sides of
the equation.
Add 12 to both sides of
the equation.
Simplify both sides of
the equation.
Alternative
Method:
15
53
53
413
143
x
x
x
x
x
EXAMPLES
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
17 Curriculum Development Division
Ministry of Education Malaysia
Solve the following equations.
1. 532
m
2. 123
b
3. 723
k
4. 3 + 2
h= 5
5. 4 +5
h = 6 6. 21
4
m
7. 54
2 h
8. 6
k+ 3 = 1 9. 2
53
h
10. 3 – 2m = 7
11. 72
3 m
12. 12 + 5h = 2
TEST YOURSELF E
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
18 Curriculum Development Division
Ministry of Education Malaysia
PART F:
FURTHER PRACTICE ON SOLVING
LINEAR EQUATIONS
LEARNING OBJECTIVE
Upon completion of Part F, pupils will be able to apply the concept of
solutions of linear equations in one unknown when solving equations of
various forms.
TEACHING AND LEARNING STRATEGIES
Pupils face difficulty when solving linear equations of various forms.
Strategy:
Teacher should emphasise the idea of balancing the linear equations. When pupils
have mastered the skills and concepts involved in solving linear equations, they
can solve the questions using the alternative method.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
19 Curriculum Development Division
Ministry of Education Malaysia
PART F:
FURTHER PRACTICE
Solve the following equations:
(i) – 4x – 5 = 2x + 7
Solution:
Method 1
2
126
126
756
756
7254
x
x
x
x
x
xx
66
55
Method 2
7254 xx
– 4x – 5 + 5 = 2x + 7 + 5
– 4x = 2x + 12
– 4x – 2x = 2x – 2x + 12
– 6x = 12
2
126
x
x
66
Subtract 2x from both sides of the equation.
Simplify both sides of the equation.
Simplify both sides of the equation.
Divide both sides of the equation by –6.
Add 5 to both sides of the equation.
Simplify both sides of the equation.
Subtract 2x from both sides of the equation.
Simplify both sides of the equation.
Divide both sides of the equation by – 6.
Alternative Method:
2
6
12
126
5724
7254
x
x
x
xx
xx
–4x – 2x – 5 = 2x – 2x + 7
Add 5 to both sides of the equation.
EXAMPLES
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
20 Curriculum Development Division
Ministry of Education Malaysia
(ii) 3(n – 2) – 2(n – 1) = 2 (n + 5)
3n – 6 – 2n + 2 = 2n + 10
n – 4 = 2n + 10
n – 2n – 4 = 2n – 2n + 10
– n – 4 = 10
– n – 4 + 4 = 10 + 4
– n = 14
14
14
n
n
11
Expand both sides of the equation.
Simplify the LHS.
Subtract 2n from both sides of the equation.
Add 4 to both sides of the equation.
Alternative Method:
14
14
1024
1022263
)5(2)1(2)2(3
n
n
nn
nnn
nnn
Divide both sides of the equation by – 1.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
21 Curriculum Development Division
Ministry of Education Malaysia
3
7
21
7
7
217
318337
1837
183364
18)1(3)32(2
)3(62
16
3
32 6
)3(62
1
3
32 6
32
1
3
32
x
x
x
x
x
xx
xx
xx
xx
xx
Add 3 to both sides of the equation.
Alternative Method:
3
7
21
217
3187
1837
183364
18)1(3)32(2
632
1
3
326
32
1
3
32
x
x
x
x
x
xx
xx
xx
xx
(iii)
Simplify LHS.
Expand the brackets.
Multiply both sides of the equation by the
LCM.
Divide both sides of the equation by 7.
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
22 Curriculum Development Division
Ministry of Education Malaysia
Solve the following equations.
1. 4x – 5 + 2x = 8x – 3 – x
2. 4(x – 2) – 3(x – 1) = 2 (x + 6)
3. –3(2n – 5) = 2(4n + 7)
2
9
4
3 .4
x
6
5
3
2
2 .5
x
253
.6 xx
6
135
2 .7
yy
2
9
4
1
3
2 .8
xx
08
43
6
52 .9
xx
12
74
9
72 .10
xx
TEST YOURSELF F
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
23 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF B:
1. x = 5
4. x = –2
7. x = 48
10. x = –111
13. x = –107
2. x = 6
5. x = –7
8. x = –45
11. x = 57
14. x = 247
3. x = 9
6. x = –3
9. x = –50
12. x = –87
15. x = –115
TEST YOURSELF C:
1. p = 3
4. l = 8
7. v = 12
10. m = 8
2. k = – 4
5. j = – 9
8. y = – 6
11. r = 20
3. h = –6
6. n = 12
9. z = 8
12. w = – 56
13. t = – 64
TEST YOURSELF D:
1. m = 2
4. m = 3
7. p = 2
10. p = −1
14. s = 108
2. p = 4
2
3 5. y
8. k = 2
11. p = 3
15. u = 30
3. k = 2
4
3 6. p
9. x = –1
12. m = 1
TEST YOURSELF E:
1. m = 4
4. h = 4
7. h = 12
10. m = −2
10. b = 9
5. h = 10
8. k = −12
11. m = −8
11. k = 15
6. m = 12
9. h = 5
12. h = −2
ANSWERS
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations
24 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF F:
1. x = − 2 2. x = − 17 3. 14
1n 4. x = 6
5. x = 3 6. x = 15 7. y = 3 8. x = 7
9. x = −8 10. x = 19
Unit 1:
Negative Numbers
UNIT 5
INDICES
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Indices I 2
1.0 Expressing Repeated Multiplication as an and Vice Versa 3
2.0 Finding the Value of an
3
3.0 Verifying nmnm aaa
4
4.0 Simplifying Multiplication of Numbers, Expressed in Index
Notation with the Same Base 4
5.0 Simplifying Multiplication of Algebraic Terms, Expressed in Index
Notation with the Same Base 5
6.0 Simplifying Multiplication of Numbers, Expressed in Index
Notation with Different Bases 5
7.0 Simplifying Multiplication of Algebraic Terms Expressed in Index
Notation with Different Bases 5
Part B: Indices II 8
1.0 Verifying nmnm aaa
9
2.0 Simplifying Division of Numbers, Expressed In Index Notation
with the Same Base 9
3.0 Simplifying Division of Algebraic Terms, Expressed in Index
Notation with the Same Base 10
4.0 Simplifying Multiplication of Numbers, Expressed in Index
Notation with Different Bases 10
5.0 Simplifying Multiplication of Algebraic Terms, Expressed in
Index Notation with Different Bases 10
Part C: Indices III 12
1.0 Verifying mnnm aa )(
13
2.0 Simplifying Numbers Expressed in Index Notation Raised
to a Power 13
3.0 Simplifying Algebraic Terms Expressed in Index Notation Raised
to a Power 14
4.0 Verifying n
n
aa
1
15
5.0 Verifying nn aa
1
16
Activity 20
Answers 22
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
1
Curriculum Development Division
Ministry of Education Malaysia
PART 1
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding on the
concept of indices.
2. This module aims to provide the basic essential skills for the learning of
Additional Mathematics topics such as:
Indices and Logarithms
Progressions
Functions
Quadratic Functions
Quadratic Equations
Simultaneous Equations
Differentiation
Linear Law
Integration
Motion Along a Straight Line
3. Teachers can use this module as part of the materials for teaching the
sub-topic of Indices in Form 4. Teachers can also use this module after
PMR as preparatory work for Form 4 Mathematics and Additional
Mathematics. Nevertheless, students can also use this module for self-
assessed learning.
4. This module is divided into three parts. Each part consists of a few learning
objectives which can be taught separately. Teachers are advised to use any
sections of the module as and when it is required.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
2
Curriculum Development Division
Ministry of Education Malaysia
LEARNING OBJECTIVES
Upon completion of Part A, pupils will be able to:
1. express repeated multiplication as an and vice versa;
2. find the value of an;
3. verify nmnm aaa ;
4. simplify multiplication of
(a) numbers;
(b) algebraic terms, expressed in index notation with the same base;
5. simplify multiplication of
(a) numbers; and
(b) algebraic terms, expressed in index notation with different bases.
PART A:
INDICES I
TEACHING AND LEARNING STRATEGIES
The concept of indices is not easy for some pupils to grasp and hence they
have phobia when dealing with multiplication of indices.
Strategy:
Pupils learn from the pre-requisite of repeated multiplication starting from
squares and cubes of numbers. Through pattern recognition, pupils make
generalisations by using the inductive method.
The multiplication of indices should be introduced by using numbers and
simple fractions first, and then followed by algebraic terms. This is intended
to help pupils build confidence to solve questions involving indices.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
3
Curriculum Development Division
Ministry of Education Malaysia
1.0 Expressing Repeated Multiplication As an
and Vice Versa
(i) 3332
(ii) )4)(4)(4(3)4(
(iii) rrrr 3
(iv) )6)(6()6( 2 mmm
2.0 Finding the Value of an
2 factors of 3
3 factors of (4)
3 factors of r
2 factors of (6+m)
32
is read as
‘three to the power of 2’
or
‘three to the second power’.
32
base
81
16
3333
2222
3
2
3
2iii)(
125
)5)(5)(5()5()ii(
32
222222)i(
4
44
3
5
LESSON NOTES A
index
(a) What is 24?
(b) What is (−1)3?
(c) What is an?
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
4
Curriculum Development Division
Ministry of Education Malaysia
3.0 Verifying nmnm aaa
325
32
213
2
437
43
)1()1(
)]1)(1)(1[()]1)(1[()1()1()iii(
77
)77(777)ii(
22
)2222()222(22)i(
yy
yyyyyyy
4.0 Simplifying Multiplication of Numbers, Expressed In Index Notation with the Same
Base
nmnm aaa
6
515
11
8383
8
14343
3
1
3
1
3
1
3
1)iii(
)5(
)5()5()5()ii(
6
6666)i(
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
5
Curriculum Development Division
Ministry of Education Malaysia
3433133
236236
10755255525
15
4
15
4
2
1
5
4
3
2)iii(
30523)ii(
)i(
qpqpqpp
rstrst
nmnmnnmm
5.0 Simplifying Multiplication of Algebraic Terms, Expressed In Index Notation with the
Same Base
6.0 Simplifying Multiplication of Numbers, Expressed In Index Notation with Different
Bases
7.0 Simplifying Multiplication of Algebraic Terms Expressed In Index Notation with
Different Bases
4133
52323
402011920119
64242
)iv(
)()()()iii(
6632)ii(
)i(
t
s
t
s
t
s
t
s
abababab
wwwww
pppp
4
4
4
4
44
,Conversely
t
s
t
s
t
s
t
s
45423423
17103147331473
312384384
5
3
2
1
5
3
2
1
5
3
2
1
2
1)iii(
75757755)ii(
2323233(i)
Note: Sum up the indices
with the same base.
numbers with
different bases
cannot be simplified.
555
555
)(
,Conversely
)(
abba
baab
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
6
Curriculum Development Division
Ministry of Education Malaysia
1. Find the value of each of the following.
(a)
243
3333335
(b) 36
(c) 4)4(
(d)
5
5
1
(e)
3
4
3
(f)
2
5
12
(g) 47
(h)
5
3
2
2. Simplify the following.
(a)
5
2323
12
1243
m
mmm
(b) bbb 42 35
(c) 342 3)3(2 xxx
(d) 323 )()2(7 ppp
EXAMPLES & TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
7
Curriculum Development Division
Ministry of Education Malaysia
3. Simplify the following.
(a)
576
96434 23
(b) 232 22)3(
(c) 343 )7()7()1(
(d)
232
5
4
3
1
3
1
(e) 423 5522
(f)
7
2
3
2
7
2
3
2223
4. Simplify the following.
(a) 2424 1234 gfgf
(b) 232 32)3( srr
(c) 343 )3()7()( vww
(d)
232
5
4
5
1
7
3kkh
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
8
Curriculum Development Division
Ministry of Education Malaysia
PART B:
INDICES II
LEARNING OBJECTIVES
Upon completion of Part B, pupils will be able to:
1. verifynmnm aaa ;
2. simplify division of
(a) numbers;
(b) algebraic terms, expressed in index notation with the same base;
3. simplify division of
(a) numbers; and
(b) algebraic terms, expressed in index notation with different bases.
TEACHING AND LEARNING STRATEGIES
Some pupils might have difficulties in when dealing with division of indices.
Strategy:
Pupils should be able to make generalisations by using the inductive method.
The divisions of indices are first introduced by using numbers and simple
fractions, and then followed by algebraic terms. This is intended to help
pupils build confidence to solve questions involving indices.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
9
Curriculum Development Division
Ministry of Education Malaysia
1.0 Verifying nmnm aaa
2. 0 Simplifying Division of Numbers, Expressed In Index Notation with the Same Base
3
5412
54
12
7
310
3
10
4
239239
6
2828
3
333
3 (iv)
5
55
5(iii)
7
7777(ii)
4
444 (i)
LESSON NOTES B
Note:
1
1
0
0
a
a
aaa
aaaa
m
mmm
mmmm
(a) What is 2
5 ÷ 2
5?
(b) What is 20?
(c) What is a0?
nmnm aaa
1
1 1
1
1
/
1
/
1
/
1 1 1
23
23
297
29
352
35
)2()2(
)2)(2(
)2)(2)(2()2()2()iii(
55
55
55555555555)ii(
22
222
2222222)i(
pp
pp
ppppp
1
/
1
/
1 1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
10
Curriculum Development Division
Ministry of Education Malaysia
3.0 Simplifying Division of Algebraic Terms, Expressed In Index Notation with the Same
Base
3
8
3
8
3
8 (iii)
445
20(ii)
(i)
23
2
3
437
3
7
24646
hhh
h
kkk
k
nnnn
4.0 Simplifying Multiplication of Numbers, Expressed In Index Notation With Different
Bases
5.0 Simplifying Multiplication of Algebraic Terms, Expressed In Index Notation with
Different Bases
REMEMBER!!!
Numbers with
different bases cannot
be simplified.
45
2638
23
68
6
11
6
11
6
415
64
156415
5
4
5
4
60
48)ii(
333
3
939 (i)
qp
qpqp
qp
k
h
k
h
k
h
kh
hkhh
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
11
Curriculum Development Division
Ministry of Education Malaysia
1. Find the value of each of the following.
(a)
144
12
121212
2
3535
(b) 999 310
(c)
3
9
8
8
(d)
1218
3
2
3
2
(e)
18
20
)5(
)5(
(f)
24
1018
3
33
2. Simplify the following.
(a)
7
512512
q
qqq
(b) 79 84 yy
(c)
8
10
15
35
m
m
(d)
88
1114
2
2
b
b
3. Simplify the following.
(a)
45
1549
4
59
2
9
2
9
8
36
nm
nmnm
nm
(b)
76
1316
12
64
dc
dc
(c)
34
96
12
64
gf
fgf
(d)
56
489
12
378
vu
uvu
EXAMPLES & TEST YOURSELF B
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
12
Curriculum Development Division
Ministry of Education Malaysia
PART C:
INDICES III
LEARNING OBJECTIVES
Upon completion of Part C of the module, pupils will be able to:
1. derivemnnm aa )(
;
2. simplify
(a) numbers;
(b) algebraic terms, expressed in index notation raised to a power;
3. verify n
n
aa
1
; and
4. verify nn aa
1
.
TEACHING AND LEARNING STRATEGIES
The concept of indices is not easy for some pupils to grasp and hence they
have phobia when dealing with algebraic terms.
Strategy:
Pupils learn from the pre-requisite of repeated multiplication starting from
squares and cubes of numbers. Through pattern recognition, pupils make
generalisations by using the inductive method.
In each part of the module, the indices are first introduced using numbers and
simple fractions, and then followed by algebraic terms. This is intended to
help pupils build confidence to solve questions involving indices.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
13
Curriculum Development Division
Ministry of Education Malaysia
1.0 Verifying mnnm aa )(
24
23
8
6
44
33
4
3
4
32
4
3
35391527
555999
595959359
236
33
3323
15
11
15
11
15
11
15
11
15
11
15
11)iii(
2323
23
)23)(23)(23()23()ii(
22
2
22)2()i(
2. 0 Simplifying Numbers Expressed In Index Notation Raised to a Power
245
396
385
31363
85
136(iv)
20715421075342)10(75
34(iii)
1593525395725)397(2(ii)
121062106)2(10(i)
mnnm aa )(
LESSON NOTES C
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
14
Curriculum Development Division
Ministry of Education Malaysia
3.0 Simplifying Algebraic Terms Expressed In Index Notation Raised to a Power
518
518
273615
23
76535
23
7653
15
20
15
20
15
205
53
5455
3
4
412
412
4
412
4143
44
3
201510
5453525432
105
52552
3
32
3
32
3
32
12
42
12
4)2()v(
32
32
)2(
)2(2)iv(
625
1
625
5
5
1
5
1)iii(
)()ii(
3
3)3((i)
qp
qp
qp
qp
qpp
qp
qpp
n
m
n
m
n
m
n
m
n
m
ba
ba
ba
baba
gfe
gfegfe
x
xx
Note:
A negative number raised to an even power is positive.
A negative number raised to an odd power is negative.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
15
Curriculum Development Division
Ministry of Education Malaysia
4. 0 Verifying n
n
aa
1
Alternative Method
n
n
10
110
10
1
100
110
10
1
10
110
110
1010
10010
100010
0001010
2
2
1
1
0
1
2
3
4
n
n
aa
1
352
3
52
2
2
264
2
64
777
1
77777
7777)ii(
3
13
333
1
333333
333333)i(
Hint: 100
?
1000
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
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Curriculum Development Division
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5.0 Verifying nn aa
1
pp
pp
p
p
pp
p
p
mm
mm
mmm
1
1
1
11
55
1
55 5
1
5
1
5
1
5
1
5
1
55
5
5
1
5
5
1
15
5
15
5
1
2
1
2
1
2
1
2
2
1
2
2
1
12
2
12
2
1
)iii(
22
222222
22
22
222(ii)
33
333
33
33
333(i)
Take square root on both sides
of the equation.
Note:
mnn
m
nn
aa
aa
1
(a) What is 2
1
4 ?
(b) What is 2
3
4 ?
(c) What is n
m
a ?
nn aa
1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
17
Curriculum Development Division
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1. Find the value of each of the following.
(a) (b)
32 ])1[(
(c)
2
2
3
7
2
(d)
32
5
3
(e)
32
5
3
(f)
2. (a) Simplify the following.
(i) 824
4246426
32
3232
(ii) 2346 52
(iii) 5132 44
(iv)
32
5
2
4
3
(v)
23
7
3
4
7
(vi)
4
422
5
43
12
5
327682
22
15
3535
4
232
EXAMPLES & TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
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Curriculum Development Division
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2. (b) Simplify the following.
(i)
15
155
535153
32
2
))(2(2
x
x
xx
(ii) 674 yx
(iii) 3122 ww
(iv) 779 84 yy
(v)
2
68
59
9
36
qp
qp
(vi)
3. Simplify the following expressions:
(a)
32
1
2
12
5
5
(b)
1
4
3
(c)
4
23y
x
(d)
51
4
6
2
ts
st
(e)
3
23
12
2 km
nm
(f)
2
63
32
2
8
ba
cab
4423 3 2 mnnm
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
19
Curriculum Development Division
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4. Find the value of each of the following.
(a)
4
6464 33
1
(b) 2
5
100
(c)
4
3
81
(d) 2
1
2
1
273
(e) m
1
m235
110 )()( aaa
(f)
3
4
27
1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
20
Curriculum Development Division
Ministry of Education Malaysia
1. 52
10
44
4
P 04 O
34 R 174 T
134
2. 2327551010
T 514510 O 65510 N 55510 B 614510
3.
2
22
4
32
D 4
22
E 2
2
2
3 N 2
2
4
3 O
3
42
4. xyxy 239 82
M 4
27xy A 4
114
x
y L
4
21xy K 2
74
x
y
5. 425 32
A 820 32 N 69 32 T
620 32 S 89 32
6. 4225 nnmm
T 87nm U
810nm L 67nm E
610nm
Solve the questions to discover the WONDERWORD!
You are given 11 multiple choice questions.
Choose the correct answer for each of the question.
Use the alphabets for each of the answer to form the WONDERWORD!
ACTIVITY
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
21
Curriculum Development Division
Ministry of Education Malaysia
7.
3243
5
2
5
2
5
2
5
2
F 12
5
2
A 2
5
2
V 6
5
2
E 5
5
2
8.
5
3
2
4
7
Y
15
10
4
7 R
8
7
4
7 M
8
10
4
7 A
15
7
4
7
9. 36
59
5
25
ba
ba
L 81515 ba I
835 ba S 235 ba T
5615 ba
10.
5232
5
2
5
2
3
1
3
1
P 105
5
2
3
1
E 76
5
2
3
1
I 75
5
2
3
1
R 106
5
2
3
1
11. 23
76
3
12
qp
qp
Y 3
53qp A
534 qp R 993
1
qp D
993 qp
Congratulations! You have completed this activity.
1 2 3 4 5 6 7 8 9 10 11
The WONDERWORD IS: ........................................................
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
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Curriculum Development Division
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TEST YOURSELF A:
1.
(a) 243
(b) 216
(c) 256 (d)
3125
1
(e)
64
27
(f)
25
214
(g) 2401 (h)
243
32
2.
(a) 512m
(b) 715b
(c) 918x
(d) 814p
3.
(a) 576
(b) 288
(c) 823543
(d)
6075
16
(e) 000250
(f)
34983
256
ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
23
Curriculum Development Division
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4.
(a) 2412 gf
(b) 2554 sr
(c) 3782764 vw (d) 52
125153
144kh
TEST YOURSELF B:
1.
(a) 144
(b) 441531
(c) 144262 (d)
729
64
(e) 25
(f) 81
2.
(a) 7q (b) 2
2
1y
(c) 2
3
7m
(d) 364b
3.
(a) 45
2
9nm
(b) 610
3
16dc
(c) 632 gf
(d) 3714 vu
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
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Curriculum Development Division
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TEST YOURSELF C:
1.
(a) (b) 1
(c)
2401
64
(d)
15625
729
5
36
(e)
125
729
5
33
6
(f)
2. (a)
(i) 8
3224
(ii) 624 52
(iii)
(iv)
)5(2
33
2
(v) 3
2
4
)3(7
(vi)
2
146
5
)4(3
2. (b)
(i) 1532x (ii) 4224 yx
(iii) 3 0
1
w
(iv)
7
14
2
y
(v) 2
16
q
p
(vi) 187162 nm
32768
21677716224
114
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 5: Indices
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Curriculum Development Division
Ministry of Education Malaysia
3.
(a)
32
1
2
15
(b)
3
4
(c) 4
8
81x
y
(d)
9
2
3
1
t
s
(e) 3368 nmk
(f)
16
64
16
1
b
ca
4.
(a) 4
(b) 000100
(c)
27
1
(d) 9
(e) 5a
(f)
81
1
ACTIVITY:
The WONDERWORD is ONEMALAYSIA
Unit 1:
Negative Numbers
UNIT 6
COORDINATES
AND
GRAPHS OF FUNCTIONS
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Coordinates 2
Part A1: State the Coordinates of the Given Points 4
Activity A1 8
Part A2: Plot the Point on the Cartesian Plane Given Its Coordinates 9
Activity A2 13
Part B: Graphs of Functions 14
Part B1: Mark Numbers on the x-Axis and y-Axis Based on the Scales Given 16
Part B2: Draw Graph of a Function Given a Table for Values of x and y 20
Activity B1 23
Part B3: State the Values of x and y on the Axes 24
Part B4: State the Value of y Given the Value x from the Graph and Vice Versa 28
Activity B2 34
Answers 35
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
Ministry of Education Malaysia
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding of the concept of
coordinates and graphs.
2. It is hoped that this module will provide a solid foundation for the studies of
Additional Mathematics topics such as:
Coordinate Geometry
Linear Law
Linear Programming
Trigonometric Functions
Statistics
Vectors
3. Basically, this module is designed to enhance the pupils’ skills in:
stating coordinates of points plotted on a Cartesian plane;
plotting points on a Cartesian plane given the coordinates of the points;
drawing graphs of functions on a Cartesian plane; and
stating the y-coordinate given the x-coordinate of a point on a graph and
vice versa.
4. This module consists of two parts. Part A deals with coordinates in two sections
whereas Part B covers graphs of functions in four sections. Each section deals
with one particular skill. This format provides the teacher with the freedom of
choosing any section that is relevant to the skills to be reinforced.
5. Activities are also included to make the reinforcement of basic essential skills
more enjoyable and meaningful.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
Ministry of Education Malaysia
LEARNING OBJECTIVES
Upon completion of Part A, pupils will be able to:
1. state the coordinates of points plotted on a Cartesian plane; and
2. plot points on the Cartesian plane, given the coordinates of the points.
PART A:
COORDINATES
TEACHING AND LEARNING STRATEGIES
Some pupils may find difficulty in stating the coordinates of a point. The
concept of negative coordinates is even more difficult for them to grasp.
The reverse process of plotting a point given its coordinates is yet another
problem area for some pupils.
Strategy:
Pupils at Form 4 level know what translation is. Capitalizing on this, the
teacher can use the translation = , where O is the origin and P
is a point on the Cartesian plane, to state the coordinates of P as (h, k).
Likewise, given the coordinates of P as ( h , k ), the pupils can carry out
the translation = to determine the position of P on the Cartesian
plane.
This common approach will definitely make the reinforcement of both the
basic skills mentioned above much easier for the pupils. This approach
of integrating coordinates with vectors will also give the pupils a head start
in the topic of Vectors.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
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PART A:
COORDINATES
1.
2. The translation must start from the origin O horizontally [left or right] and then vertically
[up or down] to reach the point P.
3. The appropriate sign must be given to the components of the translation, h and k, as shown in the
following table.
Component Movement Sign
h left –
right +
k up +
down –
4. If there is no horizontal movement, the x-coordinate is 0.
If there is no vertical movement, the y-coordinate is 0.
5. With this system, the coordinates of the Origin O are (0, 0).
Coordinates of P = (h, k)
Start from the
origin.
x
y
O
● P
h units
k units
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
Ministry of Education Malaysia
PART A1: State the coordinates of the given points.
1.
Coordinates of A = (2, 3)
1.
Coordinates of A =
2.
Coordinates of B = (–3, 1)
2.
Coordinates of B =
3.
Coordinates of C = (–2, –2)
3.
Coordinates of C =
EXAMPLES TEST YOURSELF
• A
• 4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
Start from
the origin,
move 2 units
to the right.
Next, move
3 units up. • A
•
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
4
3
2
1
-1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
Next, move
1 unit up.
• B
Start from the
origin, move 3 units
to the left.
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• B
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• C
Start from
the origin,
move 2 units
to the left.
Next, move 2
units down.
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• C
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
5
Curriculum Development Division
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PART A1: State the coordinates of the given points.
4.
Coordinates of D = (4, –3)
4.
Coordinates of D =
5.
Coordinates of E = (3, 0)
5.
Coordinates of E =
6.
Coordinates of F = (0, 3)
6.
Coordinates of F =
EXAMPLES TEST YOURSELF
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
Start from
the origin,
move 4 units
to the right.
Next, move
3 units
down.
• D
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x • E
Start from the
origin, move 3 units
to the right.
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
Start from
the origin,
move 3 units
up.
• F
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• • D
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x • E
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• F
EXAMPLES
TEST YOURSELF
Do not move
along the y-axis
since y = 0.
Do not move
along the x-axis
since x = 0.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
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PART A1: State the coordinates of the given points.
7.
Coordinates of G = (–2, 0)
7.
Coordinates of G =
8.
Coordinates of H = (0, –2)
8.
Coordinates of H =
9.
Coordinates of J = (6, 8)
9.
Coordinates of J =
EXAMPLES TEST YOURSELF
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
Start from
the origin,
move 2 units
to the left.
• G
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x • G
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• H
Start from the
origin, move 2 units
down.
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• H
8
6
4
2
–2
–4
–6
–8
y
–8 –6 –4 –2 0 2 4 6 8 x
• J
Start from
the origin,
move 6 units
to the right.
Next, move
8units up.
8
6
4
2
–2
–4
–6
–8
y
–8 –6 –4 –2 0 2 4 6 8 x
• J
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
7
Curriculum Development Division
Ministry of Education Malaysia
PART A1: State the coordinates of the given points.
10.
Coordinates of K = (– 6 , 6)
10.
Coordinates of K =
11.
Coordinates of L = (–15, –20)
11.
Coordinates of L =
12.
Coordinates of M = (3, – 4)
12.
Coordinates of M =
8
6
4
2
–2
–4
–6
–8
y
–8 –6 –4 –2 0 2 4 6 8 x
Start from
the origin,
move 6 units
to the left.
• K
Next, move
6 units up.
8
6
4
2
–2
–4
–6
–8
y
–8 –6 –4 –2 0 2 4 6 8 x
• K
20
15
10
5
–5
–10
–15
–20
y
–20 –15 –10 –5 0 5 10 15 20 x
• L
Next, move
20 units
down.
Start from the
origin, move 15 units
to the left.
20
15
10
5
–5
–10
–15
–20
y
–20 –15 –10 –5 0 5 10 15 20 x
• L
M
4
2
–2
–4
y
–4 –2 0 2 4 x
•
Start from
the origin,
move 3 units
to the right.
Next, move 4
units down.
4
2
–2
–4
y
–4 –2 0 2 4 x
• M
EXAMPLES TEST YOURSELF
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
8
Curriculum Development Division
Ministry of Education Malaysia
Write the step by step directions involving integer coordinates that
will get the mouse through the maze to the cheese.
–6 –5 –4 –3 –2 –1
7
6
5
4
3
2
1
–1
–2
–3
–4
–5
–6
0
y
1 2 3 4 5 6 7
x
ACTIVITY A1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
9
Curriculum Development Division
Ministry of Education Malaysia
PART A2: Plot the point on the Cartesian plane given its coordinates.
.
1. Plot point A (3, 4)
1. Plot point A (2, 3)
2. Plot point B (–2, 3)
2. Plot point B (–3, 4)
3. Plot point C (–1, –3)
3. Plot point C (–1, –2)
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 -1 0 1 2 3 4 x
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
EXAMPLES TEST YOURSELF
4
3
2
1
–1
–2
–3
–4
• A
• y
–4 –3 –2 –1 0 1 2 3 4 x
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• B
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• C
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
10
Curriculum Development Division
Ministry of Education Malaysia
PART A2: Plot the point on the Cartesian plane given the coordinates.
.
4. Plot point D (2, – 4)
4. Plot point D (1, –3)
5. Plot point E (1, 0)
5. Plot point E (2, 0)
6. Plot point F (0, 4)
6. Plot point F (0, 3)
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
EXAMPLES TEST YOURSELF
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• D
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x • E
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• F
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
11
Curriculum Development Division
Ministry of Education Malaysia
PART A2: Plot the point on the Cartesian plane given the coordinates.
.
7. Plot point G (–2, 0)
7. Plot point G (– 4,0)
8. Plot point H (0, – 4)
8. Plot point H (0, –2)
9. Plot point J (6, 4)
9. Plot point J (8, 6)
EXAMPLES TEST YOURSELF
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x • G
8
6
4
2
–2
–4
–6
–8
y
–8 –6 –4 –2 0 2 4 6 8 x
• J
8
6
4
2
–2
–4
–6
–8
y
–8 –6 –4 –2 0 2 4 6 8 x
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• H
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
12
Curriculum Development Division
Ministry of Education Malaysia
PART A2: Plot the point on the Cartesian plane given the coordinates.
.
10. Plot point K (– 4, 6)
10. Plot point K (– 6, 2)
11. Plot point L (–15, –10)
11. Plot point L (–20, –5)
12. Plot point M (30, –15)
12. Plot point M (10, –25)
29
10
–10
–20
y
–20 –10 0 10 20 x
• L
EXAMPLES TEST YOURSELF
8
4
–4
–8
y
–8 –4 0 4 8 x
• K
8
4
–4
–8
y
-8 -4 0 4 8 x
–20 –10 0 10 20
20
10
–10
–20
y
x
20
10
–10
–20
y
–40 –20 0 20 40 x
20
10
–10
–20
y
–40 –20 0 20 40 x
• M
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
13
Curriculum Development Division
Ministry of Education Malaysia
1. Plot the following points on the Cartesian plane.
P(3, 3) , Q(6, 3) , R(3, 1) , S(6, 1) , T(6, –2) , U(3, –2) ,
A(–3, 3) , B(–5, –1) , C(–2, –1) , D(–3, – 2) , E(1, 1) , F(2, 1).
2. Draw the following line segments:
AB, AD, BC, EF, PQ, PR, RS, UT, ST
YAKOMI ISLANDS
2 4 –2 –4 x
2
4
–2
y
0
–4
,
Exclusive News:
A group of robbers stole RM 1 million from a bank. They hid the money
somewhere near the Yakomi Islands. As an expert in treasure hunting, you
are required to locate the money! Carry out the following tasks to get the
clue to the location of the money.
Mark the location with the symbol.
Enjoy yourself !
ACTIVITY A2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
14
Curriculum Development Division
Ministry of Education Malaysia
LEARNING OBJECTIVES
Upon completion of Part B, pupils will be able to:
1. understand and use the concept of scales for the coordinate axes;
2. draw graphs of functions; and
3. state the y-coordinate given the x-coordinate of a point on a graph and
vice versa.
PART B:
GRAPHS OF FUNCTIONS
TEACHING AND LEARNING STRATEGIES
Drawing a graph on the graph paper is a challenge to some pupils. The concept
of scales used on both the x-axis and y-axis is equally difficult. Stating the
coordinates of points lying on a particular graph drawn is yet another
problematic area.
Strategy:
Before a proper graph can be drawn, pupils need to know how to mark numbers
on the number line, specifically both the axes, given the scales to be used.
Practice makes perfect. Thus, basic skill practices in this area are given in Part
B1. Combining this basic skills with the knowledge of plotting points
on the Cartesian plane, the skill of drawing graphs of functions, given the
values of x and y, is then further enhanced in Part B2.
Using a similar strategy, Stating the values of numbers on the axes is
done in Part B3 followed by Stating coordinates of points on a graph in
Part B4.
For both the skills mentioned above, only the common scales used in the
drawing of graphs are considered.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
15
Curriculum Development Division
Ministry of Education Malaysia
PART B:
GRAPHS OF FUNCTIONS
1. For a standard graph paper, 2 cm is represented by 10 small squares.
2. Some common scales used are as follows:
Scale Note
2 cm to 10 units
10 small squares represent 10 units
1 small square represents 1 unit
2 cm to 5 units
10 small squares represent 5 units
1 small square represents 0.5 unit
2 cm to 2 units
10 small squares represent 2 units
1 small square represents 0.2 unit
2 cm to 1 unit
10 small squares represent 1 unit
1 small square represents 0.1 unit
2 cm to 0.1 unit
10 small squares represent 0.1 unit
1 small square represents 0.01 unit
2 cm
2 cm
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
16
Curriculum Development Division
Ministry of Education Malaysia
PART B1: Mark numbers on the x-axis and y-axis based on the scales given.
1. Mark – 4. 7, 16 and 27on the x-axis.
Scale: 2 cm to 10 units.
[ 1 small square represents 1 unit ]
1. Mark – 6 4, 15 and 26 on the x-axis.
Scale: 2 cm to 10 units.
[ 1 small square represents 1 unit ]
2. Mark –7, –2, 3 and 8on the x-axis.
Scale: 2 cm to 5 units.
[ 1 small square represents 0.5 unit ]
2. Mark –8, –3, 2 and 6, on the x-axis.
Scale: 2 cm to 5 units.
[ 1 small square represents 0.5 unit ]
3. Mark –3.4, – 0.8, 1 and 2.6, on the x-axis.
Scale: 2 cm to 2 units.
[ 1 small square represents 0.2 unit ]
3. Mark –3.2, –1, 1.2 and 2.8 on the x-axis.
Scale: 2 cm to 2 units.
[ 1 small square represents 0.2 unit ]
4. Mark –1.3, – 0.6, 0.5 and 1.6 on the x-axis.
Scale: 2 cm to 1 unit.
[ 1 small square represents 0.1 unit ]
4. Mark –1.7, – 0.7, 0.7 and 1.5 on the x-axis.
Scale: 2 cm to 1 unit.
[ 1 small square represents 0.1 unit ]
0 –10 10 20 30
x
7 16 27 –4
x
x
–5 –10 0 5 10
x
–2 3 8 –7
x
–2 –4 2 4
x
1 –3.4 0 –0.8 2.6
x
– 1 –2 1 2
x
0.5 –1.3 0 –0.6 1.6
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
17
Curriculum Development Division
Ministry of Education Malaysia
PART B1: Mark numbers on the x-axis and y-axis based on the scales given.
5. Mark – 0.15, – 0.04, 0.03 and 0.17 on the
x-axis.
Scale: 2 cm to 0.1 unit
[ 1 small square represents 0.01 unit ]
5. Mark – 0.17, – 0.06, 0.04 and 0.13 on the
x-axis.
Scale: 2 cm to 0.1 unit
[ 1 small square represents 0.01 unit ]
6. Mark –13, –8, 2 and 14 on the y-axis.
Scale: 2 cm to 10 units
[ 1 small square represents 1 unit ]
6. Mark –16, – 4, 5 and 15 on the y-axis.
Scale: 2 cm to 10 units
[ 1 small square represents 1 unit ]
x
0 –0.1 –0.2 0.1 0.2 0.03 0.17 –0.04 –0.15
x
y y
0
10
–20
20
–10
–13
–8
2
14
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
18
Curriculum Development Division
Ministry of Education Malaysia
PART B1: Mark numbers on the x-axis and y-axis based on the scales given.
7. Mark –9, –3, 1 and 7 on the y-axis.
Scale: 2 cm to 5 units.
[ 1 small square represents 0.5 unit ]
7. Mark –7, – 4, 2 and 6 on the y-axis.
Scale: 2 cm to 5 units.
[ 1 small square represents 0.5 unit ]
8. Mark –3.2, – 0.6, 1.4 and 2.4 on the y-axis.
Scale: 2 cm to 2 units.
[ 1 small square represents 0.2 unit ]
8. Mark –3.4, –1.4, 0.8 and 2.8 on the y-axis.
Scale: 2 cm to 2 units.
[ 1 small square represents 0.2 unit ]
y
y
y
0
5
–10
10
–9
–3
1
7
–5
y
0
–4
4
–2
–3.2
–0.6
2
1.4
2.4
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
19
Curriculum Development Division
Ministry of Education Malaysia
PART B1: Mark numbers on the x-axis and y-axis based on the scales given.
9. Mark –1.6, – 0.4, 0.4 and 1.5 on the y-axis.
Scale: 2 cm to 1 unit.
[ 1 small square represents 0.1 unit ]
9. Mark –1.5, – 0.8, 0.3 and 1.7 on the y-axis.
Scale: 2 cm to 1 unit.
[ 1 small square represents 0.1 unit ]
10. Mark – 0.17, – 0.06, 0.08 and 0.16 on the
y-axis.
Scale: 2 cm to 0.1 unit.
[ 1 small square represents 0.01 unit ]
10. Mark – 0.18, – 0.03, 0.05 and 0.14 on the
y-axis.
Scale: 2 cm to 0.1 units.
[ 1 small square represents 0.01 unit ]
y
y
y
0
1
–2
2
–1
0.4
1.5
– 0.4
–1.6
y
0
0.2
– 0.17
–0.1
– 0.06
0.1
0.08
0.16
–0.2
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
20
Curriculum Development Division
Ministry of Education Malaysia
PART B2: Draw graph of a function given a table for values of x and y.
1. The table shows some values of two variables, x and y,
of a function.
x –2 –1 0 1 2
y –2 0 2 4 6
By using a scale of 2 cm to 1 unit on the x-axis and
2 cm to 2 units on the y-axis, draw the graph of the
function.
1. The table shows some values of two variables, x and y,
of a function.
x –3 –2 –1 0 1
y –2 0 2 4 6
By using a scale of 2 cm to 1 unit on the x-axis and
2 cm to 2 units on the y-axis, draw the graph of the
function.
2. The table shows some values of two variables, x and y,
of a function.
x –2 –1 0 1 2
y 5 3 1 –1 –3
By using a scale of 2 cm to 1 unit on the x-axis and
2 cm to 2 units on the y-axis, draw the graph of the
function.
2. The table shows some values of two variables, x and y,
of a function.
x –2 –1 0 1 2
y 7 5 3 1 –1
By using a scale of 2 cm to 1 unit on the x-axis and
2 cm to 2 units on the y-axis, draw the graph of the
function.
–1 1 x –2 2
–2
6
4
2
y
0
–1 1 x –2 2
–2
6
4
2
y
0
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
21
Curriculum Development Division
Ministry of Education Malaysia
PART B2: Draw graph of a function given a table for values of x and y.
3. The table shows some values of two variables, x and y,
of a function.
x – 4 –3 –2 –1 0 1 2
y 15 5 –1 –3 –1 5 15
By using a scale of 2 cm to 1 unit on the x-axis and
2 cm to 5 units on the y-axis, draw the graph of the
function.
3. The table shows some values of two variables, x and y,
of a function.
x –1 0 1 2 3 4 5
y 19 4 –5 –8 –5 4 19
By using a scale of 2 cm to 1 unit on the x-axis and
2 cm to 5 units on the y-axis, draw the graph of the
function.
4. The table shows some values of two variables, x and y,
of a function.
x –2 –1 0 1 2 3 4
y –7 –2 1 2 1 –2 –7
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of the
function.
4. The table shows some values of two variables, x and y,
of a function.
x –2 –1 0 1 2 3
y –8 –4 –2 –2 – 4 –8
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis, draw the graph of the
function.
y
10
5
15
–5
x –3 1 –4 2 0 –1 –2
0
y
2
–6
–2
–4
x 3 4 2 1 –1 –2
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
22
Curriculum Development Division
Ministry of Education Malaysia
PART B2: Draw graph of a function given a table for values of x and y.
5. The table shows some values of two variables, x and y,
of a function.
x –2 –1 0 1 2
y –7 –1 1 3 11
By using a scale of 2 cm to 1 unit on the x-axis and
2 cm to 5 units on the y-axis, draw the graph of the
function.
5. The table shows some values of two variables, x and y,
of a function.
x –2 –1 0 1 2
y –6 2 4 6 16
By using a scale of 2 cm to 1 unit on the x-axis and
2 cm to 5 units on the y-axis, draw the graph of the
function.
6. The table shows some values of two variables, x and y,
of a function.
x –3 –2 –1 0 1 2 3
y 22 5 0 1 2 –3 –20
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 10 units on the y-axis, draw the graph of the
function.
6. The table shows some values of two variables, x and y,
of a function.
x –3 –2 –1 0 1 2 3
y 21 4 –1 0 1 – 4 –21
By using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 10 units on the y-axis, draw the graph of the
function.
x 2 3 1 –2 –3 –1 0
y
20
–20
–10
10
y
10
5
15
–5
x
–2 1 2 –1
0
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
23
Curriculum Development Division
Ministry of Education Malaysia
Each table below shows the values of x and y for a certain function.
The graphs of all these functions, when drawn on the same axes, form a beautiful logo. Draw the logo on
the graph paper provided by using a scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis.
FUNCTION 1 FUNCTION 2
x – 4 –3 –2 –1 0 x 0 1 2 3 4
y 16 17 18 19 20 y 20 19 18 17 16
FUNCTION 3
x – 4 –3 –2 –1 0 1 2 3 4
y 16 9 4 1 0 1 4 9 16
FUNCTION 4
x –3 –2 –1 0 1 2 3
y 9 14 17 18 17 14 9
FUNCTION 5
x –3 –2 –1.5 –1 – 0.5 0
y 9 8 7.9 7 4.6 0
FUNCTION 6
x 0 0.5 1 1.5 2 3
y 0 4.6 7 7.9 8 9
x
y
0
ACTIVITY B1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
24
Curriculum Development Division
Ministry of Education Malaysia
PART B3: State the values of x and y on the axes.
1. State the values of a, b, c and d on the x-axis
below.
Scale: 2 cm to 10 units.
[ 1 small square represents 1 unit ]
a = 7, b = 13, c = – 4, d = –14
1. State the values of a, b, c and d on the x-axis
below.
2. State the values of a, b, c and d on the x-axis
below.
Scale: 2 cm to 5 units.
[ 1 small square represents 0.5 unit ]
a = 2, b = 7.5, c = –3, d = –8.5
2. State the values of a, b, c and d on the x-axis
below.
3. State the values of a, b, c and d on the x-axis
below.
Scale: 2 cm to 2 units.
[ 1 small square represents 0.2 unit ]
a = 0.6, b = 3.4, c = –1.2, d = –2.6
3. State the values of a, b, c and d on the x-axis
below.
–20 10 20
x
c d 0 –10 a b –20 10 20
x
c d 0 –10 a b
–5 –10 0 5 10
x
c a b d –5 –10 0 5 10
x
c a b d
c –2 – 4 2 4
x
a d 0 b c –2 – 4 2 4
x
a d 0 b
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
25
Curriculum Development Division
Ministry of Education Malaysia
PART B3: State the values of x and y on the axes.
4. State the values of a, b, c and d on the x-axis
below.
Scale: 2 cm to 1 unit.
[ 1 small square represents 0.1 unit ]
a = 0.8, b = 1.4, c = – 0.3, d = –1.6
4. State the values of a, b, c and d on the x-axis
below.
5. State the values of a, b, c and d on the x-axis
below.
Scale: 2 cm to 0.1 unit.
[ 1 small square represents 0.01 unit ]
a = 0.04, b = 0.14, c = – 0.03, d = – 0.16
5. State the values of a, b, c and d on the x-axis
below.
6. State the values of a, b, c and d on the y-axis
below.
Scale: 2 cm to 10 units.
[ 1 small square
represents 1 unit ]
a = 3, b = 17
c = – 6, d = –15
6. State the values of a, b, c and d on the y-axis
below.
–1 –2 1 2
x
a d 0 c b –1 –2 1 2
x
a d 0 c b
c
x
0 –0.1 –0.2 0.1 0.2 a b d c
x
0 –0.1 – 0.2 0.1 0.2 a b d
y
0
10
–20
20
–10
d
c
a
b
y
0
10
–20
20
–10
d
c
a
b
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
26
Curriculum Development Division
Ministry of Education Malaysia
PART B3: State the values of x and y on the axes.
7. State the values of a, b, c and d on the y-axis
below.
Scale: 2 cm to 5 units.
[ 1 small square
represents 0.5 unit ]
a = 4, b = 9.5
c = –2, d = –7.5
7. State the values of a, b, c and d on the y-axis
below.
8. State the values of a, b, c and d on the y-axis
below.
Scale: 2 cm to 2 units.
[ 1 small square
represents 0.2 unit ]
a = 0.8, b = 3.2
c = –1.2, d = –2.6
8. State the values of a, b, c and d on the y-axis
below.
y
0
5
–10
10
d
c
a
b
–5
y
0
5
–10
10
d
c
a
b
–5
y
0
–4
4
–2
d
c
2
a
b
y
0
–4
4
–2
d
c
2
a
b
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
27
Curriculum Development Division
Ministry of Education Malaysia
PART B3: State the values of x and y on the axes.
9. State the values of a, b, c and d on the y-axis
below.
Scale: 2 cm to 1 unit.
[ 1 small square
represents 0.1 unit ]
a = 0.7, b = 1.2
c = – 0.6, d = –1.4
9. State the values of a, b, c and d on the y-axis
below.
10. State the values of a, b, c and d on the y-axis
below.
Scale: 2 cm to 0.1 unit.
[ 1 small square
represents 0.01 unit ]
a = 0.03, b = 0.07
c = – 0.04, d = – 0.18
10. State the values of a, b, c and d on the y-axis
below.
y
0
1
–2
2
–1
a
b
c
d
y
0
1
–2
2
–1
a
b
c
d
y
0
d
–0.1
c
a
–0.2
0.2
b
0.1
y
0
d
c
a
–0.2
0.2
b
0.1
–0.1
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
28
Curriculum Development Division
Ministry of Education Malaysia
PART B4: State the value of y given the value x from the graph and vice versa.
1. Based on the graph below, find the value of y
when (a) x = 1.5
(b) x = –2.8
(a) 7 (b) –1.6
1. Based on the graph below, find the value of y
when (a) x = 0.6
(b) x = –1.7
(a) (b)
2. Based on the graph below, find the value of y
when ( a ) x = 0.14
( b ) x = – 0.26
(a) 1.5 (b) 11.5
2. Based on the graph below, find the value of y
when ( a ) x = 0.07
( b ) x = – 0.18
(a) (b)
–1 1 x –2 2
–2
6
4
2
y
0
– 2.8
1.5
7
– 1.6
–1 1 x –2 2
–2
6
4
2
y
0
– 0.26
1.5
0.14
11.5
x –0.1 – 0. 2 0.1 0.2
y
10
–10
5
–5
0 x –0.1 –0. 2 0.1 0.2
y
10
–10
5
– 5
0
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
29
Curriculum Development Division
Ministry of Education Malaysia
PART B4: State the value of y given the value x from the graph and vice versa.
3. Based on the graph below, find the value of y
when ( a ) x = 0.6
( b ) x = –2.7
( a ) 11 ( b ) –3.5
3. Based on the graph below, find the value of y
when ( a ) x = 1.2
( b ) x = –1.8
( a ) ( b )
4. Based on the graph below, find the value of y
when (a) x = 1.4
(b) x = –1.5
(a) 3 (b) –5.8
4. Based on the graph below, find the value of y
when (a) x = 2.7
(b) x = –2.1
(a) (b)
y
10
5
15
–5
x –3 1 – 4 2 0 –1 –2
11
0.6
– 2.7
– 3.5
y
10
5
15
–5
x –3 1 – 4 2 0 –1 –2
x 3 4 2 1 –1 –2 0
y
2
– 6
– 2
– 4
1.4
3
– 1.5
– 5.8
x 3 4 2 1 –1 –2 0
y
2
– 6
– 2
– 4
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
30
Curriculum Development Division
Ministry of Education Malaysia
PART B4: State the value of y given the value x from the graph and vice versa.
5. Based on the graph below, find the value of y
when (a) x = 1.7
(b) x = –1.3
(a) 5.5 (b) –3.5
5. Based on the graph below, find the value of y
when (a) x = 1.2
(b) x = –1.9
(a) (b)
6. Based on the graph below, find the value of y
when (a) x = 1.6
(b) x = –2.3
(a) –9 (b) 25
6. Based on the graph below, find the value of y
when (a) x = 2.8
(b) x = –2.6
(a) (b)
y
10
5
15
–5
–2 x 1 2 –1 0
5.5
1.7
– 1.3
– 3.5
y
10
5
15
–5
–2 x 1 2 –1 0
x 2 3 1 –2 –3 –1 0
y
20
–20
–10
10
1.6
– 9
– 2.3
25
x 2 3 1 –2 –3 –1 0
y
20
–20
–10
10
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
31
Curriculum Development Division
Ministry of Education Malaysia
PART B4: State the value of y given the value x from the graph and vice versa.
7. Based on the graph below, find the value of x
when (a) y = 5.4
(b) y = –1.6
(a) 1.4 (b) –2.8
7. Based on the graph below, find the value of x
when (a) y = 2.8
(b) y = –2.4
(a) (b)
8. Based on the graph below, find the value of x
when ( a ) y = 4
( b ) y = –7.5
(a) – 0.07 (b) 0.08
8. Based on the graph below, find the value of x
when ( a ) y = 6.5
( b ) y = –7
(a) (b)
–1 1 x –2 2
–2
6
4
2
y
0
x –0.1 –0. 2 0.1 0.2
y
10
–10
5
– 5
0
–1 1 x –2 2
–2
6
4
2
y
0
– 2.8
1.4
5.4
– 1.6
– 0.07
4
0.08
– 7.5
x –0.1 –0. 2 0.1 0.2
y
10
–10
5
–5
0
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
32
Curriculum Development Division
Ministry of Education Malaysia
PART B4: State the value of y given the value x from the graph and vice versa.
9. Based on the graph below, find the values of x
when (a) y = 8.5
(b) y = 0
(a) –3.1 , 2.1 (b) –2 , 1
9. Based on the graph below, find the values of x
when (a) y = 3.5
(b) y = 0
(a) (b)
10. Based on the graph below, find the values of x
when (a) y = 2.6
(b) y = – 4.8
(a) 0.6 , 2.1 (b) –1.2 , 3.9
10. Based on the graph below, find the values of x
when (a) y = 1.2
(b) y = – 4.4
(a) (b)
x 3 4 2 1 –1 –2 0
y
2
– 6
– 2
– 4
x –3 1 – 4 2 –1 –2 2.1 – 3.1
8.5
0
y
10
5
15
–5
x 3 4 2 1 –1 –2 0
y
2
– 6
– 2
– 4
0.6 2.1
– 1.2 3.9
2.6
– 4.8
x –3 1 – 4 2 –1 –2 0
y
10
5
15
–5
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
33
Curriculum Development Division
Ministry of Education Malaysia
PART B4: State the value of y given the value x from the graph and vice versa.
11. Based on the graph below, find the value of x
when (a) y = 14
(b) y = –17
(a) 2.6 (b) –2.3
11. Based on the graph below, find the value of x
when (a) y = 11
(b) y = –23
(a) (b)
12. Based on the graph below, find the value of x
when (a) y = 6.5
(b) y = 0
(c) y = –6
(a) – 0.8 (b) 1.3 (c) 2.3
12. Based on the graph below, find the value of x
when (a) y = 7.5
(b ) y = 0
(c) y = –9
(a) (b) (c)
x 2 3 1 –2 –3 –1 0
y
20
–20
–10
10
2.6
– 2.3
– 17
14
x 2 3 1 –2 –3 –1 0
y
20
–20
–10
10
y
10
5
15
–5
–2 x 1 2 –1 0
y
10
5
15
–5
–2 x 1 2 –1 0
6.5
– 6
1.3 – 0.8 2.3
EXAMPLES
TEST YOURSELF
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
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Task 1: Two points on the graph given are (6.5, k) and (h, 45).
Find the values of h and k.
Task 2: Smuggling takes place at the locations with coordinates (h, k).
State each location in terms of coordinates.
0
5
10
15
20
25
30
35
40
45
50
55
60
y
1 2 3 4 5 6 7 8 9 x
There is smuggling at sea and you know two possible locations.
As a responsible citizen, you need to report to the marine police these two locations.
ACTIVITY B2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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PART A:
PART A1:
1. A (4, 2) 2. B (– 4, 3)
2.
3. C (–3, –3) 4. D (3, – 4)
5. E (2, 0) 6. F (0, 2)
7. G (–1, 0) 8. H (0, –1)
9. J (8, 6) 10. K (– 4, 8)
11. L (–10, –15) 12. M (4, –3)
ACTIVITY A1:
Start at (5, 3).
Then, move in order to (4, 3), (4, –3), (3, –3), (3, 2), (1, 2) , (1, –3) , (–3, –3) , (–3, 3),
(– 4, 3), (–
4, 5), (–3, 5) and (–3, 6).
ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
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PART A2:
1.
4.
2.
5.
3.
6.
4
3
2
1
–1
–2
–3
-–4
–4 –3 –2 –1 0 1 2 3 4
y
x
• B
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• A
4
3
2
1
–1
–2
–3
–4
–4 –3 –2 –1 0 1 2 3 4
y
x
• D
4
3
2
1
–1
–2
–3
–4
–4 –3 –2 –1 0 1 2 3 4
y
x • E
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• C
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
• F
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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7.
10.
8.
11.
9.
12.
4
3
2
1
–1
–2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x • G •
K
8
4
–4
–8
y
–8 –4 0 4 8 x
4
3
2
1
–1
-2
–3
–4
y
–4 –3 –2 –1 0 1 2 3 4 x
–
• H
–20 –10 0 10 20
20
10
–10
–20
y
x
• L
8
6
4
2
–2
–4
–6
–8
y
–8 –6 –4 –2 0 2 4 6 8 x
• J
• M
20
10
–10
–20
y
–40 –20 0 20 40 x
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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ACTIVITY A2:
YAKOMI ISLANDS
2
4
–2
y
O
–4 RM 1 million
U
A
B C
D
E F
P Q
R S
T
2 4 –2 –4 x
,
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
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PART B1:
1
2.
3.
4.
5.
6.
7.
8.
9.
10.
0 –10 10 20 30
x
4 15 26 –6 –5 –10 0 5 10
x
–3 2 6 –8
–2 –4 2 4
x
–3.2 0 –1 2.8 1.2 –1 –2 1 2
x
0.7 –1.7 0 –0.7 1.5
x
0 –0.1 –0.2 0.1 0.2 0.04 0.13 –0.06 –0.16
y
0
10
–20
20
–10
–16
–4
5
15
y
0
5
–10
10
–7
–4
2
6
–5
y
0
1
–2
2
–1
0.3
1.7
–0.8
–1.5
y
0
0.2
– 0.18
– 0.1
– 0.03
0.1
0.05
0.14
– 0.2
y
0
–4
4
–2
–3.4
–1.4
2
0.8
2.8
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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Curriculum Development Division
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PART B2:
1.
2.
3.
4.
5.
6.
–2
6
4
2
y
0 x –3 1 –1 –2 –1 1 x –2 2
–2
6
4
2
y
0
x 4 –1 5 1 0
y
10
5
15
–5
2 3
y
–4
–8
–2
–6
0 x 3 2 1 –1 –2
y
10
5
15
–5
–2 x 1 2 –1 0
x 2 3 1 –2 –3 – 1 0
y
20
–20
–10
10
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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ACTIVITY B1:
–4 –3 –2 –1 0 1 2 3 4 x
2
4
6
8
10
12
14
16
18
20
y
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 6: Coordinates and Graphs of Functions
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PART B3:
1. a = 3, b = 16, c = – 3, d = – 18
2. a = 3.5, b = 7, c = – 2.5, d = – 8
3. a = 1.4, b = 2.4, c = – 1.6, d = – 3.8
4. a = 0.7, b = 1.8, c = – 0.5, d = – 1.4
5. a = 0.08, b = 0.16, c = – 0.02, d = – 0.17
6. a = 6, b = 15, c = – 3, d = – 17
7. a = 2, b = 8, c = – 0.5, d = – 8.5
8. a = 1.4, b = 3.6, c = – 0.8, d = – 3.4
9. a = 0.5, b = 1.7, c = – 0.4, d = – 1.6
10. a = 0.06, b = 0.16, c = – 0.07, d = – 0.15
PART B4:
1. (a) 6.4 (b) – 2.8
2. (a) – 12 (b) 13
3. (a) – 2.5 (b) 9
4. (a) 0.6 (b) – 5.4
5. (a) 8 (b) – 6.5
6. (a) – 16 (b) 22
7. (a) 0.7 (b) – 1.3
8. (a) – 0.08 (b) 0.12
9. (a) – 3.5, 1.5 (b) – 3 , 1
10. (a) – 1.6, 0.6 (b) – 2.7, 1.7
11. (a) 2.2 (b) – 3.5
12. (a) – 2.3 (b) – 0.6 (c) 1.4
ACTIVITY B2:
k =15, h = 1.1, 8.9
Two possible locations: (1.1, 15), (8.9, 15)
Unit 1:
Negative Numbers
UNIT 7
LINEAR INEQUALITIES
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Linear Inequalities 2
1.0 Inequality Signs 3
2.0 Inequality and Number Line 3
3.0 Properties of Inequalities 4
4.0 Linear Inequality in One Unknown 5
Part B: Possible Solutions for a Given Linear Inequality in One Unknown 7
Part C: Computations Involving Addition and Subtraction on Linear Inequalities 10
Part D: Computations Involving Division and Multiplication on Linear Inequalities 14
Part D1: Computations Involving Multiplication and Division on
Linear Inequalities 15
Part D2: Perform Computations Involving Multiplication of Linear
Inequalities 19
Part E: Further Practice on Computations Involving Linear Inequalities 21
Activity 27
Answers 29
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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MODULE OVERVIEW
1. The aim of this module is to reinforce pupils‟ understanding of the concept involved
in performing computations on linear inequalities.
2. This module can be used as a guide for teachers to help pupils master the basic skills
required to learn this topic.
3. This module consists of six parts and each part deals with a few specific skills.
Teachers may use any parts of the module as and when it is required.
4. Overall lesson notes given in Part A stresses on important facts and concepts required
for this topic.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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PART A:
LINEAR INEQUALITIES
LEARNING OBJECTIVE
Upon completion of Part A, pupils will be able to understand and use the
concept of inequality.
TEACHING AND LEARNING STRATEGIES
Some pupils might face problems in understanding the concept of linear
inequalities in one unknown.
Strategy:
Teacher should ensure that pupils are able to understand the concept of inequality
by emphasising the properties of inequalities. Linear inequalities can also be
taught using number lines as it is an effective way to teach and learn inequalities.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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PART A:
LINEAR INEQUALITY
1.0 Inequality Signs
a. The sign “<” means „less than‟.
Example: 3 < 5
b. The sign “>” means „greater than‟.
Example: 5 > 3
c. The sign “ ” means „less than or equal to‟.
d. The sign “ ” means „greater than or equal to‟.
2.0 Inequality and Number Line
−3 < − 1
−3 is less than − 1
and
−1 > − 3
−1 is greater than − 3
1 < 3
1 is less than 3
and
3 > 1
3 is greater than 1
OVERALL LESSON NOTES
−1 − 2 − 3 x
0 1 2 3
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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3.0 Properties of Inequalities
(a) Addition Involving Inequalities
Arithmetic Form Algebraic Form
812 so 48412
92 so 6962
If a > b, then cbca
If a < b, then cbca
(b) Subtraction Involving Inequalities
Arithmetic Form Algebraic Form
7 > 3 so 5357
2 < 9 so 6962
If a > b, then cbca
If a < b, then cbca
(c) Multiplication and Division by Positive Integers
When multiply or divide each side of an inequality by the same positive number, the
relationship between the sides of the inequality sign remains the same.
Arithmetic Form Algebraic Form
5 > 3 so 5 (7) > 3(7)
12 > 9 so 12 9
3 3
If a > b and c > 0 , then ac > bc
If a > b and c > 0, then a b
c c
52 so )3(5)3(2
128 so 2
12
2
8
If ba and 0c , then bcac
If ba and 0c , then c
b
c
a
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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(d) Multiplication and Division by Negative Integers
When multiply or divide both sides of an inequality by the same negative number, the
relationship between the sides of the inequality sign is reversed.
Arithmetic Form Algebraic Form
8 > 2 so 8(−5) < 2(−5)
6 < 7 so 6(−3) > 7(−3)
16 > 8 so 16 8
4 4
10 <15 so 10 15
5 5
If a > b and c < 0, then ac < bc
If a < b and c < 0, then ac > bc
If a > b and c < 0, then a b
c c
If a < b and c < 0, then a b
c c
Note: Highlight that an inequality expresses a relationship. To maintain the same
relationship or „balance‟, pupils must perform equal operations on both sides of
the inequality.
4.0 Linear Inequality in One Unknown
(a) A linear inequality in one unknown is a relationship between an unknown and a
number.
Example: x > 12
m4
(b) A solution of an inequality is any value of the variable that satisfies the inequality.
Examples:
(i) Consider the inequality 3x
The solution to this inequality includes every number that is greater than 3.
What numbers are greater than 3? 4 is greater than 3. And so are 5, 6, 7, 8, and
so on. What about 5.5? What about 5.99? And 5.000001? All these numbers are
greater than 3, meaning that there are infinitely many solutions!
But, if the values of x are integers, then 3x can be written as
,...8 ,7 ,6 ,5 ,4x
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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A number line is normally used to represent all the solutions of an inequality.
(ii) x > 2
(iii) 3x
The solid dot
means the value
3 is included.
The open dot
means the value
2 is not
included.
3 − 2 − 1 1 0 2 x
4
o
0 − 1 − 2 x
1 2 3 4
To draw a number line representing 3x , place an
open dot on the number 3. An open dot indicates that
the number is not part of the solution set. Then, to
show that all numbers to the right of 3 are included in
the solution, draw an arrow to the right of 3.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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PART B:
POSSIBLE SOLUTIONS FOR A
GIVEN LINEAR INEQUALITY IN
ONE UNKNOWN
TEACHING AND LEARNING STRATEGIES
Some pupils might have difficulties in finding the possible solution for a given
linear inequality in one unknown and representing a linear inequality on a number
line.
Strategy:
Teacher should emphasise the importance of using a number line in order to solve
linear inequalities and should ensure that pupils are able to draw correctly the
arrow that represents the linear inequalities.
LEARNING OBJECTIVES
Upon completion of Part B, pupils will be able to solve linear
inequalities in one unknown by:
(i) determining the possible solution for a given linear inequality in one
unknown:
(a) x h
(b) x h
(c) hx
(d) x h
(ii) representing a linear inequality:
(a) x h
(b) x h
(c) hx
(d) x h on a number line and vice versa.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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PART B:
POSSIBLE SOLUTIONS FOR
A GIVEN LINEAR INEQUALITY IN ONE UNKNOWN
List out all the possible integer values for x in the following inequalities: (You can use the
number line to represent the solutions)
(1) x > 4
Solution:
The possible integers are: 5, 6, 7, …
(2) 3x
Solution:
The possible integers are: – 4, − 5, −6, …
(3) 13 x
Solution:
The possible integers are: −2, −1, 0, and 1.
−2 −5 −8 x
−1 0 2 1 −7 −4 −6 −3 3 4
EXAMPLES
4 1 −2 x
5 6 8 7 −1 2 0 3 9 10
−2 −5 −8 x
−1 0 2 1 −7 −4 −6 −3 3 4
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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Draw a number line to represent the following inequalities:
(a) x > 1
(b) 2x
(c) 2x
(d) 3x
TEST YOURSELF B
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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TEACHING AND LEARNING STRATEGIES
Some pupils might have difficulties when dealing with problems involving
addition and subtraction on linear inequalities.
Strategy:
Teacher should emphasise the following rule:
1) When a number is added or subtracted from both sides of the inequality,
the inequality sign remains the same.
LEARNING OBJECTIVES
Upon completion of Part C, pupils will be able perform computations
involving addition and subtraction on inequalities by stating a new
inequality for a given inequality when a number is:
(a) added to; and
(b) subtracted from
both sides of the inequalities.
PART C:
COMPUTATIONS INVOLVING
ADDITION AND SUBTRACTION ON
LINEAR INEQUALITIES
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PART C:
COMPUTATIONS INVOLVING ADDITION AND SUBTRACTION
ON LINEAR INEQUALITIES
Operation on Inequalities
1) When a number is added or subtracted from both sides of the inequality, the inequality
sign remains the same.
Examples:
(i) 2 < 4
Adding 1 to both sides of the inequality:
The inequality
sign is
unchanged.
LESSON NOTES
1 x
2 3 4
2 < 4
4 x
2 3 5
2 + 1 < 4 + 1
3 < 5
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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(ii) 4 > 2
Subtracting 3 from both sides of the inequality:
(1) Solve 145x .
Solution:
9
51455
145
x
x
x
(2) Solve 3 2.p
Solution:
3 2
3 3 2 3
5
p
p
p
Subtract 5 from both sides
of the inequality.
Simplify.
Add 3 to both sides of the
inequality.
Simplify.
The inequality
sign is
unchanged.
EXAMPLES
x −1 0 1 2
1 x
2 3 4
4 > 2
4 − 3 > 2 − 3
1 > − 1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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Solve the following inequalities:
(1) 24 m
(2) 3.4 2.6x
(3) 613 x
(4) 65.4 d
(5) 1723 m
(6) 78 54y
(7) 9 5d
(8) 2 1p
(9) 1
32
m
(10) 3 8x
TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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TEACHING AND LEARNING STRATEGIES
The computations involving division and multiplication on inequalities can be
confusing and difficult for pupils to grasp.
Strategy:
Teacher should emphasise the following rules:
1) When both sides of the inequality is multiplied or divided by a positive
number, the inequality sign remains the same.
2) When both sides of the inequality is multiplied or divided by a negative
number, the inequality sign is reversed.
3)
LEARNING OBJECTIVES
Upon completion of Part D, pupils will be able perform computations
involving division and multiplication on inequalities by stating a new
inequality for a given inequality when both sides of the inequalities are
divided or multiplied by a number.
PART D:
COMPUTATIONS INVOLVING
DIVISION AND MULTIPLICATION
ON LINEAR INEQUALITIES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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PART D1:
COMPUTATIONS INVOLVING
MULTIPLICATION AND DIVISION ON LINEAR INEQUALITIES
1. When both sides of the inequality is multiplied or divided by a positive number, the
inequality sign remains the same.
Examples:
(i) 2 < 4
Multiplying both sides of the inequality by 3:
LESSON NOTES
x
The inequality
sign is
unchanged.
1 x
2 3 4
2 < 4
2 3 < 4 3
6 < 12
x 6 8 10 12 14
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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(ii) − 4 < 2
Dividing both sides of the inequality by 2:
2. When both sides of the inequality is multiplied or divided by a negative number, the
inequality sign is reversed.
Examples:
(i) 4 < 6
Dividing both sides of the inequality by −1:
The inequality
sign is reversed.
x −6 −5 −4 −3
3 x
4 5 6
The inequality
sign is
unchanged.
−4 x
− 2 0 2
4 < 6
4 (−1) > 6 (−1)
− 4 > − 6
− 4 < 2
− 4 2 < 2 2
− 2 < 1
−2 − 1 0 1 2
x
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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(ii) 1 > −3
Multiply both sides of the inequality by −1:
Solve the inequality 3 12q .
Solution:
(i) 3 12q
3
12
3
3
q
4q
Divide each side of the
inequality by −3.
Simplify.
The inequality
sign is reversed.
EXAMPLES
The inequality
sign is reversed.
1 > −3
x −3 −2 −1 0 1
(− 1) (1) < (−1) (−3)
31
x −1 0 1 2 3
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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Solve the following inequalities:
(1) 7 49p
(2) 6 18x
(3) −5c > 15
(4) 200 < −40p
(5) 243 d
(6) 82 x
(7) x312
(8) y525
(9) 162 m
(10) 276 b
TEST YOURSELF D1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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PART D2:
PERFORM COMPUTATIONS INVOLVING
MULTIPLICATION OF LINEAR INEQUALITIES
Solve the inequality 32
x .
Solution:
32
x .
3)2()2
(2 x
6x
Multiply both sides of the
inequality by −2.
Simplify.
The inequality
sign is reversed.
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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1. Solve the following inequalities:
(1) − 38
d
(2) 82
n
(3) 5
10y
(4) 67
b
(5) 0 128
x
(6) 8 06
x
TEST YOURSELF D2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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TEACHING AND LEARNING STRATEGIES
Pupils might face problems when dealing with problems involving linear
inequalities.
Strategy:
Teacher should ensure that pupils are given further practice in order to enhance
their skills in solving problems involving linear inequalities.
LEARNING OBJECTIVES
Upon completion of Part E, pupils will be able perform computations
involving linear inequalities.
PART E:
FURTHER PRACTICE ON
COMPUTATIONS INVOLVING
LINEAR INEQUALITIES
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PART E:
FURTHER PRACTICE ON COMPUTATIONS
INVOLVING LINEAR INEQUALITIES
Solve the following inequalities:
1.
(a) 05 m
(b) 62 x
(c) 3 + m > 4
2.
(a) 3m < 12
(b) 2m > 42
(c) 4x > 18
TEST YOURSELF E1
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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3.
(a) m + 4 > 4m + 1
(b) mm 614
(c) mm 433
4.
(a) 64 x
(b) 12315 m
(c) 54
3 x
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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(d) 1835 x
(e) 1031 p
(f) 432
x
(g) 85
3 x
(h) 43
2
p
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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What is the smallest integer for x if 1835 x ?
Solution:
1835 x
3185 x
155 x O
3x
x = 4, 5, 6,…
Therefore, the smallest integer for x is 4.
3x
A number line can
be used to obtain the
answer.
2 1 0 x
3 4 5 6
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
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1. If ,1413 x what is the smallest integer for x?
2. What is the greatest integer for m if 147 mm ?
3. If 43
2
x, find the greatest integer value of x.
4. If 4
3
2
p, what is the greatest integer for p?
5. What is the smallest integer for m if 9
2
3
m?
TEST YOURSELF E2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
______________________________________________________________________________
27 Curriculum Development Division Ministry of Education Malaysia
1
2
3
4
5
6
7
8
9
10
11
12
ACTIVITY
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
______________________________________________________________________________
28 Curriculum Development Division Ministry of Education Malaysia
HORIZONTAL:
4. 31 is an ___________.
5. An inequality can be represented on a number __________.
7. 62 is read as 2 is __________ than 6.
9. Given 912 x , 5x is a _____________ of the inequality.
11. 123 x
4x
The inequality sign is reversed when divided by a ____________ integer.
VERTICAL:
1.
2
12
x
x
The inequality sign remains unchanged when multiplied by a ___________ integer.
2. 246 x equals to 4x when both sides are _____________ by 6.
3. 5x equals to 153 x when both sides are _____________ by 3.
6. ___________ inequalities are inequalities with the same solution(s).
8. 2x is represented by a ____________ dot on a number line.
10. 63 x is an example of ____________ inequality.
12. 35 is read as 5 is _____________ than 3.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
______________________________________________________________________________
29 Curriculum Development Division Ministry of Education Malaysia
TEST YOURSELF B:
(a)
(b)
(c)
(d)
TEST YOURSELF C:
(1) 6m (2) 6x (3) 19x (4) 5.1d (5) 6m
(6) 24y (7) 4d (8) 3p (9) 2
5m (10) 5x
TEST YOURSELF D1:
(1) 7p (2) 3x (3) 3c (4) 5p (5) 8d
(6) 4x (7) 4x (8) 5y (9) 8m (10) 2
9b
TEST YOURSELF D2:
(1) 24d (2) 16n (3) 50y (4) 42b (5) 96x 48 (6) x
0 − 2 − 3
x
1 2 3 − 1
0 − 2 − 3
x
1 2 3 − 1
0 − 2 − 3
x
1 2 3 − 1
0 − 2 − 3
x
1 2 3 − 1
ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 7: Linear Inequalities
______________________________________________________________________________
30 Curriculum Development Division Ministry of Education Malaysia
TEST YOURSELF E1:
1. 5 )( ma 8 )( xb 1 )( mc
2. 4 )( ma 21 )( mb 2
9 )( xc
3. 1
( ) 1 ( ) 4 (c) 2
a m b m m
4. ( ) 10 (b) 1 (c) 8 (d) 3 (e) 3 (f) 2 (g) 25 (h) 10a x m x x p x x p
TEST YOURSELF E2:
(1) 6x (2) 1m (3) 13x (4) 9p (5) 14m
ACTIVITY:
1. positive
2. divided
3. multiplied
4. inequality
5. line
6. Equivalent
7. less
8. solid
9. solution
10. linear
11. negative
12. greater
Unit 1:
Negative Numbers
UNIT 8
TRIGONOMETRY
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Trigonometry I 2
Part B: Trigonometry II 6
Part C: Trigonometry III 11
Part D: Trigonometry IV 15
Part E: Trigonometry V 19
Part F: Trigonometry VI 21
Part G: Trigonometry VII 25
Part H: Trigonometry VIII 29
Answers 33
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
1 Curriculum Development Division
Ministry of Education Malaysia
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding of the concept
of trigonometry and to provide pupils with a solid foundation for the study
of trigonometric functions.
2. This module is to be used as a guide for teacher on how to help pupils to
master the basic skills required for this topic. Part of the module can be
used as a supplement or handout in the teaching and learning involving
trigonometric functions.
3. This module consists of eight parts and each part deals with one specific
skills. This format provides the teacher with the freedom of choosing any
parts that is relevant to the skills to be reinforced.
4. Note that Part A to D covers the Form Three syllabus whereas Part E to H
covers the Form Four syllabus.
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
2 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
Some pupils may face difficulties in remembering the definition and
how to identify the correct sides of a right-angled triangle in order to
find the ratio of a trigonometric function.
Strategy:
Teacher should make sure that pupils can identify the side opposite to
the angle, the side adjacent to the angle and the hypotenuse side
through diagrams and drilling.
PART A:
TRIGONOMETRY I
LEARNING OBJECTIVE
Upon completion of Part A, pupils will be able to identify opposite,
adjacent and hypotenuse sides of a right-angled triangle with reference
to a given angle.
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
3 Curriculum Development Division
Ministry of Education Malaysia
Opposite side is the side opposite or facing the angle .
Adjacent side is the side next to the angle .
Hypotenuse side is the side facing the right angle and is the longest side.
LESSON NOTES
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
4 Curriculum Development Division
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Example 1:
AB is the side facing the angle , thus AB is the opposite side.
BC is the side next to the angle , thus BC is the adjacent side.
AC is the side facing the right angle and it is the longest side, thus AC is the
hypotenuse side.
Example 2:
QR is the side facing the angle , thus QR is the opposite side.
PQ is the side next to the angle , thus PQ is the adjacent side.
PR is the side facing the right angle or is the longest side, thus PR is the
hypotenuse side.
EXAMPLES
θ
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
5 Curriculum Development Division
Ministry of Education Malaysia
Identify the opposite, adjacent and hypotenuse sides of the following right-angled triangles.
1.
Opposite side =
Adjacent side =
Hypotenuse side =
2.
Opposite side =
Adjacent side =
Hypotenuse side =
3.
Opposite side =
Adjacent side =
Hypotenuse side =
4.
Opposite side =
Adjacent side =
Hypotenuse side =
5.
Opposite side =
Adjacent side =
Hypotenuse side =
6.
Opposite side =
Adjacent side =
Hypotenuse side =
TEST YOURSELF A
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
6 Curriculum Development Division
Ministry of Education Malaysia
PART B:
TRIGONOMETRY II
TEACHING AND LEARNING STRATEGIES
Some pupils may face problem in
(i) defining trigonometric functions; and
(ii) writing the trigonometric ratios from a given right-angled
triangle.
Strategy:
Teacher must reinforce the definition of the trigonometric functions
through diagrams and examples. Acronyms SOH, CAH and TOA can
be used in defining the trigonometric ratios.
LEARNING OBJECTIVE
Upon completion of Part B, pupils will be able to state the definition
of the trigonometric functions and use it to write the trigonometric
ratio from a right-angled triangle.
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
7 Curriculum Development Division
Ministry of Education Malaysia
Definition of the Three Trigonometric Functions
(i) sin = opposite side
hypotenuse side
(ii) cos = adjacent side
hypotenuse side
(iii) tan = opposite side
adjacent side
sin = opposite side
hypotenuse side
= AB
AC
cos = adjacent side
hypotenuse side =
BC
AC
tan = opposite side
adjacent side=
AB
BC
LESSON NOTES
Acronym:
SOH:
Sine – Opposite - Hypotenuse
Acronym:
CAH:
Cosine – Adjacent - Hypotenuse
Acronym:
TOA:
Tangent – Opposite - Adjacent
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
8 Curriculum Development Division
Ministry of Education Malaysia
Example 1:
AB is the side facing the angle , thus AB is the opposite side.
BC is the side next to the angle , thus BC is the adjacent side.
AC is the side facing the right angle and is the longest side, thus AC is the hypotenuse
side.
Thus sin = opposite side
hypotenuse side =
AB
AC
cos = adjacent side
hypotenuse side =
BC
AC
tan = opposite side
adjacent side =
AB
BC
EXAMPLES
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
9 Curriculum Development Division
Ministry of Education Malaysia
Example 2:
WU is the side facing the angle, thus WU is the opposite side.
TU is the side next to the angle, thus TU is the adjacent side.
TW is the side facing the right angle and is the longest side, thus TW is the hypotenuse
side.
Thus, sin = opposite side
hypotenuse side =
WU
TW
cos = adjacent side
hypotenuse side =
TU
TW
tan = opposite side
adjacent side =
WU
TU
You have to identify the
opposite, adjacent and
hypotenuse sides.
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
10 Curriculum Development Division
Ministry of Education Malaysia
Write the ratios of the trigonometric functions, sin , cos and tan , for each of the diagrams
below:
1.
sin =
cos =
tan =
2.
sin =
cos =
tan =
3.
sin =
cos =
tan =
4.
sin =
cos =
tan =
5.
sin =
cos =
tan =
6.
sin =
cos =
tan =
TEST YOURSELF B
θ
θ
θ
θ
θ
θ
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
11 Curriculum Development Division
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PART C:
TRIGONOMETRY III
TEACHING AND LEARNING STRATEGIES
Some pupils may face problem in finding the angle when given
two sides of a right-angled triangle and they also lack skills in
using calculator to find the angle.
Strategy:
1. Teacher should train pupils to use the definition of each
trigonometric ratio to write out the correct ratio of the sides
of the right-angle triangle.
2. Teacher should train pupils to use the inverse trigonometric
functions to find the angles and express the angles in degree
and minute.
LEARNING OBJECTIVE
Upon completion of Part C, pupils will be able to find the angle of
a right-angled triangle given the length of any two sides.
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
12 Curriculum Development Division
Ministry of Education Malaysia
Find the angle in degrees and minutes.
Example 1:
sin = 2
5
o
h
= sin-1
2
5
= 23o 34 4l
= 23o 35
(Note that 34 41 is rounded off to 35)
Example 2:
cos = a
h =
3
5
= cos-1
3
5
= 53o 7 48
= 53o 8
(Note that 7 48 is rounded off to 8)
Since sin = opposite
hypotenuse
then = sin-1
opposite
hypotenuse
Since cos = adjacent
hypotenuse
then = cos-1 adjacent
hypotenuse
Since tan = opposite
adjacent
then = tan-1
opposite
adjacent
1 degree = 60 minutes 1 minute = 60 seconds
1o = 60 1 = 60
Use the key D M S or on your calculator to express the angle in degree and minute.
Note that the calculator expresses the angle in degree, minute and second. The angle in
second has to be rounded off. ( 30, add 1 minute and < 30, cancel off.)
LESSON NOTES
EXAMPLES
θ
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
13 Curriculum Development Division
Ministry of Education Malaysia
Example 3:
tan = o
a =
7
6
= tan-1
7
6
= 49o 23 55
= 49o 24
Example 4:
cos = a
h =
5
7
= cos-1
5
7
= 44o 24 55
= 44o 25
Example 5:
sin = o
h =
4
7
= sin-1
4
7
= 34o 50 59
= 34o 51
Example 6:
tan = o
a =
5
6
= tan-1
5
6
= 39o 48 20
= 39o 48
θ
θ
θ
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
14 Curriculum Development Division
Ministry of Education Malaysia
Find the value of in degrees and minutes.
1.
2.
3.
4.
5.
6.
TEST YOURSELF C
θ θ
θ
θ
θ
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
15 Curriculum Development Division
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PART D:
TRIGONOMETRY IV
TEACHING AND LEARNING STRATEGIES
Pupils may face problem in finding the length of the side of a
right-angled triangle given one angle and any other side.
Strategy:
By referring to the sides given, choose the correct trigonometric
ratio to write the relation between the sides.
1. Find the length of the unknown side with the aid of a
calculator.
LEARNING OBJECTIVE
Upon completion of Part D, pupils will be able to find the
angle of a right-angled triangle given the length of any two
sides.
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
16 Curriculum Development Division
Ministry of Education Malaysia
Find the length of PR.
With reference to the given angle, PR is the
opposite side and QR is the adjacent side.
Thus tangent ratio is used to form the
relation of the sides.
tan 50o =
5
PR
PR = 5 tan 50o
Find the length of TS.
With reference to the given angle, TR is the
adjacent side and TS is the hypotenuse
side.
Thus cosine ratio is used to form the
relation of the sides.
cos 32o =
8
TS
TS cos 32o = 8
TS = 8
cos32o
LESSON NOTES
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
17 Curriculum Development Division
Ministry of Education Malaysia
Find the value of x in each of the following.
Example 1:
tan 25o =
3
x
x = 3
tan 25o
= 6.434 cm
Example 2:
sin 41.27o =
5
x
x = 5 sin 41.27o
= 3.298 cm
Example 3:
cos 34o 12 =
6
x
x = 6 cos 34o 12
= 4.962 cm
Example 4:
tan 63o =
9
x
x = 9 tan 63o
= 17.66 cm
EXAMPLES
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
18 Curriculum Development Division
Ministry of Education Malaysia
Find the value of x for each of the following.
1.
2.
3.
4.
5.
6.
TEST YOURSELF D
10 cm
6 cm
13 cm
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
19 Curriculum Development Division
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PART E:
TRIGONOMETRY V
TEACHING AND LEARNING STRATEGIES
Pupils may face problem in relating the coordinates of a given
point to the definition of the trigonometric functions.
Strategy:
Teacher should use the Cartesian plane to relate the coordinates
of a point to the opposite side, adjacent side and the hypotenuse
side of a right-angled triangle.
LEARNING OBJECTIVE
Upon completion of Part E, pupils will be able to state the
definition of trigonometric functions in terms of the
coordinates of a given point on the Cartesian plane and use
the coordinates of the given point to determine the ratio of the
trigonometric functions.
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
20 Curriculum Development Division
Ministry of Education Malaysia
In the diagram, with reference to the angle , PR is the opposite side, OP is the adjacent side
and OR is the hypotenuse side.
r
y
OR
PR
hypotenuse
oppositesin
r
x
OR
OP
hypotenuse
adjacentcos
x
y
OP
PR
adjacent
oppositetan
LESSON NOTES
θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
21 Curriculum Development Division
Ministry of Education Malaysia
PART F:
TRIGONOMETRY VI
TEACHING AND LEARNING STRATEGIES
Pupils may face difficulties in determining that the sign of the x-coordinate
and y-coordinate affect the sign of the trigonometric functions.
Strategy:
Teacher should use the Cartesian plane and use the points on the four
quadrants and the values of the x-coordinate and y-coordinate to show how the
sign of the trigonometric ratio is affected by the signs of the x-coordinate and
y-coordinate.
Based on the A – S – T – C, the teacher should guide the pupils to determine
on which quadrant the angle is when given the sign of the trigonometric ratio
is given.
(a) For sin to be positive, the angle must be in the first or second
quadrant.
(b) For cos to be positive, the angle must be in the first or fourth
quadrant.
(c) For tan to be positive, the angle must be in the first or third quadrant.
LEARNING OBJECTIVE
Upon completion of Part F, pupils will be able to relate the sign of the
trigonometric functions to the sign of x-coordinate and y-coordinate and to
determine the sign of each trigonometric ratio in each of the four quadrants.
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
22 Curriculum Development Division
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First Quadrant
sin = y
r (Positive)
cos = x
r(Positive)
tan = y
x(Positive)
(All trigonometric ratios are positive in the
first quadrant)
Second Quadrant
sin = y
r (Positive)
cos = x
r
(Negative)
tan = y
x(Negative)
(Only sine is positive in the second
quadrant)
Third Quadrant
sin = y
r
(Negative)
cos = x
r
(Negative)
tan = y y
x x
(Positive)
(Only tangent is positive in the third
quadrant)
Fourth Quadrant
sin = y
r
(Negative)
cos = x
r (Positive)
tan = y
x
(Negative)
(Only cosine is positive in the fourth
quadrant)
LESSON NOTES
θ θ
θ θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
23 Curriculum Development Division
Ministry of Education Malaysia
Using acronym: Add Sugar To Coffee (ASTC)
sin is positive
sin is negative
cos is positive
cos is negative
tan is positive
tan is negative
A – All positive
C – only cos is positive T – only tan is positive
S – only sin is positive
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
24 Curriculum Development Division
Ministry of Education Malaysia
State the quadrants the angle is situated and show the position using a sketch.
1. sin = 0.5
2. tan = 1.2
3. cos = −0.16
4. cos = 0.32
5. sin = −0.26 6. tan = −0.362
TEST YOURSELF F
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
25 Curriculum Development Division
Ministry of Education Malaysia
PART G:
TRIGONOMETRY VII
TEACHING AND LEARNING STRATEGIES
Pupils may face problem in calculating the length of the sides of a
right-angled triangle drawn on a Cartesian plane and determining the
value of the trigonometric ratios when a point on the Cartesian plane is
given.
Strategy:
Teacher should revise the Pythagoras Theorem and help pupils to
recall the right-angled triangles commonly used, known as the
Pythagorean Triples.
LEARNING OBJECTIVE
Upon completion of Part G, pupils will be able to calculate the length
of the side of right-angled triangle on a Cartesian plane and write the
value of the trigonometric ratios given a point on the Cartesian plane
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
26 Curriculum Development Division
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The Pythagoras Theorem:
(a) 3, 4, 5 or equivalent (b) 5, 12, 13 or equivalent (c) 8, 15, 17 or equivalent
The sum of the squares of two sides of
a right-angled triangle is equal to the
square of the hypotenuse side.
PR2 + QR
2 = PQ
2
LESSON NOTES
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
27 Curriculum Development Division
Ministry of Education Malaysia
1. Write the values of sin , cos and tan
from the diagram below.
OA2 = (−6)
2 + 8
2
= 100
OA = 100
= 10
sin = 8 4
10 5
y
r
cos = 6 3
10 5
x
r
tan = 8 4
6 3
y
x
2. Write the values of sin , cos and tan
from the diagram below.
OB2 = (−12)
2 + (−5)
2
= 144 + 25
= 169
OB = 169
= 13
sin = 5
13
y
r
cos = 12
13
x
r
tan = 5 5
12 12
EXAMPLES
θ θ
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
28 Curriculum Development Division
Ministry of Education Malaysia
Write the value of the trigonometric ratios from the diagrams below.
1.
sin =
cos =
tan =
2.
sin =
cos =
tan =
3.
sin =
cos =
tan =
4.
sin =
cos =
tan =
5.
sin =
cos =
tan =
6.
sin =
cos =
tan =
TEST YOURSELF G
θ θ θ
θ
θ
θ θ
B(5,4)
B(5,12)
x
y
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
29 Curriculum Development Division
Ministry of Education Malaysia
PART H:
TRIGONOMETRY VIII
TEACHING AND LEARNING STRATEGIES
Pupils may find difficulties in remembering the shape of the
trigonometric function graphs and the important features of the
graphs.
Strategy:
Teacher should help pupils to recall the trigonometric graphs which
pupils learned in Form 4. Geometer’s Sketchpad can be used to
explore the graphs of the trigonometric functions.
LEARNING OBJECTIVE
Upon completion of Part H, pupils will be able to sketch the
trigonometric function graphs and know the important features of the
graphs.
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
30 Curriculum Development Division
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(a) y = sin x
The domain for x can be from 0o to 360
o or 0 to 2 in radians.
Important points: (0, 0), (90o, 1), (180
o, 0), (270
o, −1) and (360
o, 0)
Important features: Maximum point (90o, 1), Maximum value = 1
Minimum point (270o, −1), Minimum value = −1
(b) y = cos x
Important points:(0o, 1), (90
o, 0), (180
o, −1), (270
o, 0) and (360
o, 1)
Important features: Maximum point (0o, 1) and (360
o, 1),
Maximum value = 1 Minimum point (180o, −1)
Minimum value = 1
LESSON NOTES
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
31 Curriculum Development Division
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(c) y = tan x
Important points: (0o, 0), (180
o, 0) and (360
o, 0)
Is there any
maximum or
minimum point
for the tangent
graph?
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
32 Curriculum Development Division
Ministry of Education Malaysia
1. Write the following trigonometric functions to the graphs below:
y = cos x y = sin x y = tan x
2. Write the coordinates of the points below:
(a)
(b)
A(0,1)
TEST YOURSELF H
y = cos x y = sin x
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
33 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF A:
1. Opposite side = AB
Adjacent side = AC
Hypotenuse side = BC
2. Opposite side = PQ
Adjacent side = QR
Hypotenuse side = PR
3. Opposite side = YZ
Adjacent side = XZ
Hypotenuse side = XY
4. Opposite side = LN
Adjacent side = MN
Hypotenuse side = LM
5. Opposite side = UV
Adjacent side = TU
Hypotenuse side = TV
6. Opposite side = RT
Adjacent side = ST
Hypotenuse side = RS
TEST YOURSELF B:
1. sin = AB
BC
cos = AC
BC
tan = AB
AC
2. sin = PQ
PR
cos = QR
PR
tan = PQ
QR
3. sin = YZ
YX
cos = XZ
XY
tan = YZ
XZ
4. sin = LN
LM
cos = MN
LM
tan = LN
MN
5. sin = UV
TV
cos = UT
TV
tan = UV
UT
6. sin = RT
RS
cos = ST
RS
tan = RT
TS
ANSWERS
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
34 Curriculum Development Division
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TEST YOURSELF C:
1. sin = 1
3
= sin-1
1
3 = 19
o 28
2. cos = 1
2
= cos-1
1
2 = 60
o
3. tan = 5
3
= tan-1
5
3 = 59
o 2
4. cos = 5
8
= cos-1
5
8 = 51
o 19
5. tan = 7.5
9.2
= tan-1
7.5
9.2 = 39
o 11
6. sin = 6.5
8.4
= sin-1
6.5
8.4= 50
o 42
TEST YOURSELF D:
1. tan 32o =
4
x
x = 4
tan 32o = 6.401 cm
2. sin 53.17o =
7
x
x = 7 sin 53.17o = 5.603 cm
3. cos 74o 25 =
10
x
x = 10 cos 74o 25
= 2.686 cm
4. sin 551
3
o
= 6
x
x = 13
6
sin55o
= 7.295 cm
5. tan 47o =
13
x
x = 13 tan 47o = 13.94 cm
6. cos 61o =
10
x
x = 10
cos61o= 20.63 cm
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
35 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF F:
1. 1ST
and 2nd
2. 1st and 3
rd
3. 2nd
and 3rd
4. 1st and 4
th
5. 3rd
and 4th
6. 2nd
and 4th
TEST YOURSELF G:
1. sin = 4
5
cos = 3
5
tan = 4
3
2. sin = 12
13
cos = 5
13
tan = 12
5
3. sin = 4
5
cos = 3
5
tan = 4
3
4. sin = 4
5
cos = 3
5
tan = 4
3
5. sin = 8
17
cos = 15
17
tan = 8
15
6. sin = 5
13
cos = 12
13
tan = 5
12
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry
36 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF H:
1.
y = tan x y = sin x y = cos x
2. (a) A (0, 1), B (90o, 0), C (180
o, 1), D (270
o, 0)
(b) P (90o, 1), Q (180
o, 0), R (270
o, 1), S (360
o, 0)