BEAMS

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


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


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


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

           Unit 1:    Negative Numbers
           Unit 2:    Fractions
           Unit 3:    Algebraic Expressions and Algebraic Formulae
           Unit 4:    Linear Equations
           Unit 5:    Indices
           Unit 6:    Coordinates and Graphs of Functions
           Unit 7:    Linear Inequalities
           Unit 8:    Trigonometry




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

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

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

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




                                                                                                  iv
PANEL OF CONTRIBUTORS

                 Advisors:

      Haji Ali bin Ab. Ghani AMN
                 Director
     Curriculum Development Division

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

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


            Editorial Advisor:

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

                 Editors:

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

          Aszunarni binti Ayob
             Assistant Director
     Curriculum Development Division

          Rosita binti Mat Zain
             Assistant Director
     Curriculum Development Division
                                         Writers:



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

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

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

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

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

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

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

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

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

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

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

                                                          Noorliah binti Ahmat
                                                          SM Teknik, Kuala Lumpur



      Ali Akbar bin Asri                                  Nor A’idah binti Johari

      SM Sains, Labuan                                    SMK Teknik Setapak, Selangor



      Amrah bin Bahari                                    Nor Dalina binti Idris

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


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

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

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

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

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

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

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

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

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



                                 Layout and Illustration:


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

Additional Mathematics Skills




              UNIT 1
     NEGATIVE NUMBERS


 Unit 1:
 Negative Numbers




           Curriculum Development Division
           Ministry of Education Malaysia
                             TABLE OF CONTENTS


Module Overview                                                                 1

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

           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




                                     MODULE OVERVIEW

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

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

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

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

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

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

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                               1
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          PART A:
                     ADDITION AND SUBTRACTION
                              OF INTEGERS USING
                                   NUMBER LINES



                                   LEARNING OBJECTIVE

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




                        TEACHING AND LEARNING STRATEGIES

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

        Strategy:

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                            2
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



                                                  PART A:
                           ADDITION AND SUBTRACTION OF INTEGERS
                                           USING NUMBER LINES

                                             LESSON NOTES



1.0   Representing Integers on a Number Line

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

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

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



2.0   Addition and Subtraction of Positive Integers

                             Rules for Adding and Subtracting Positive Integers

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




                                –3    –2    –1     0      1       2   3    4

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



                                –3    –2    –1     0      1       2   3    4



           Curriculum Development Division
           Ministry of Education Malaysia
                                                                                                         3
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                               EXAMPLES




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




                       –5   –4     –3     –2     –1     0      1     2      3         4   5   6


                                               Adding a positive integer:

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

                                                       2+3=5




           Alternative Method:


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




                       –5    –4    –3     –2     –1     0     1      2      3     4       5   6

                                                      Adding a positive integer:

                                                Start at 2 and move 3 units to the right:

                                                                  2+3=5



        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                  4
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



   (ii)     –2 + 5

                                                    Add a
                                                  positive 5




                     –5   –4    –3     –2    –1     0      1      2   3     4       5   6


                                              Adding a positive integer:

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

                                                         –2 + 5 = 3




            Alternative Method:

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




                     –5    –4    –3    –2    –1      0       1    2    3        4   5   6


                            Adding a positive integer:

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

                                     –2 + 5 = 3




          Curriculum Development Division
          Ministry of Education Malaysia
                                                                                            5
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



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




               –5   –4    –3    –2     –1      0       1        2    3     4      5     6


                                     Subtracting a positive integer:

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

                                                   2 – 5 = –3



          Alternative Method:



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




               –5   –4    –3    –2     –1      0       1        2    3     4     5      6


                                            Subtracting a positive integer:

                                       Start at 2 and move 5 units to the left:

                                                        2 – 5 = –3




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                            6
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



     (iv) –3 – 2 = –5

                             Subtract a
                             positive 2




                        –5     –4     –3      –2     –1     0     1          2       3   4   5   6



                                                Subtracting a positive integer:

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

                                                          –3 – 2 = –5


           Alternative Method:



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




                       –5     –4     –3       –2    –1     0     1       2       3       4   5   6


                            Subtracting a positive integer:

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

                                          –3 – 2 = –5




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                     7
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



3.0   Addition and Subtraction of Negative Integers


      Consider the following operations:


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


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

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

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

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


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




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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                               8
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                    Rules for Adding and Subtracting Negative Integers

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




                       –3    –2    –1     0     1     2        3   4

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




                        –3   –2    –1     0     1     2        3   4




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                              9
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                                 EXAMPLES




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




                –5     –4     –3    –2     –1    0     1      2     3     4       5   6


                            Adding a negative integer:

                     Start by drawing an arrow from 0 to –2, and
                       then, draw an arrow of 1 unit to the left:

                                    –2 + (–1) = –3



             Alternative Method:                     Make sure you start from
                                                      the position of the first
                                                              integer.




                –5     –4     –3    –2     –1    0     1      2     3     4       5   6


                        Adding a negative integer:

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

                                –2 + (–1) = –3



           Curriculum Development Division
           Ministry of Education Malaysia
                                                                                          10
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



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




                     –5    –4    –3    –2     –1     0       1     2       3    4     5        6


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




              Alternative Method:
                                                                 Make sure you start from
                                                                  the position of the first
                                                                          integer.




                     –5    –4    –3    –2     –1     0       1     2       3    4      5       6


                                              Adding a negative integer:

                                         Start at 1 and move 3 units to the left:

                                                           1 + (–3) = –2




            Curriculum Development Division
            Ministry of Education Malaysia
                                                                                                   11
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



     (iii)     3 – (–3) = 6

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




               –5    –4    –3     –2     –1      0      1     2     3       4       5       6



                              Subtracting a negative integer:

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

                                        3 – (–3) = 6




       Alternative Method:

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




              –5    –4    –3      –2    –1      0      1     2     3    4       5       6


                                                     Subtracting a negative integer:

                                                Start at 3 and move 3 units to the right:

                                                              3 – (–3) = 6




         Curriculum Development Division
         Ministry of Education Malaysia
                                                                                                12
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



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

                                     Subtract a               3+3=6
                                     negative 8




              –5    –4   –3     –2      –1        0   1       2      3        4   5   6



                               Subtracting a negative integer:

                         Start by drawing an arrow from 0 to –5, and
                         then, draw an arrow of 8 units to the right:

                                          –5 – (–8) = 3




         Alternative Method:




              –5   –4     –3    –2      –1        0   1       2      3        4   5   6



                         Subtracting a negative integer:

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

                                     –5 – (–8) = 3




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                          13
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          TEST YOURSELF A



Solve the following.

       1.      –2 + 4




                    –5     –4   –3    –2    –1     0    1      2   3   4   5   6



       2.      3 + (–6)



                    –5     –4   –3   –2     –1    0     1      2   3   4   5   6


       3.       2 – (–4)




                    –5     –4   –3    –2   –1     0     1      2   3   4   5   6


       4.      3 – 5 + (–2)




                    –5     –4   –3    –2    –1    0     1      2   3   4   5   6


       5.      –5 + 8 + (–5)



                    –5     –4   –3    –2    –1    0     1      2   3   4   5   6




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                   14
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          PART B:
                    ADDITION AND SUBTRACTION
                        OF INTEGERS USING
                         THE SIGN MODEL



                                   LEARNING OBJECTIVE

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




                        TEACHING AND LEARNING STRATEGIES

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

      Strategy:

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                        15
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



                                              PART B:

                       ADDITION AND SUBTRACTION OF INTEGERS
                               USING THE SIGN MODEL

                                        LESSON NOTES


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


                                            The Sign Model

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


                                             EXAMPLES



   Example 1

   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




          Curriculum Development Division
          Ministry of Education Malaysia
                                                                                                 16
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



   Example 2

   What is the value of  3  5 ?


                        NUMBER                                     SIGN

                            –3                                 _ _ _

                            –5                                 – – – – –

                       WORKINGS

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

                          Answer                                    –8


   Example 3

   What is the value of  3  5 ?


                        NUMBER                                     SIGN

                            –3                                 – – –
                            +5                                 + + + + +

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




            Curriculum Development Division
            Ministry of Education Malaysia
                                                                                      17
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                           TEST YOURSELF B



  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




         Curriculum Development Division
         Ministry of Education Malaysia
                                                                                 18
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          PART C:
                         FURTHER PRACTICE ON
                       ADDITION AND SUBTRACTION
                              OF INTEGERS



                                   LEARNING OBJECTIVE

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




                        TEACHING AND LEARNING STRATEGIES

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

      Strategy:

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                               19
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



                                             PART C:
     FURTHER PRACTICE ON ADDITION AND SUBTRACTION OF INTEGERS

                                          LESSON NOTES



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




                                Steps for Adding and Subtracting
                                            Integers


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

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

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

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                          20
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



     Examples:

     i)         34 + 37 =

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

                                 +71



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

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




     iii)       –73 + 22 =

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


     iv)        228 – 338 =

                         +                    –
                                                               Subtract the smaller number from
                        228                   338              the larger number and put the sign
                                                                   of the larger number in the
                                 –110                                        answer.




            Curriculum Development Division
            Ministry of Education Malaysia
                                                                                                    21
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



       v)      –428 – 316 =

                        +                    –

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




     vi)       –863 – 127 + 225 =

                        +                     –

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

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




     vii)        234 – 675 – 567 =

                            +                    –

                         234                  675                Add the two numbers in the „–‟
                                                               column and bring down the number
                                              567
                                                                       in the „+‟ column.

                         234                 1242               Subtract the smaller number from
                                                                the larger number in the third row
                                 –1008                             and put the sign of the larger
                                                                      number in the answer.




           Curriculum Development Division
           Ministry of Education Malaysia
                                                                                                     22
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



     viii)       –482 + 236 – 718 =

                          +                    –

                         236                  482                Add the two numbers in the „–‟
                                                               column and bring down the number
                                              718                      in the „+‟ column.

                         236                 1200              Subtract the smaller number from
                                                               the larger number in the third row
                                  –964                            and put the sign of the larger
                                                                     number in the answer.

     ix)         –765 – 984 + 432 =


                          +                    –

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

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

                          +                    –

                         436                  1782               Add the two numbers in the „+‟
                                                               column and bring down the number
                         652                                           in the „–‟ column.

                                             1782
                        1088                                   Subtract the smaller number from
                                                               the larger number in the third row
                                  –694                            and put the sign of the larger
                                                                     number in the answer.




           Curriculum Development Division
           Ministry of Education Malaysia
                                                                                                    23
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



                                         TEST YOURSELF C



  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




         Curriculum Development Division
         Ministry of Education Malaysia
                                                                                       24
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          PART D:
                     ADDITION AND SUBTRACTION
                     OF INTEGERS INCLUDING THE
                          USE OF BRACKETS




                                  LEARNING OBJECTIVE

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




                       TEACHING AND LEARNING STRATEGIES

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

    Strategy:

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                           25
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers


                                       PART D:
                        ADDITION AND SUBTRACTION OF INTEGERS
                           INCLUDING THE USE OF BRACKETS

                                           LESSON NOTES




                                        The Accept - Reject Model

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




                                  To Accept or To Reject?              Answer

               +(5)                        Accept +5                     +5

               –(2)                        Reject +2                     –2

               + (–4)                      Accept –4                     –4

               – (–8)                      Reject –8                     +8




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                26
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                              EXAMPLES



   i) 5 + (–1) =


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

                                                                           + + + + +
                                                                           –


            5 + (–1) =                                                            4



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




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



       5 + (–1) = 5 – 1


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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                    27
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



   ii) –6 + (–3) =



               Number                To Accept or To Reject?                    Answer

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

                                                                         – – – – – –
                                                                          – – –
            –6 + (–3) =                                                           –9



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




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


        –6 + (–3) = –6 – 3 =


                 +                        –

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

                            –9




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                  28
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



   iii)    –7 – (–4) =


                Number              To Accept or To Reject?                      Answer

                   –7                            Reject 7                            –7
                 – (–4)                          Reject –4                           +4

                                                                      – – – – – – –
                                                                      + + + +

              –7 – (–4) =                                                            –3



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



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


          –7 – (–4) = –7 + 4 =



                   +                      –
                                                               Subtract the smaller number from
                    4                        7                 the larger number and put the sign
                                                                  of the larger number in the
                              –3                                           answer.




           Curriculum Development Division
           Ministry of Education Malaysia
                                                                                                    29
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



   iv) –5 – (3) =


              Number                 To Accept or To Reject?                   Answer

                –5                              Reject 5                         –5
               – (3)                            Reject 3                         –3

                                                                        – – – – –
                                                                        – – –
            – 5 – (3) =                                                          –8



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




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


        –5 – (3) = –5 – 3 =



                 +                         –

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

                              –8




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                  30
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



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




       Using the Table Model:



                 +                    –

                                      35
                                      57                       Add the numbers and then put the
                                                                 negative sign in the answer.
                           –92




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



       Using the Table Model:



                 +                    –

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                     31
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          TEST YOURSELF D


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)




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                             32
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          PART E:
                               MULTIPLICATION OF
                                   INTEGERS



                                   LEARNING OBJECTIVE

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




                        TEACHING AND LEARNING STRATEGIES

      This part emphasises the multiplication rules of integers.

      Strategy:

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                         33
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

                                               PART E:
                               MULTIPLICATION OF INTEGERS

                                             LESSON NOTES


   Consider the following pattern:
          3×3=9

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

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

        3  (2)  6

        3  (3)  9



        (3)  3  9

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

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

       (3)  (2)  6

       (3)  (3)  9


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



        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                    34
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                            EXAMPLES




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

       (a)     4 × 3 = 12

       (b)     –8 × –6 = 48




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

       (a)     –4 × (3) = –12

       (b)     8 × (–6) = –48




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

       (a)     (4) × 0 = 0

       (b)     (–8) × 0 = 0

       (c)     0 × (5) = 0

       (d)     0 × (–7) = 0




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                              35
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          TEST YOURSELF E


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)




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                        36
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          PART F:
                  MULTIPLICATION OF INTEGERS
                             USING
                   THE ACCEPT-REJECT MODEL




                                  LEARNING OBJECTIVE


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




                       TEACHING AND LEARNING STRATEGIES

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

     Strategy:

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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                         37
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers


                                      PART F:
                            MULTIPLICATION OF INTEGERS
                          USING THE ACCEPT-REJECT MODEL

                                          LESSON NOTES



The Accept-Reject Model

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



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




Multiplication Rules:


                   Sign              To Accept or To Reject            Answer

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


                                            EXAMPLES



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

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

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



        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                               38
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                            TEST YOURSELF F


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) =




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                             39
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          PART G:
                          DIVISION OF INTEGERS



                                   LEARNING OBJECTIVE


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




                        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.




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                       40
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



                                                   PART G:
                                          DIVISION OF INTEGERS

                                            LESSON NOTES



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

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

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

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



Rules of Division

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

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


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


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

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

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

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




          Curriculum Development Division
          Ministry of Education Malaysia
                                                                                                        41
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                           EXAMPLES




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

       (a)     (12) ÷ (3) = 4

       (b)     (–8) ÷ (–2) = 4



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

       (a)     (–12) ÷ (3) = –4

       (b)     (+8) ÷ (–2) = –4



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

       (a)     0 ÷ (5) = 0

       (b)     0 ÷ (–7) = 0




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                42
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                           TEST YOURSELF G


  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)




         Curriculum Development Division
         Ministry of Education Malaysia
                                                                                         43
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




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




                                  LEARNING OBJECTIVE


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




                      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.




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                           44
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers


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


                                          LESSON NOTES


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




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




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




Division Rules:

                   Sign             To Accept or To Reject          Answer

               (+) ÷ (+)                   Accept +                    +


               (–) ÷ (–)                   Reject –                    +


               (+) ÷ (–)                   Accept –                    –


               (–) ÷ (+)                   Reject +                    –




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                 45
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



                                            EXAMPLES




                                   To Accept or To Reject      Answer

                (6) ÷ (3)                 Accept +               2

              (–6) ÷ (–3)                  Reject –              2

              (+6) ÷ (–3)                 Accept –              –2

               (–6) ÷ (3)                  Reject +             –2




Division [Fraction Form]:

                  Sign             To Accept or To Reject      Answer


                   ()                    Accept +               +
                   ()


                   ()
                                           Reject –              +
                   ()

                   ()
                                          Accept –               –
                   ()

                   ()
                                           Reject +              –
                   ()




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                        46
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          EXAMPLES




                                   To Accept or To Reject      Answer

                  (  8)
                                          Accept +               4
                  (  2)

                  (  8)
                                           Reject –              4
                  (  2)

                  (  8)
                                          Accept –              –4
                  ( 2)

                  (  8)
                                           Reject +             –4
                  ( 2)




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                        47
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          TEST YOURSELF H


Solve the following.

 1.    18 ÷ (–6)                           12                        24
                                  2.                           3.
                                           2                        8




         25                              6                   6.    – (–35) ÷ 7
 4.                               5.
        5                                 3




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




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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                    48
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                          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.




                       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.




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                           49
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers


                                      PART I:
                       COMBINED OPERATIONS INVOLVING INTEGERS

                                             LESSON NOTES


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


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

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

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


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

                                                  Brackets
                                                  power of
                                                  Division
                                                  Multiplication
                                                  Addition
                                                  Subtraction



                                               EXAMPLES



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




        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                                   50
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers




                                            TEST YOURSELF I



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)




         Curriculum Development Division
         Ministry of Education Malaysia
                                                                                            51
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



                                                ANSWERS


TEST YOURSELF A:

       1.      2




                    –5    –4    –3    –2   –1     0     1      2   3   4   5   6



       2.      –3




                    –5     –4   –3    –2   –1     0     1      2   3   4   5   6




       3.      6




                    –5    –4    –3    –2   –1     0     1      2   3   4   5   6




       4.      –4



                    –5    –4    –3    –2   –1     0     1      2   3   4   5   6




       5.      –2



                    –5    –4    –3    –2   –1     0     1      2   3   4   5   6


        Curriculum Development Division
        Ministry of Education Malaysia
                                                                                   52
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



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



          Curriculum Development Division
          Ministry of Education Malaysia
                                                                             53
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers



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




              Curriculum Development Division
              Ministry of Education Malaysia
                                                                                     54
       Basic Essential

Additional Mathematics Skills




             UNIT 2

           FRACTIONS

 Unit 1:
 Negative Numbers




           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




                                      MODULE OVERVIEW

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

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

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

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




        Curriculum Development Division
        Ministry of Education Malaysia                                                       1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                                          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.




        Curriculum Development Division
        Ministry of Education Malaysia                                               2
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                      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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                 3
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                                           LESSON NOTES




Fraction is written in the form of:

                                           a        numerator
                                           b        denominator

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

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

           Examples:                           Examples:                   Examples:

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




                           Rules for Adding or Subtracting Fractions

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

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



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




         Curriculum Development Division
         Ministry of Education Malaysia                                                       4
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                                            EXAMPLES



1.0 Addition And Subtraction of Fractions with the Same Denominator


    1.1 Addition of Fractions with the Same Denominators


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




                    1                               4                                5
                    8                                8                                 8




                                            Add only the numerators and keep the
                 1 3 4                            denominator the same.
          ii)      
                 8 8 8
                      1                     Write the fraction in its simplest form.
                    
                      2



                                             Add only the numerators and keep the
                  1 5 6
          iii)                                   denominator the same.
                  f  f  f




        Curriculum Development Division
        Ministry of Education Malaysia                                                     5
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




    1.2 Subtraction of Fractions with The Same Denominators



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




                                                                                     4   1
                      5
                                                        1                 
                                                                                     8
                                                                                       
                                                                                         2
                      8                                  8




                                               Subtract only the numerators and keep
                 1 5   4
          ii)                                    the denominator the same.
                 7 7   7




                                             Subtract only the numerators and keep
                  3 1 2
          iii)                                  the denominator the same.
                  n n n




        Curriculum Development Division
        Ministry of Education Malaysia                                                       6
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




     1.3 Addition and Subtraction Involving Whole Numbers and Fractions

                          1
          i) Calculate 1  .
                          8




                                                 1
                   1                 +
                                                 8

                                                                        9
                   8                             1                
                                    +                                  8
                   8                             8
                                                                          1
                                                                       1
                                                                          8



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




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

                            29                              18                              12  y
                                                                                      
                            7                                5                                3

                              1                               3
                        4                               3
                              7                               5




        Curriculum Development Division
        Ministry of Education Malaysia                                                               7
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




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




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

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




        Curriculum Development Division
        Ministry of Education Malaysia                                                      8
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



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




                        1                       +          4
                    1
                        8                                  8


                   9                           +          4                  13                    5
                                                                                                1
                    8                                      8                    8                    8




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



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




         Curriculum Development Division
         Ministry of Education Malaysia                                                          9
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions


2.0 Addition and Subtraction of Fractions with Different Denominators

                    1 1
     i) Calculate     .                               The denominators are not the same.
                    8 2                                 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!




                                                                                ?

                    1          +                   1                           ?
                    8                              2
    To make the denominators the same, multiply both the numerator and the denominator of
    the second fraction by 4:
                                          4

                                    1          4
                                                                                 Now, the denominators
                                    2          8                                  are the same. Therefore,
                                                                                  we can add the fractions
                                          4
                                                                                          together!

    Now, the question can be visualized like this:




                    1           +                  4                            5
                    8                              8                             8




        Curriculum Development Division
        Ministry of Education Malaysia                                                                   10
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



        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.



                                      Change the first fraction to an equivalent
              1   5                        fraction with denominator 6.
  (i)                                 (Multiply both the numerator and the
              3   6
                                       denominator of the first fraction by 2):

              2   5                                      2
               
              6   6                                    1
                                                         
                                                            2
                                                       3    6
              7                                          2
         
              6
              1                       Add only the numerators and keep the
         =1
              6                             denominator the same.

                                     Convert the fraction to a mixed number.




                                       Change the second fraction to an equivalent
              7   3                         fraction with denominator 12.
  (ii)                                  (Multiply both the numerator and the
             12   4                     denominator of the second fraction by 3):

              7    9                                        3
                                                        3    9
             12   12                                        
                                                          4   12
               2                                            3
         
              12
                                        Subtract only the numerators and keep the
             1
                                                denominator the same.
             6

                                          Write the fraction in its simplest form.




             Curriculum Development Division
             Ministry of Education Malaysia                                                 11
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                                            Change the first fraction to an equivalent
                    1   9
  (iii)                                        fraction with denominator 5v.
                    v   5v                   (Multiply both the numerator and the
                                             denominator of the first fraction by 5):

                    5    9                                     5
                                                          1    5
                    5v   5v                                   
                                                            v    5v
                                                              5

                    14
                
                    5v                       Add only the numerators and keep the
                                                   denominator the same.




          Curriculum Development Division
          Ministry of Education Malaysia                                                 12
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




2.2   Addition and Subtraction of Fractions When the Denominators Are Not Multiple of
      One Another



                          Method I                                       Method II

                          1         3                                    1         3
                                                                             
                          6         4                                    6         4

  (i) Find the Least Common Multiple (LCM)            (i) Multiply the numerator and the
                                                          denominator of the first fraction with
      of the denominators.
                                                          the denominator of the second fraction
                                                          and vice versa.
                      2) 4 , 6
                      2) 2 , 3
                                                               1 4           3 6
                      3) 1 , 3                             =        
                         - , 1                                 6 4           4 6


      LCM = 2  2  3 = 12                                    4             18
                                                           =         
                                                               24            24
      The LCM of 4 and 6 is 12.
                                                               22
                                                           =
  (ii) Change each fraction to an equivalent                   24
       fraction using the LCM as the
       denominator.                                            11             Write the fraction in its
                                                           =
       (Multiply both the numerator and the                    12                 simplest form.
       denominator of each fraction by a whole
       number     that   will    make     their
       denominators the same as the LCM
       value).
                                                              This method is preferred but you
                                                               must remember to give the
            1    2           33                              answer in its simplest form.
      =               
            6    2           43

            2               9
      =               
           12              12

            11
      =
            12



          Curriculum Development Division
          Ministry of Education Malaysia                                                                  13
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



                                               EXAMPLES




       2                 1
 1.             
       3                 5

         2 5                1 3
                                                 Multiply the first fraction with the second denominator
       =                 +                       and
         3 5                5 3                multiply the second fraction with the first denominator.
                                               Multiply the first fraction by the
                                            denominator of the second fraction and
           10                 3
                                            multiply the second fraction by the
           15                15                denominator of the first fraction.

           13
       =                                    Add only the numerators and keep the
           15
                                                  denominator the same.




       5             3
 2.             
       6             8

                8                6
        5                    3
      =                  –
        6                    8
                8                6
                                           Multiply the first fraction by the
                                        denominator of the second fraction and
           40   18
      =                                  multiply the second fraction by the
           48   48                         denominator of the first fraction.


           22                           Subtract only the numerators and keep
      =                                       the denominator the same.
           48

           11                               Write the fraction in its simplest form.
      =
           24




          Curriculum Development Division
          Ministry of Education Malaysia                                                                    14
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




              2             1
     3.         g       
              3             7

                  2g  7                 1 3
             =                      
                   3 7                  7 3          Multiply the first fraction by the
                                                    denominator of the second fraction and
                                                      multiply the second fraction by the
                  14 g 3
             =                                        denominator of the first fraction.
                   21 21
                                                          Write as a single fraction.
                  14 g  3
             =
                      21


              2g                h
     4.                 
               3                5

                        5                     3
                2g                        h
                                   
                 3                        5
                        5                     3       Multiply the first fraction by the
                                                     denominator of the second fraction and
                  10 g 3h                              multiply the second fraction by the
                                                      denominator of the first fraction.
                   15 15
                                                           Write as a single fraction.
                 10 g  3h
             
                     15



              6             4
     5.             
              c             d

                  6 d                  4 c
              =        
                  c d                  d c
                                                         Multiply the first fraction by the
                                                      denominator of the second fraction and
                  6d 4c                                 multiply the second fraction by the
                   
                  cd cd                                  denominator of the first fraction.

                  6d  4c                                   Write as a single fraction.
              =
                     cd




          Curriculum Development Division
          Ministry of Education Malaysia                                                       15
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



2.3   Addition or Subtraction of Mixed Numbers with Different Denominators



          1                    3
1.    2                   2                             Convert the mixed numbers to improper fractions.
          2                    4
                                                Convert the mixed numbers to improper fractions.
              5                    11
      =                
              2                     4

              5 2                 11             Change the first fraction to an equivalent fraction
      =            
              2 2                  4                              with denominator 4.
                                                 (Multiply both the numerator and the denominator
          10                   11                               of the first fraction by 2)
      =                
           4                    4
              21                                      Add only the numerators and keep the
      =
              4                                             denominator the same.

              1
      5                                           Change the fraction back to a mixed number.
              4

          5                    3
2.    3                   1                      Convert the mixed numbers to improper fractions.
          6                    4
              23                   7
      =                                       Convert the mixed numbers to improper fractions.
              6                    4
                                               The denominators are not multiples of one another:
        23  4                          7 6
      =        
        6 4                            4 6    Multiply the first fraction by the denominator
                                                 of the second fraction.
          92 42                                 Multiply the second fraction by the
      =                                         denominator of the first fraction.
          24 24

              50                                     Add only the numerators and keep the
      =
              24                                           denominator the same.

          25
      =                                               Write the fraction in its simplest form.
          12

               1                                  Change the fraction back to a mixed number.
      = 2
              12




          Curriculum Development Division
          Ministry of Education Malaysia                                                                    16
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



2.4 Addition or Subtraction of Algebraic Expression with Different Denominators
       m                 m
1.                                               The denominators are not multiples one another:
                                                 The denominators are not multiples of of one another
      m2                2                         Multiply the first fraction with the second denominator
                                                   Multiply the second fraction with the first denominator
                    2                 ( m2)    Multiply the first fraction by the denominator
        m                         m
     =                                            of the second fraction.
       m2          2            2    ( m2)    Multiply the second fraction by the
                                                   denominator of the first fraction.


           2m      mm  2
     =                                              Remember to use
         2m  2 2m  2                              brackets

         2m  m(m  2)                              Write the above fractions as a single fraction.
     =
            2(m  2)

         2m  m 2  2 m                            Expand:
     =
            2(m  2)
                                                                 m (m – 2) = m2 – 2m

           m2
     =
         2(m  2)

        y                y 1
2.                                               The denominators are not multiples of one another:
      y 1                 y                       The denominators are not multiples of one another
                                                    Multiply the fraction with by the denominator
                                                  Multiply the first first fractionthe second denominator
           y y                 y  1 ( y 1)     Multiply the second fraction with the first denominator
                                                    of the second fraction.
     =                   
         y 1  y                 y  ( y 1)      Multiply the second fraction by the
                                                    denominator of the first fraction.

         y 2  ( y  1)( y  1)                          Write the fractions as a single fraction.
     =
              y ( y  1)
                                                    Expand:
       y 2  ( y 2  1)                                        (y – 1) (y + 1) = y2 + y – y – 12
     =
           y ( y  1)
                                                                               = y2 – 1

         y2  y2  1
     =                                            Expand:
           y ( y  1)                                                 – (y2 – 1) = –y2 + 1

              1
     =
         y ( y  1)



         Curriculum Development Division
         Ministry of Education Malaysia                                                                      17
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




         3   5n
   3.                                          The denominators are not multiples of one another:
        8n   4n 2
                                                 Multiply the first fraction multiples of one another
                                                  The denominators are not by the denominator
             3        4n 2        5  n  8n     Multiply the first fraction
                                                  of the second fraction. with the second denominator
        =                                        Multiply the second fraction with the first denominator
                                                 Multiply the second fraction by the
            8n  4n 2              4 n 2  8n     denominator of the first fraction.

             12n 2    8n (5  n)
        =           
                  2
            8n(4n ) 8n(4n 2 )

            12 n 2       8n (5  n)                           Write as a single fraction.
        =
                      8n(4n 2 )
                                                  Expand:
            12 n 2      40 n  8n 2
        =                                                       – 8n (5 + n) = –40n – 8n2
                       8n(4n 2 )
            4n 2       40 n                                    Subtract the like terms.
        =
                8n ( 4 n 2 )

            4n (n  10 )                           Factorise and simplify the fraction by canceling
        =                                                      out the common factors.
                4n(8n 2 )

            n  10
        =
              8n 2




        Curriculum Development Division
        Ministry of Education Malaysia                                                                 18
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



                                           TEST YOURSELF A


Calculate each of the following.

       2 1                                             11 5
  1.                                           2.         
       7 7                                             12 12




       2 1                                             2 5
  3.                                           4.        
       7 14                                            3 12




       2 4
  5.                                                 1 5
       7 5                                       6.      
                                                       2 7




          2
  7. 2      3                                      2   7
         13                                      8. 4  2 
                                                     5   9




       2 1                                             11 5
  9.                                           10.      
       s s                                             w w




         Curriculum Development Division
         Ministry of Education Malaysia                          19
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                                                       2   5
  11.
        2 1
                                               12.        
        a 2a                                           f 3f




                                                       1 5
  13.
        2 4
                                               14.     
        a b                                            p q




                                                       p 1
      5  2   2   3                               16.         (2  p) 
  15. m  n  m  n                                    2
     7   5   7   5




        2 x  3 y 3x  y                               12  4 x  5
  17.                                          18.             
            2       5                                    2x      x




          x    x 1
  19.                                                 x   x4
        x 1     x                               20.           
                                                       x2 x2




        Curriculum Development Division
        Ministry of Education Malaysia                                    20
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




        6x  3 y 4x  8 y                               2   4n
  21.                                          22.            
           2        4                                  3n   9n 2




      r 5  2r 2                                       p3 p2
  23.                                          24.           
      5   15 r                                          p2   2p




        2n  3 4n  3                                  3m  n n  3
  25.                                          26.               
         5n 2   10n                                     mn      n




        5m mn                                        m3 nm
  27.                                          28.          
         5m   mn                                       3m   mn




         3 5n
  29.                                                 p 1 p
        8n 4n 2                                  30.          
                                                       3m   m




         Curriculum Development Division
         Ministry of Education Malaysia                                 21
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                                           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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                     22
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                         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
                      (+)  (+) =           +           (+)      (+)     =   +
                      (+)  (–) =           –           (+)      (–)     =   –
                      (–)  (+) =           –           (–)      (+)     =   –
                      (–)  (–) =           +           (–)      (–)     =   +




        Curriculum Development Division
        Ministry of Education Malaysia                                                       23
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions


                                          LESSON NOTES


1.0 Multiplication of Fractions


       Recall that multiplication is just repeated addition.
       Consider the following:
                                                 2  3


       First, let’s assume this box            as 1 whole unit.


       Therefore, the above multiplication 2 3 can be represented visually as follows:


              2 groups of 3 units




                                           3      +     3         =    6

     This means that 3 units are being repeated twice, or mathematically can be written as:
                                                23  3  3
                                                      6

     Now, let’s calculate 2 x 2. This multiplication can be represented visually as:


              2 groups of 2 units




                                           2      +     2         =        4
     This means that 2 units are being repeated twice, or mathematically can be written as:
                                                   2 2  2  2
                                                        4




        Curriculum Development Division
        Ministry of Education Malaysia                                                        24
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



       Now, let’s calculate 2 x 1. This multiplication can be represented visually as:



               2 groups of 1 unit

                                             1    +      1     =         2

       This means that 1 unit is being repeated twice, or mathematically can be written as:
                                           2 1  1  1  2



       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?

                                1
       Let’s consider 2          .
                                2

                                                                   1
       Since                  represents 1 whole unit, therefore     unit can be represented by the
                                                                   2
       following shaded area:



                                                                    1
       Then, we can represent visually the multiplication of 2       as follows:
                                                                    2

                              1
                2 groups of     unit
                              2
                                                 1   1                   2
                                                   +           =           1
                                                 2   2                   2
                         1
       This means that     unit is being repeated twice, or mathematically can be written as:
                         2
                                               1 1 1
                                             2  
                                               2 2 2
                                                  2
                                                
                                                  2
                                                1




        Curriculum Development Division
        Ministry of Education Malaysia                                                          25
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                              1                                   1
       Let’s consider again      2. What does it mean? It means ‘ out of 2 units’ and the
                              2                                   2
       visualization will be like this:

                                                               1                     1
                                                                 out of 2 units        2 1
                                                               2                     2




                                            1    1
       Notice that the multiplications 2     and  2 will give the same answer, that is, 1.
                                            2    2

                    1
       How about      2?
                    3

                                                                     1
       Since              represents 1 whole unit, therefore           unit can be represented by the
                                                                     3
       following shaded area:


                                                               1
                                          The shaded area is     unit.
                                                               3


                                                                1
       Then, we can represent visually the multiplication          2 as follows:
                                                                3




                                  1   1                         2
                                    +              =
                                  3   3                         3
                         1
       This means that     unit is being repeated twice, or mathematically can be written as:
                         3
                                                1     1 1
                                                  2 
                                                3     3 3
                                                      2
                                                    
                                                      3




        Curriculum Development Division
        Ministry of Education Malaysia                                                            26
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



                        1                                    1
       Let’s consider      2 . What does it mean? It means ‘ out of 2 units’ and the visualization
                        3                                    3
       will be like this:


                                                         1                        1      2
                                                           out of 2 units           2 
                                                         3                        3      3


                                            1    1                                        2
       Notice that the multiplications 2     and  2 will give the same answer, that is,   .
                                            3    3                                        3

       Consider now the multiplication of a fraction by a fraction, like this:

                                                1 1
                                                 
                                                3 2

                      1       1
       This means ‘     out of units’ and the visualization will be like this:
                      3       2

                                                      1        1                 1 1 1
                                                        out of   units             
                           1                          3        2                 3 2 6
                             unit
                           2




       Consider now this multiplication:

                                                2 1
                                                 
                                                3 2

                      2       1
       This means ‘     out of units’ and the visualization will be like this:
                      3       2


                           1
                             unit
                           2

                                                        2        1                2 1 2
                                                          out of   units            
                                                        3        2                3 2 6


        Curriculum Development Division
        Ministry of Education Malaysia                                                          27
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions


       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:



                        1   1 1                                2   1 2
                                                                 
                        3   2 6                                3   3 9



                                               1 1              1
       So, what do you think the answer for      ? Do you get    as the answer?
                                               4 3             12

       The steps to multiply a fraction by a fraction can therefore be summarized as follows:


                  Steps to Multiply Fractions:                            Remember!!!


          1) Multiply the numerators together and                   (+)      (+)    =   +
               multiply the denominators together.                  (+)      (–)    =   –
                                                                    (–)      (+)    =   –
          2) Simplify the fraction (if needed).                     (–)      (–)    =   +



1.1    Multiplication of Simple Fractions
       Examples:



                  2   3         6                               2    3    6
         a)                                            b)          
                  5   7         35                              7    5    35




                      6   2     12                                  6     2         12
          c)                                         d)              
                      7   5     35                                  7     5         35



                 Multiply the two numerators together and the two denominators together.



        Curriculum Development Division
        Ministry of Education Malaysia                                                          28
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



1.2    Multiplication of Fractions with Common Factors

        12   5    12  5 
              or     
         7   6     7 6 

          First Method:                               Second Method:

         (ii) Multiply the two numerators            (i) Simplify the fraction by canceling
              together    and     the two                out the common factors.
              denominators together:
                                                               2 12         5
                                                                        
                                                                    7       61
                12   5   60
                      =
                 7   6   42                          (i) Then,    multiply    the    two
                                                         numerators together and the two
                                                         denominators    together,   and
         (ii) Then, simplify.                            convert to a mixed number, if
                                                         needed.
                          6010 10    3
                                 1
                          42    7    7                         2
                                7                                  12   5            10    3
                                                                                      1
                                                                    7   6             7    7
                                                                             1



1.3    Multiplication of a Whole Number and a Fraction


                           2  5 
                                 1
          Remember              6
            2= 2
                                  
               1
                                    2  31 
                           =                        Convert the mixed number to improper
                                    1  6                           fraction.

                                                       Simplify by canceling out the common
                               31 
                               12
                                                                      factors.
                          =      
                            1  6 
                                 3
                                                      Multiply the two numerators together and
                                                           the two denominators together.
                               31
                          =                                   Remember: (+)  (–) = (–)
                                3
                                 1
                          =  10                    Change the fraction back to a mixed number.
                                 3



        Curriculum Development Division
        Ministry of Education Malaysia                                                            29
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



                                            EXAMPLES




                   5   15
 1. Find            
                  12 10
                    1
                      5   15 5             Simplify by canceling out the common factors.
     Solution:         
                     12   10 2
                   4
                                            Multiply the two numerators together and the
                                                    two denominators together.
                            5
                   = 
                            8                        Remember: (+)  (–) = (–)




            21   2
 2. Find                                       Simplify by canceling out the common
            6    5
                                                               factors.
                     21   2 1
     Solution :                                            21
                   3
                     6    5                     Note that      can be further simplified.
                                                            3
                        21   2 1
                    = 7                          Simplify further by canceling out the
                        6    5                              common factors.
                                3
                            1
                                7
                                              Multiply the two numerators together and
                                5
                                                    the two denominators together.
                                  2
                    =           1
                                  5                   Remember: (+)  (–) = (–)

                                                  Change the fraction back to a mixed
                                                               number.




        Curriculum Development Division
        Ministry of Education Malaysia                                                      30
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions


1.4    Multiplication of Algebraic Fractions



                             2   5x
        1.    Simplify         
                             x    4


                          2  5x 1
              Solution : 1                      Simplify the fraction by canceling out the x’s.
                          x   4
                          1           2

                                                   Multiply the two numerators together and
                           5                            the two denominators together.
                         =
                           2
                                  1                   Change the fraction back to a mixed
                         = 2                                       number.
                                  2




                         n 9      
        2. Simplify           4m 
                         2 n      

                         n 9      
             Solution:        4m 
                         2 n      
                                                               Simplify the fraction by canceling the
                              1              2                      common factor and the n.
                           n9        n  4m 
                         =                
                           2n  1   1
                                       2 1 
                                                            Multiply the two numerators together
                           9       n ( 2m)
                         =                                 and the two denominators together.
                           2          1

                              9
                         =       2nm                        Write the fraction in its simplest form.
                              2




        Curriculum Development Division
        Ministry of Education Malaysia                                                                  31
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



                                          TEST YOURSELF B1



                9   25                                            45    3    14
1. Calculate                                   2. Calculate –             
                5   27                                            12    7    20




                11                                              1 1   
3. Calculate 2                                4. Calculate     4    
               4                                                3 5   




                    m                                         n
5. Simplify     3                           6. Simplify      (5m) 
                    k                                         2




                1  3x                                         n
7. Simplify    1                              8. Simplify      (2a  3d ) 
                6  14                                         2




                   2          9                               x     1
9. Simplify            5x    y              10. Simplify     20   
                   3         10                               4     x




        Curriculum Development Division
        Ministry of Education Malaysia                                              32
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions


                                          LESSON NOTES



2.0    Division of Fractions

       Consider the following:
                                                 6  3


       First, let’s assume this circle         as 1 whole unit.




       Therefore, the above division can be represented visually as follows:


                                                               6 units are being divided into a group of 3
                                                                                  units:

                                                                             6  3  2




       This means that 6 units are being divided into a group of 3 units, or mathematically
       can be written as:
                                              6  3  2

       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:


                                                                            1
                                1 1                            How many       is in
                                  .                                        8
                                2 8                                   1
                                                                        ?
                                                                      2



        Curriculum Development Division
        Ministry of Education Malaysia                                                                33
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




       This means ‘How many                             is in                    ?




                                           1                         1
                                           8                         2


       The answer is 4:




       Consider now this division:
                                                                           1      3
                                3 1                             How many     is in ?
                                  .                                       4      4
                                4 4




       This means ‘How many                             is in                    ?




                                           1                          3
                                           4                          4


                                                                       But, how do you
       The answer is 3:
                                                                     calculate the answer?




        Curriculum Development Division
        Ministry of Education Malaysia                                                       34
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions


       Consider again 6  3  2.

       Actually, the above division can be written as follows:

                                                     6               These operations are the same!
                                             63 
                                                     3
                                                         1
                                                   6               The reciprocal
                                                         3                       1
                                                                       of 3 is     .
                                                                                 3




       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:

                                     1 1
                            (i)                            Change the second fraction to its
                                     2 8
                                                         reciprocal and change the sign  to .
                                       1 8
                                      
                                       2 1
                                       8                              The reciprocal
                                     
                                       2                                     1   8
                                                                        of     is .
                                     4                                      8   1




                                                                    The reciprocal of a
                                                                    fraction is found by
                                                                       inverting the
                                                                          fraction.




        Curriculum Development Division
        Ministry of Education Malaysia                                                                35
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                                    3 1
                           (ii)                             Change the second fraction to its
                                    4 4                   reciprocal and change the sign  to .
                                     3 4
                                     
                                     4 1
                                    3                                  The reciprocal
                                                                              1   4
                                                                         of     is .
                                                                              4   1




           The steps to divide fractions can therefore be summarized as follows:


                       Steps to Divide Fractions:                                    Tips:
             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).



2.1    Division of Simple Fractions

       Example:



               2   3
                                     Change the second fraction to its reciprocal
               5   7                       and change the sign  to  .
               2   7
            =    
               5   3                      Multiply the two numerators together and
              14                               the two denominators together.
            =
              15




        Curriculum Development Division
        Ministry of Education Malaysia                                                                 36
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




2.2       Division of Fractions With Common Factors

        Examples:



                 10       2
                                          Change the second fraction to its reciprocal and
                 21       9
                                                       change the  sign to  .
                 10         9
              =         
                 21         2              Simplify by canceling out the common factors.
                 10          9
              =5          3
               7 21          21             Multiply the two numerators together and the
                    15
              =                                    two denominators together.
                     7
                    1                                Remember: (+)  (–) = (–)
              = 2
                    7
                                            Change the fraction back to a mixed number.




               3
               5
               6
                                              Express the fraction in division form.
               7
               3   6
                
               5   7                        Change the second fraction to its reciprocal
                                                 and change the  sign to  .
                 1
                     3   7
                                          Then, simplify by canceling out the common
                     5   62                                   factors.

                      7
                                          Multiply the two numerators together and the
                     10
                                                   two denominators together.




        Curriculum Development Division
        Ministry of Education Malaysia                                                         37
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                                            EXAMPLES


            35   25
 1. Find       
            12   6

                        35   25
     Solution :            
                        12   6
                                                     Change the second fraction to its reciprocal
                                                             and change the  sign to .
                    35   61
                = 7                                 Then, simplify by canceling out the common
                  2 12   25                                            factors.
                      7           5
               =                                     Multiply the two numerators together and the
                     10
                                                             two denominators together.


                   2       5x
 2. Simplify –       
                   x        4
                                                    Change the second fraction to its reciprocal
                       2          4
     Solution :      –                                     and change the  sign to .
                       x         5x


                           8                    Multiply the two numerators together and the two
                    = –                                      denominators together.
                          5x 2

                   y
 3. Simplify       x
                   2
    Solution :
                                                      Express the fraction in division form.
     Method I       y
                        2
                    x                               Change the second fraction to its reciprocal
                   
                      y
                          
                             1                                and change  to  .
                      x      2
                        y
                                              Multiply the two numerators together and the two
                       2x                                    denominators together.

                                                           Remember: (+)  (–) = (–)




        Curriculum Development Division
        Ministry of Education Malaysia                                                              38
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




   Method II
                                           The given fraction.
                    y
                                           The numerator is also
                    x
                    2                         a fraction with
                                              denominator x

                         y
               =         x         
                                       x    Multiply the numerator and the denominator of
                                           Multiply the numerator and the denominator of the
                        2             x                    given x
                                           the given fraction withfraction by x.


                          y
                            x
               =          x
                          2 x

                          y
               =    
                         2x




                    (1  1 )
 4. Simplify                   r
                         5


   Solution:
                                                                                 1
                    (1  1 )                           r is the denominator of
                                                                                 r
                                                                                   .
                              r
                         5
                             1                                                         r
                        (1    )                   Multiply the given fraction with
                                                                                       r
                                                                                         .
                =            r  r
                           5     r
                        r 1
                =                                                Note that:
                         5r
                                                                1
                                                            (1  )  r  r  1
                                                                r




        Curriculum Development Division
        Ministry of Education Malaysia                                                         39
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



                                          TEST YOURSELF B2




                   3     21                                        5    7   5
1.   Calculate                                2.   Calculate        
                   7     2                                         9    8  16




                8     4y                                          16
3.   Simplify                                  4.   Simplify
                y     3                                            2
                                                                   k




                    2                                              4m     2m 2
5.   Simplify                                         Simplify        
                   5 x                          6.
                                                                    n      3n
                    3




                     4                                              x
                                                 8.   Simplify
                   y 1                                            1
                                                                        1
7.   Simplify
                     8                                                  x




        Curriculum Development Division
        Ministry of Education Malaysia                                           40
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions




                  3 (1  1 )                                       5 1
9.    Calculate             4                                           x
                                                 10.    Simplify
                        5                                           y




                    
                  x 1 4
                            9
                                                                   1
                                                                    p
11.    Simplify
                        2                        12.   Simplify
                                                                   1
                        3                                            1
                                                                   5




         Curriculum Development Division
         Ministry of Education Malaysia                                     41
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



                                                 ANSWERS



TEST YOURSELF A:

             3                          1                             5
       1.                         2.                           3.
             7                          2                            14

             1                          38      3                         3
       4.                         5.       or 1                6. 
             4                          35      35                       14

             67       2                 73      28                   3
       7.       or 5              8.       or 1                9.
             13      13                 45      45                   s

                 6                           5                         1
       10.                        11.                          12.
                 w                          2a                        3f

                 2b  4a              q  5p                   15. m  n
       13.
                   ab             14.
                                        pq

                 3p  3                  16 x  17 y                     2x  1
       16.                        17.                          18.
                   2                         10                            x

                     1                                                8x  y
       19.                        20. 2                        21.
                 x( x  1)                                              2

           7n  4                     r 2 1                        p2  6
       22.                        23.                          24.
            9n 2                        3r                           2 p2

                 7 n  4n 2  6          1 m                              n5
       25.                        26.                          27.
                     10 n 2               m                                 5n

                 n3                     n  10                       4p 3
       28.                        29.                          30.
                  3n                      8n 2                         3m




        Curriculum Development Division
        Ministry of Education Malaysia                                            42
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions



TEST YOURSELF B1:



               5      2                          9        1          11      1
       1.        or 1                 2.          or  1      3.       or 5
               3      3                          8        8           2      2

                   7        2               3m                       5mn
       4.           or  1           5.                       6.
                   5        5                k                        2

               x                                     3                   10   3
       7.                             8.    na        nd      9.          x y
               4                                     2                    3   5

                       1
       10.    5x 
                       4




TEST YOURSELF B2:



              2                                 14        5               6
       1.                             2.          or  1      3.    
              49                                 9        9              y2

                                             6                       6
                                      5.                       6.
       4.    8k                             5 x                     m

                  1                          x2                      9
       7.                                                      9.
              2( y  1)               8.
                                            x 1
                                                                     20

              5x  1                        13x                           5
       10.                            11.                      12.   
               xy                            6                           4p




        Curriculum Development Division
        Ministry of Education Malaysia                                             43
       Basic Essential

Additional Mathematics Skills




           UNIT 3
ALGEBRAIC EXPRESSIONS
              AND
 Unit 1:
  ALGEBRAIC FORMULAE
 Negative Numbers




        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




                                    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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                             1
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                             PART A:
                      PERFORMING OPERATIONS ON
                            ALGEBRAIC EXPRESSIONS

                                      LEARNING OBJECTIVES

       Upon completion of Part A, pupils will be able to perform operations on algebraic
       expressions.


                            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.



           Curriculum Development Division
           Ministry of Education Malaysia                                                          2
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                          LESSON NOTES



                            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 – n2,    3
                                                                                            g


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.


                  Coefficient                            Unknowns
                                          6 xy



     Examples:     Algebraic expression with one term:                 2r,    3
                                                                             g

                   Algebraic expression with two terms:                3x + 2y, 6s – 7t

                   Algebraic expression with three terms:              6x2 +7x + 10, 8c + 3a – n2



4.   Like terms are terms with the same unknowns and the same powers.

     Examples:         3ab,      –5ab are like terms.

                                 2 2
                       3x2,        x   are like terms.
                                 5

5.   Unlike terms are terms with different unknowns or different powers.

     Examples: 1.5m,       9k,     3xy,   2x2y are all unlike terms.


        Curriculum Development Division
        Ministry of Education Malaysia                                                              3
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




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.

                   3m 2    4r 2 g     x2  y2
     Examples:       ,  ,         , 2             .
                   15 6h 2rg  g 2 x  2 xy  y 2

10. To simplify an algebraic fraction, identify the common factor of both the numerator and the
    denominator. Then, simplify it by elimination.




        Curriculum Development Division
        Ministry of Education Malaysia                                                        4
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae



                                            EXAMPLES


Simplify the following algebraic expressions and algebraic fractions:

                                                              s t
(a)    5x – (3x – 4x)                                 ( e)     
                                                              4 6

                                                               5x 3 y
(b)    –3r –9s + 6r + 7s                               (f )      
                                                               6 2z

         4r 2 g                                                e
(c)                                                    (g )       2g
       2rg  g 2                                               f

                                                                       1
         3 4                                                    3x 
(d )                                                                  2
         p q                                             (h)
                                                                 3x

Solutions:                          Algebraic expression with like terms can be simplified by
(a)    5x – (3x – 4x)                adding or subtracting the coefficients of the unknown.

       = 5x – (– x)                           Perform the operation in the bracket.

       = 5x + x

       = 6x



(b)    –3r –9s + 6r + 7s
                                    Arrange the algebraic terms according to the like terms.
       = –3r + 6r –9s + 7s
                                                                  .
       = 3r – 2s                              Unlike terms cannot be simplified.
                                        Leave the answer in the simplest form as shown.


          4r 2 g
(c)
        2rg  g 2

               4r 2 g 1               Simplify by canceling out the common factor and the
                                        same unknowns in both the numerator and the
         1
           g ( 2r  g )
                                                         denominator.
           4r 2
        
          2r  g


          Curriculum Development Division
          Ministry of Education Malaysia                                                        5
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae


           3 4
 (d )        
           p q
           3q 4 p                             The LCM of p and q is pq.
              
            pq pq
           3q  4 p
         
              pq

       s t
(e)      
       4 6
          3s     2t                           The LCM of 4 and 6 is 12.
             
         43 6 2
         3s  2t
       
           12


             1                           Simplify by canceling out the common
       5x 3 y 5x  y
(f )                                    factor, then multiply the numerators
       6 2z 2  2z                            together and followed by the
     2
               5 xy
                                                     denominators.
                4z


        e        e 1                     Change division to multiplication of the
(g )       2g                                   reciprocal of 2g.
        f        f 2g
                  e
               
                 2 fg


                                                 Equate the denominator.
               1 3 x(2) 1
          3x            
  (h )         2 2        2
            3x       3x
                  6x  1
                 2
                   3x
                  6x  1 1
                        
                    2      3x
                  6x  1
                
                    6x




           Curriculum Development Division
           Ministry of Education Malaysia                                           6
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae


                                          ALTERNATIVE METHOD

Simplify the following algebraic fractions:


             1                  1
      3x                  3x                  The denominator of
                                                                       1
                                                                          is 2 . Therefore,
             2                  2 2
(a)                  =                                                2
        3x                    3x      2                                              2
                                                  multiply the algebraic fraction by .
                                                                                     2
                             1
                    3 x(2)  (2)
                             2                    Each of the terms in the numerator and
                  =
                        3 x(2)                    denominator of the algebraic fraction is
                                                  multiplied by 2.

                         6x  1
                  =
                          6x



                                                                       3
       3            3                           The denominator of       is x. Therefore,
         2           2                                              x
       x            x               x
(b)              =                                                                   x
         5               5            x           multiply the algebraic fraction by .
                                                                                      x
                   3
                     ( x )  2( x )
                  x                              Each of the terms in the numerator and
                        5( x)                     denominator is multiplied by x.
                   3  2x
                 
                     5x




                                                                       3
                                                   The denominator of     is 2x. Therefore,
       3       3                                                2x
    8       8   2 x 
       2x       2 x                                                              2x
(c)                                              multiply the algebraic fraction by     .
       2             2         2x                                                      2x
                                                   Each of the terms in the numerator and
                                                   denominator is multiplied by 2x.
                             3 
                   8(2 x)   (2 x)
                             2x                  .
                 
                          2( 2 x )

                     16 x  3
                 
                       4x




        Curriculum Development Division
        Ministry of Education Malaysia                                                        7
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




           3           3           7                                   8 x
(d )                                           The denominator of         is 7.
       8 x     8 x           7                                    7
            4        4                     Therefore, multiply the algebraic
        7        7 
                                                               7
                         3(7)                    fraction by     .
                                                              7
                  8 x
                         ( 7 )  4( 7 )
                   7                           Each of the terms in the numerator
                      21                         and denominator is multiplied by 7.
                
                  8  x  28
                    21
                                                     Simplify the denominator.
                  36  x




          Curriculum Development Division
          Ministry of Education Malaysia                                               8
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                             TEST YOURSELF A


Simplify the following algebraic expressions:
  1. 2a –3b + 7a – 2b                               2. − 4m + 5n + 2m – 9n




  3. 8k – ( 4k – 2k )                               4. 6p – ( 8p – 4p )




          3 1                                             4h 2k
  5.                                               6.       
          y 5x                                             3   5




         4a 3b                                            4c  d     8
  7.                                               8.           
          7 2c                                              2      3c  d




         xy                                                u   uv
  9.         yz                                    10.      
          z                                               vw 2w




             2                                          4
  11 .                                                   2
          5
                                                    12.  
           6                                           x
           x                                          4
                                                         5
                                                         x




           Curriculum Development Division
           Ministry of Education Malaysia                                    9
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                           PART B:
                       EXPANSION OF ALGEBRAIC
                                     EXPRESSIONS



                                   LEARNING OBJECTIVE

           Upon completion of Part B, pupils will be able to expand algebraic
           expressions.




                          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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                      10
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                       LESSON NOTES




                                              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 – 15b2

                                              = 12a2 + 8ab – 15b2



4.   Useful expansion tips:

            (i)    (a + b)2 = a2 + 2ab + b2

            (ii)   (a – b)2 = a2 – 2ab + b2

            (iii) (a – b)(a + b) = (a + b)(a – b)

                                  = a2 – b2




        Curriculum Development Division
        Ministry of Education Malaysia                                                    11
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                                EXAMPLES



Expand each of the following algebraic expressions:

(a)    2(x + 3y)                                (d ) ( a  3) 2

(b) – 3a (6b + 5 – 4c)                          (e)  32k  5
                                                                   2



                                                (f ) ( p  2)( p  5)
( c)
       2
         9 y  12
       3

Solutions:


                                               When expanding a bracket, each term
(a)    2 (x + 3y)                            within the bracket is multiplied by the term
                                                         outside the bracket.
        = 2x + 6y


                                               When expanding a bracket, each term
(b)     –3a (6b + 5 – 4c)                    within the bracket is multiplied by the term
                                                         outside the bracket.
       = –18ab – 15a + 12ac


       2
(c)      9 y  12
       3                                       Simplify by canceling out the common
          2 3        2 4
       =     9 y   12                         factor, then multiply the numerators
        1 3        1 3                       together and followed by the denominators.
       =    6y + 8


(d ) (a  3) 2

                                              When expanding two brackets, each term
       =   (a + 3) (a + 3)
                                              within the first bracket is multiplied by
                                               every term within the second bracket.
       = a2 + 3a + 3a + 9
       = a2 + 6a + 9




           Curriculum Development Division
           Ministry of Education Malaysia                                                   12
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




 (e)  32k  5
                 2




       = –3(2k + 5) (2k + 5)                  When expanding two brackets, each term
                                              within the first bracket is multiplied by
                                               every term within the second bracket.
       = –3(4k2 + 20k + 25)

       = –12k2 – 60k – 75




 (f ) ( p  2) (q  5)
                                              When expanding two brackets, each term
       = pq – 5p + 2q – 10                    within the first bracket is multiplied by
                                               every term within the second bracket.


 ALTERNATIVE METHOD

 Expanding two brackets



                                                           When expanding two
 (a)    (a + 3) (a + 3)                                   brackets, write down the
                                                         product of expansion and
                                                           then, simplify the like
        = a2 + 3a + 3a + 9                             (c) (4x – 3y)(6x – 5y)
                                                                   terms.
        = a2 + 6a + 9

                                                                – 18 xy
                                                                – 20 xy
                                                                – 38 xy
 (b)    (2p + 3q) (6p – 5q)

                                                          = 24x2 – 38 xy + 15y2
        = 12p2 – 10 pq + 18 pq – 15q2
        = 12p2 + 8 pq – 15q2




         Curriculum Development Division
         Ministry of Education Malaysia                                                   13
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                            TEST YOURSELF B


Simplify the following expressions and give your answers in the simplest form.


               3                                          1
                                                              6q  1
  1.    4 2n                                     2.
               4                                          2




  3.  6 x2 x  3 y                                4. 2a  b  2(a  b)




       2( p  3)  ( p  6)
                                                              6 x  y    x  2 y 
  5.                                                        1
                                                     6.                             
                                                            3                    3 




  7.   e  12  2e  1                             8.     m  n 2  m2m  n 




  9.   f    g  f  g   g 2 f  g             10 .   h  i h  i   2ih  3i 




        Curriculum Development Division
        Ministry of Education Malaysia                                                       14
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                                 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.



                              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:

      1.    y  2 x 3 (7 x  5) 4
                dy
                   2 x 3 [28(7 x  5) 3 ]  (7 x  5) 4 (6 x 2 )
                dx
                     2 x 2 (7 x  5) 3 (49 x  15)


                  (3  x) 3
       2.   y
                   7  2x
                dy (7  2 x)[3(3  x) 2 ]  (3  x) 3 (2)
                  
                dx              (7  2 x ) 2
                        (3  x) 2 (4 x  15)
                    
                            (7  2 x ) 2
      Strategy
      1. Pupils revise the techniques of factorisation.


        Curriculum Development Division
        Ministry of Education Malaysia                                                      15
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae



                                            LESSON NOTES


                                           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 + b2 = (a + b)2

                 a2 – 2ab + b2 = (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 – b2 = (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.




          Curriculum Development Division
          Ministry of Education Malaysia                                                        16
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                             EXAMPLES



  (a)   Factorising the Common Factors

                                                   Factorise the common factor m.
        i)    mn + m = m (n +1)
                                               .
                                                   Factorise the common factor p.
        ii)   3mp + pq = p (3m + q)
                                               .
                                                   Factorise the common factor 2n.
        iii) 2mn – 6n = 2n (m – 3)
                                               .

  (b)   Factorising Algebraic Expressions with Four Terms

                                                        Factorise the first and the second terms
                                                        with the common factor y, then factorise
         i)   vy + wy + vz + wz
                                                           the third and fourth terms with the
              = y (v + w) + z (v + w)                               common factor z.

              = (v + w)(y + z)                      .
                                                             (v + w) is the common factor.




        ii)   21bm – 7bs + 6cm – 2cs
                                                   Factorise the first and the second terms with
              = 7b(3m – s) + 2c(3m – s)             common factor 7b, then factorise the third
                                                    and fourth terms with common factor 2c.
              = (3m – s)(7b + 2c)
                                                            (3m – s) is the common factor.




        Curriculum Development Division
        Ministry of Education Malaysia                                                             17
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




(c)   Factorising the Algebraic Expressions by Using Difference of Two Squares

                                          a2 – b2 = (a + b)(a – b)



      i)     x2 – 16 = x2 – 42
                       = (x + 4)(x – 4)


      ii)    4x2 – 25 = (2x)2 – 52

                         = (2x + 5)(2x – 5)


(d)   Factorising the Expressions by Using the Cross Method

                i)           x2 – 5x + 6
                                                                      The summation of the cross
                                                                     multiplication products should
                              x           3                         equal to the middle term of the
                              x           2                           quadratic expression in the
                               3 x  2 x  5 x                              general form.




                           x2 – 5x + 6 = (x – 3) (x – 2)




                 ii)         3x2 + 4x – 4
                                                                        The summation of the cross
                                                                       multiplication products should
                              3x            2
                                                                       equal to the middle term of the
                              x            2                            quadratic expression in the
                               2x  6x   4x                                  general form.




                          3x2 + 4x – 4 = (3x – 2) (x + 2)




           Curriculum Development Division
           Ministry of Education Malaysia                                                                18
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                       ALTERNATIVE METHOD



Factorise the following quadratic expressions:                          REMEMBER!!!

                                                                   An algebraic expression can
                          2
i)                      x – 5x + 6                                 be represented in the general
                                                                    form of ax2 + bx + c, where
                 a=+1         b= –5          c =+6                    a, b, c are constants and
                                                                         a ≠ 0, b ≠ 0, c ≠ 0.



                                       ac              b
           +1  (+ 6) = + 6                                       –2  (–3) = +6
                                       +6             –5
                                                                  –2 + (–3) = –5
                                       –2             –3



                                      (x – 2)        (x – 3)

                     x 2  5x  6  ( x  2)(x  3)



     ii)                x 2 – 5x – 6

                     a=+1         b= –5              c = –6




               +1  (–6) = –6
                                        ac             b

                                       –6             –5
                                                               +1  (–6) = –6
                                        +1            –6
                                                               +1 – 6 = –5



                                      (x + 1)    (x– 6)

                       x 2  5x  6  ( x  1)(x  6)




           Curriculum Development Division
           Ministry of Education Malaysia                                                          19
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae



 (iii)                        2x2 – 11x + 5


                    a=+2         b = –11             c =+5



           (+2)  (+5) = +10                   ac             b

                                            + 10         –11

                                               –1        – 10                –1  (–10) = +10

                                                 1            10             –1 + (–10) = –11
                                                        
                                                 2             2

                                               1                         The coefficient of x2 is 2,
                                                        5              divide each number by 2.
                                               2

                                                                         The coefficient of x2 is 2,
                                                                              multiply by 2:
                                       (2x – 1)         (x – 5)
                                                                              x  12 x  5
                                                                               2x  1 x  5
                                                                                         2

                                                                               2 x  1)(x  5
                  2x 2  11x  5  (2x  1)(x  5)
                                               TEST YOURSELF C

 (iv)                           3x2 + 4x – 4

                     a =+ 3        b=+ 4             c = –4




                                            ac                b             –2 + 6 = 4
        3  (– 4) = –12                    – 12           +4

                                            –2            +6         The coefficient of x2 is 3, divide each
                                                                                 number by 3.
                                            2                6
                                                         
                                             3                3
                                                                   The coefficient of x2 is 3, multiply by 3:

                                           
                                               2
                                                         2
                                                                              x  2 x  2
                                                                                   3
                                               3                               3x  2 x  2
                                                                                        3

                                                                               3x  2)(x  2
                                      (3x – 2)          (x + 2)
                    3x 2  4x  4  (3x  2)(x  2)

           Curriculum Development Division
           Ministry of Education Malaysia                                                                       20
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                           TEST YOURSELF C


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




        Curriculum Development Division
        Ministry of Education Malaysia                                21
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




 11.   2x 2 + x – 10                               12.    3x 2 + 2x – 8




 13.   3p 2 – 5p – 12                              14.    4p2 – 3p – 1




 15.
           2
        2x – 3x – 5                                16.    4x 2 – 12x + 5




 17.    5p 2 + p – 6                               18.    2x
                                                                2
                                                                    – 11x + 12




 19.    3p + k + 9pr + 3kr                         20.        4c2 – 2ct – 6cw + 3tw




        Curriculum Development Division
        Ministry of Education Malaysia                                                22
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic 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.




                      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 A  1 r 2 .
                                                                  2
             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
                                              1               y, A and V are the
                                         A  bh                 subjects of the
                                              2
                                                                   formulae.
                                          V  r 2 h



        Curriculum Development Division
        Ministry of Education Malaysia                                                    23
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                            LESSON NOTES




                                                   PART D:
                               CHANGING THE SUBJECT OF A FORMULA


    1. An algebraic formula is an equation which connects a few unknowns with an equal
       sign.

                            1
                        A  bh
         Examples:          2
                        V  r 2 h

    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.

                                1               A is the subject of the formula because it is
         Examples:        A      bh
                                2                 expressed in terms of other unknowns.


                                                    a2 is not the subject of the formula
                          a2 = b2 + c2                     because the power ≠ 1


                                                   T is not the subject of the formula
                                1 2              because it is found on both sides of the
                          T      Tr h                           equation.
                                2



    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.




         Curriculum Development Division
         Ministry of Education Malaysia                                                              24
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae



                                                  EXAMPLES



                              Steps to Change the Subject of a Formula
                                           (i)     Fraction
                                          (ii)     Brackets
                                          (iii)    Group
                                          (iv)     Factorise
                                          (v)      Solve



1.   Given that 2x + y = 2, express x in terms of y.
     Solution:                                      No fraction and brackets.
               2x + y = 2
                                                            Group:
               2x = 2 – y                 Retain the x term on the left hand side of the
                                           equation by grouping all the y term to the
                       2 y
                  x=                            right hand side of the equation.
                        2

                                                              Solve:
                                             Divide both sides of the equation by 2 to
                                               make the coefficient of x equal to 1.



                   3x  y
2.    Given that           5 y , express x in terms of y.
                     2

      Solution:

           3x  y
                   5y                                      Fraction:
             2
                                             Multiply both sides of the equation by 2.
           3x + y = 10y
                                                             Group:
               3x = 10y – y
                                           Retain the x term on the left hand side of the
               3x = 9y                      equation by grouping all the y term to the
                                                 right hand side of the equation.
                       9y
                  x=
                        3                                     Solve:
                                             Divide both sides of the equation by 3 to
                  x = 3y                       make the coefficient of x equal to 1.



        Curriculum Development Division
        Ministry of Education Malaysia                                                      25
 Basic Essential Additional Mathematics Skills (BEAM) Module
 Unit 3: Algebraic Expressions and Algebraic Formulae




 3.   Given that     x  2 y , express x in terms of y.

      Solution:

                   x  2y                                       Solve:
                                             Square both sides of the equation to make the
                                2                       power of x equal to 1.
                   x = (2y)
                            2
                   x = 4y



                     x
4.    Given that        p , express x in terms of p.
                    3

      Solution:

                    x
                      p
                   3
                                                             Fraction:
                   x  3p                     Multiply both sides of the equation by 3.

                    x  (3 p ) 2
                    x  9 p2                                      Solve:

                                              Square both sides of the equation to make
                                                      the power of x equal to1.



 5.   Given that 3 x  2  x  y , express x in terms of y.

      Solution:
                                                               Group:
            3 x 2         xy                           Group the like terms
          3 x  x  y2
                                                           Simplify the terms.
                2 x  y2
                      y2                                     Solve:
                  x
                       2                     Divide both sides of the equation by 2 to
                       y 2
                                    2          make the coefficient of x equal to 1.
                    x     
                       2 
                                                              Solve:
                                             Square both sides of equation to make the
                                                      power of x equal to 1.




         Curriculum Development Division
         Ministry of Education Malaysia                                                      26
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae


                  11x
6.   Given that       – 2(1 – y) = 2 xp , express x in terms of y and p.
                   4

     Solution:
                                                                 Fraction:
                 11x
                     – 2 (1 – y) = 2 xp              Multiply both sides of the equation
                  4
                                                                    by 4.
                  11x – 8(1 – y) = 8 xp
                                                                  Bracket:
                   11x – 8 + 8y = 8xp                         Expand the bracket.

                      11x – 8xp = 8 – 8y
                                                                    Group:
                                                              Group the like terms.

                     x(11 – 8p) = 8 – 8y
                                                                   Factorise:
                                     8  8y                   Factorise the x term.
                               x=
                                    11  8 p
                                                                   Solve:
                                                      Divide both sides by (11 – 8p) to
                                                     make the coefficient of x equal to 1.




                  2 p  3x
7.   Given that            = 1 – p , express p in terms of x and n.
                     5n

     Solution:

                     2 p  3x
                              =1–p                                  Fraction:
                        5n                           Multiply both sides of the equation by
                      2p – 3x = 5n – 5pn                              5n.

                     2p + 5pn = 5n + 3x                               Group:
                                                              Group the like p terms.
                    p(2 + 5n) = 5n + 3x

                                 5n  3x                           Factorise:
                            p=                                Factorise the p terms.
                                  2  5n

                                                                      Solve:
                                                      Divide both sides of the equation by
                                                      (2 + 5n) to make the coefficient of p
                                                                   equal to 1.



        Curriculum Development Division
        Ministry of Education Malaysia                                                        27
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae



                                           TEST YOURSELF D


1.   Express x in terms of y.

     a) x  y  2  0                                b) 2 x  y  3  0




     c) 2 y  x  1
                                                     d)
                                                          1
                                                            x  y   2
                                                          2




     e) 3x  y  5                                   f) 3 y  x  4




         Curriculum Development Division
         Ministry of Education Malaysia                                    28
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




2. Express x in terms of y.


     a) y      x                                    b) 2 y  x




                 x                                   d) y  1  3 x
     c) 2 y 
                3




     e) 3 x  y  x  1                              f)   x 1  y




        Curriculum Development Division
        Ministry of Education Malaysia                                29
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




3. Change the subject of the following formulae:
                      xa                                                 1 x
     a) Given that         2 , express x in terms   b) Given that y          , express x in terms
                      xa                                                 1 x
        of a .                                           of y .




     c) Given that 1  1  1 , express u in          d) Given that 2 p  q  3 , express p in
                      f   u   v                                        2p  q   4
        terms of v and f .                               terms of q.




     e) Given that p  3m  2mn , express m in       f) Given that A  B C  1  , express C in
                                                                               
        terms of n and p .                                                   C 
                                                        terms of A and B .




                      2y  x                                                        l
     g) Given that            2 y , express y in    h) Given that T  2             , express g in
                        x                                                           g
        terms of x.
                                                        terms of T and l.




        Curriculum Development Division
        Ministry of Education Malaysia                                                                 30
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                          ACTIVITIES

                                    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)     V  r 2 h , then V is the                        of the formula.

7)     An                            can be represented by a letter.

10)                             x 2  3x  2  x  1x  2 .




        Curriculum Development Division
        Ministry of Education Malaysia                                               31
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




VERTICAL

2)     An algebraic                    consists of two or more algebraic terms combined by
       addition or subtraction or both.

6)                             2 x  1x  2  2 x 2  5 x  2 .

8)                             terms are terms with different unknowns.

9)     The number attached in front of an unknown is called                   .




        Curriculum Development Division
        Ministry of Education Malaysia                                                  32
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae


                                            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
                       2
   1. Calculate              5.
                         3

              1
         D)                                   O) 1
              5

              11                                    11
         W)                                   N)
               3                                    15


   2. Simplify  3x  9 y  6 x  7 y .

         F) 3x  2 y                          W)  9 x  16 y

         E) 3x  2 y                          X) 9 x  2 y



                       p q
   3. Simplify           .
                       3 2

              2 p  3q                              2 p  3q
         L)                                   A)
                  6                                     6

              3q  2 p                              3 p  2q
         N)                                   R)
                 6                                      6




          Curriculum Development Division
          Ministry of Education Malaysia                                                 33
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae


   4. Expand 2( x  4)  ( x  7) .

       A) x  1                                  D) x  15

       U) 3x  1                                 C) 3x  15


   5. Expand  3a(2b  5c) .

       S )  6ab  15ac                          C) 6ab  15ac

       T)  6ab  15ac                           R) 6ab  15ac



   6. Factorise x 2  25 .

       E) ( x  5)(x  5)                        T) ( x  5)(x  5)

       I) ( x  5)(x  5)                        C) ( x  25)(x  25)


   7. Factorise pq  4q .

       D) pq(1  4q)                              E) q( p  4)

       T) p(q  4)                                S) q( p  4)



   8. Factorise x 2  8x  12 .

       I ) ( x  2)(x  6)                       W) ( x  2)(x  6)

       F) ( x  4)(x  3)                        C) ( x  4)(x  3)



                      3x  y
   9. Given that              4 , express x in terms of y.
                        2x

                  y                                       y
       L) x                                    C) x 
                  5                                       5

                 y                                        8 y
       T) x                                     N) x 
                11                                         3



        Curriculum Development Division
        Ministry of Education Malaysia                                  34
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae


   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




                   5
                     1
   1.    Calculate x    .
                     3

            5 x                                  5 x
         A)                                    O)
              3                                    3x
             3x                                    3
         I)                                    N)
            x5                                   x5

                          3p q
   2. Simplify               .
                           4 5r

              15 pr                                  4q
         F)                                    R)
               4q                                   15 pr
               3 pq                                 3 pq
         W)                                    B)
               20r                                   5r

                         x xy
   3. Simplify                 .
                         yz 2 z

              2                                      x2
         N)                                    D)
              y2                                    2z 2
             x                                    x2
         L)                                    I) 2
            2z 2                                  z


   4. Solve x  y 2  x(3x  y).
         E)         2 x 2  y 2  xy                      D) 2 x 2  y 2  xy

         I)        x 2  y 2  3x 2  xy                   N) 2 x 2  y 2  xy



          Curriculum Development Division
          Ministry of Education Malaysia                                                     35
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae


   5. Expand      p  5 2 .

       I) p 2  25                            N) p 2  25

       D) p 2  10 p  25                     L) p 2  10 p  25


   6. Factorise 2 y 2  7 y  15 .


       F) (2 y  3)( y  5)                    D) (2 y  3)( y  5)

       W) (2 y  3)( y  5)                    L) ( y  3)(2 y  5)




   7. Factorise 2 p 2  11 p  5 .


       R) (2 p  1)( p  5)                   B) (2 p  1)( p  5)

       F) ( p  1)( p  5)                    W) ( p  1)(2 p  5)


                     B
   8. Given that       (C  1)  A , express C in terms of A and B.
                     C


                 B                                       1
       L) C                                   R) C 
                BA                                     BA
                 AB                                      AB
       C) C                                  N) C 
                BA                                     BA


   9. Given that 5 x  y  x  2 , express x in terms of y.
                y2  4                                  y2  4
       O) x                                  B) x 
                 16                                      24

             y 1                                   y  2
                         2                                       2

       I) x                                U) x        
             2                                      4 




        Curriculum Development Division
        Ministry of Education Malaysia                                36
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                            FURTHER
                                          EXPLORATION



                                   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




        Curriculum Development Division
        Ministry of Education Malaysia                                               37
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae




                                           ANSWERS



TEST YOURSELF A:

    1. 9a – 5b
                                                        2.    – 2m – 4n
    3. 6k
                                                        4.    2p

         15 x  y                                             20h  6k
    5.                                                  6.
           5 xy                                                 15


         6ab                                                  4(4c  d )
    7.                                                  8.
          7c                                                   3c  d

          x
    9.                                                         2
          z2                                            10.
                                                               v2
                                                              4  2x
          2x                                            12.
    11.                                                       4  5x
        5  6x



TEST YOURSELF B:


    1. – 8n + 3                                         6. x + y

                1                                       7. e 2
    2. 3q +
                2

    3. – 12x2 + 18xy                                    8. n 2  m 2  mn

    4. – 3b                                             9. f 2  2 fg

    5.    p                                             10. h 2  2ih  5i 2




         Curriculum Development Division
         Ministry of Education Malaysia                                        38
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae


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:

                                                3 y
                                    (b) x 
 1.    (a) x = 2 – y                             2              (c)    x = 2y – 1

                                                5 y
       (d) x = 4 – y                (e)    x                   (f)   x = 3y – 4
                                                 3

 2.    (a) x = y2                   (b) x  4 y 2               (c) x  36 y 2

                                              1 y 
                                                          2
                y  1
                             2

       (d) x                      ( e) x                  (f) x  y 2  1
                3                            2 

                                                 y 1                      fv
 3.    (a)   x  3a                 (b)    x                   (c) u 
                                                 y 1                     v f

                                                   p
                 7q                 (e)    m                               B
       (d)   p                                 2n  3         (f) C 
                  2                                                        B A

       (g) y 
                     x                           4 2 l
                                    (h)    g
                 2( x  1)                        T2


        Curriculum Development Division
        Ministry of Education Malaysia                                                    39
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae



                                            ACTIVITIES

                                   CROSSWORD PUZZLE




                                             RIDDLES
RIDDLE 1
  2           3          1          5         4        7          6       8       9
      F           A          N          T         A           S       T       I       C


 RIDDLE 2
  2           1          3          5         4        7          6       9       8
      W           O          N          D         E           R       F       U       L




          Curriculum Development Division
          Ministry of Education Malaysia                                                  40
       Basic Essential

Additional Mathematics Skills




             UNIT 4

   LINEAR EQUATIONS

 Unit 1:
 Negative Numbers




           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

                                                                  x
  Part C:   Solving Linear Equations in the Forms of ax = b and     =b          9
                                                                  a

  Part D:   Solving Linear Equations in the Form of ax + b = c                 12

                                                      x
  Part E:   Solving Linear Equations in the Form of     +b=c                   15
                                                      a

  Part F:   Further Practice on Solving Linear Equations                       18

Answers                                                                        23
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                  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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                   1
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                          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.




                       TEACHING AND value is a solution of a given
                a. determine if a numericalLEARNING STRATEGIES linear equation
                       in one unknown;
       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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                    2
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                 OVERALL LESSON NOTES


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
                             Arithmetic                              Algebra

                              8 = (4) (2)                              a=b

                          8 – 3 = (4) (2) – 3                       a–c=b–c

           (b) Addition
                             Arithmetic                              Algebra

                              8 = (4) (2)                               ;
                                                                       a=b

                          8 + 3 = (4) (2) + 3                      a+c=b+c


           (c) Division

                             Arithmetic                              Algebra

                              8=6+2                                    a=b

                              8 62                                   a b
                                                                               c≠0
                              3   3                                   c c


             (d) Multiplication
                             Arithmetic                               Algebra

                              8 = (6 +2)                                a=b

                          (8)(3) = (6+2) (3)                           ac = bc




          Curriculum Development Division
          Ministry of Education Malaysia                                                  3
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations


                                        PART A:
                                   LINEAR EQUATIONS


                                      LESSON NOTES



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?



       Solving an equation is like solving a puzzle to find the value of the unknown.



        Curriculum Development Division
        Ministry of Education Malaysia                                                   4
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



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.


   (a) x + 2 = 5
                                                                       x=3
                              x=?
  2.




2. The equality in an equation can also be explained by using tiles (preferably coloured tiles).




       x                                                    x
                                                            x




                     x+2=5
                     x+2=5                                             + = 5–
                                                                x + 2x – 2 – 25= – 2 2
                                                                            x 3
                                                                          x == 3




           Curriculum Development Division
           Ministry of Education Malaysia                                                    5
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                          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.




                        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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                    6
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



                                            PART B:
                   SOLVING LINEAR EQUATIONS IN THE FORM OF
                                 x+a=b         OR       x–a=b


                                           EXAMPLES



Solve the following equations.

(i) x  2  5                                  (ii) x  3  5



Solutions:


(i)     x25                       Subtract 2 from both        Alternative Method:
                                    sides of the equation.
       x+2–2=5–2                                                   x25
                                                                     x 52
             x=5–2                   Simplify the LHS.
                                                                       x3
             x=3                    Simplify the RHS.




(ii)     x35
                                   Add 3 to both sides of
                                                                 Alternative Method:
                                      the equation.
       x–3+3=5+3
                                                                     x 35
             x=5+3                   Simplify the LHS.                  x 53
             x=8                     Simplify the RHS.                  x 8




         Curriculum Development Division
         Ministry of Education Malaysia                                                7
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                  TEST YOURSELF B



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




         Curriculum Development Division
         Ministry of Education Malaysia                                            8
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                          PART C:
                  SOLVING LINEAR EQUATIONS IN
                         THE FORMS OF
                                                           x
                              ax = b         AND             b
                                                           a




                                 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
                                          x
                                      (b)  b
                                          a

                              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
                                            x
                                        (b)  b
                                            a
      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.


        Curriculum Development Division
        Ministry of Education Malaysia                                                   9
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



                                              PART C:
                               SOLVING LINEAR EQUATION
                                                          x
                                     ax = b AND             b
                                                          a


                                            EXAMPLES



Solve the following equations.

                                                   m
(i) 3m = 12                                 (ii)     4
                                                   3



Solutions:



(i)    3  m = 12
                                                                 Alternative Method:
       3  m 12
                              Divide both sides of
         3    3                 the equation by 3.                    3m  12
                                                                          12
               12                                                      m
          m                    Simplify the LHS.                          3
                3                                                      m4
          m=4                   Simplify the RHS.




         m
(ii)       4
         3
                               Multiply both sides of            Alternative Method:
       m                        the equation by 3.
         3  43                                                     m
       3                                                                4
                                                                      3
                                 Simplify the LHS.                    m  3 4
          m = 4 3
                                                                      m  12
          m = 12                 Simplify the RHS.




          Curriculum Development Division
          Ministry of Education Malaysia                                               10
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                         TEST YOURSELF C

Solve the following equations.

1.    2p = 6                       2.    5k = – 20         3.    – 4h = 24




4.    7l  56                      5.     8 j  72        6.     5n  60




7.    6v  72                      8.    7 y  42         9.    12z  96




       m                                     r                     w
10.      4                        11.         =5          12.       = –7
       2                                     4                     8




           t                                  s                       u
13.         8                    14.          9         15.          6
           8                                 12                       5




           Curriculum Development Division
           Ministry of Education Malaysia                                      11
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                           PART D:
                    SOLVING LINEAR EQUATIONS IN
                           THE FORM OF
                                            ax + b = c



                                  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.




                      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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                   12
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



                                           PART D:
            SOLVING LINEAR EQUATIONS IN THE FORM OF ax + b = c


                                          EXAMPLES



Solve the equation 2x – 3 = 11.

Solution:

Method 1

                   2x – 3 = 11               Add 3 to both sides of
                                                                       Alternative Method:
                                                the equation.
               2x – 3 + 3 = 11 + 3
                                                                          2 x  3  11
                       2x = 14               Simplify both sides of          2 x  11  3
                                                the equation.
                                                                             2 x  14
                       2 x 14
                                                                               14
                        2   2                  Divide both sides of           x
                                                the equation by 2.               2
                                 14                                           x2
                            x
                                  2            Simplify the LHS.

                            x=7                Simplify the RHS.



Method 2

                    2x  3  11

                    2 x 3 11                   Divide both sides of
                                                                     Alternative Method:
                     2 2 2                      the equation by 2.
                                                                         2 x  3  11
                            3 11                                         2 x 3 11
                       x                      Simplify the LHS.             
                            2 2                                           2 2 2
                                                                                   11 3
                       3 3 11 3                      3                         x 
                  x                       Add
                                                     2
                                                       to both sides                2 2
                       2 2 2 2
                                                                                   14
                                                  of the equation.             x
                               14                                                   2
                            x                                                  x7
                                2
                                              Simplify both sides of
                            x7                  the equation.



        Curriculum Development Division
        Ministry of Education Malaysia                                                   13
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                      TEST YOURSELF D



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




         Curriculum Development Division
         Ministry of Education Malaysia                                      14
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                          PART E
                  SOLVING LINEAR EQUATIONS IN
                         THE FORM OF

                                          x
                                            bc
                                          a



                                 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
           x
        of  b where a, b, c are integers and x is an unknown.
           a




                      TEACHING AND LEARNING STRATEGIES

      Pupils face difficulty when solving linear equations in one unknown by solving
                               x
      equations in the form of  b where a, b, c are integers and x is an unknown.
                               a


      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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                   15
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



                                           PART E:
                                                                          x
            SOLVING LINEAR EQUATIONS IN THE FORM OF                         bc
                                                                          a
                                          EXAMPLES



                     x
Solve the equation      4  1.
                     3

Solution:

Method 1

                        x
                           4 1
                        3

                     x
                       44 = 1 + 4            Add 4 to both sides of         Alternative
                     3                            the equation.                Method:

                            x                                                 x
                              5               Simplify both sides of            4 1
                            3                                                 3
                                                  the equation.                   x
                        x                                                           1 4
                           3  5 3                                              3
                        3                      Multiply both sides of             x
                                                the equation by 3.                  5
                                                                                  3
                            x  5 3
                                                                                  x  3 5
                            x = 15           Simplify both sides of the           x  15
                                                     equation.
Method 2
                                              Multiply both sides of
                 x    
                   4   3  1 3            the equation by 3.
                 3    

                x                                Expand the LHS.
                   3  4  3  1 3
                3
                                              Simplify both sides of
                        x  12  3               the equation.

                 x – 12 + 12 = 3 + 12         Add 12 to both sides of
                                                  the equation.
                            x  3  12
                                              Simplify both sides of
                            x  15               the equation.



        Curriculum Development Division
        Ministry of Education Malaysia                                                       16
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



                                         TEST YOURSELF E


Solve the following equations.

      m                                  b                       k
1.      35                       2.      2 1           3.      27
      2                                  3                       3




            h                                 h                  m
4.    3+      =5                   5.    4+     =6         6.      1  2
            2                                 5                  4




           h                             k                            h
7.    2     5                    8.      +3=1            9.    3     2
           4                             6                            5




10.   3 – 2m = 7                                  m        12.   12 + 5h = 2
                                   11.       3     7
                                                  2




           Curriculum Development Division
           Ministry of Education Malaysia                                      17
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                          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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                   18
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



                                           PART F:
                                  FURTHER PRACTICE


                                          EXAMPLES




Solve the following equations:                                     Alternative Method:
(i) – 4x – 5 = 2x + 7                                                  4x  5  2x  7
                                                                     4x  2x  7  5
                                                                          6 x  12
Solution:                                                                        12
                                                                             x
                                                                                 6
                                                                             x  2
Method 1

        4x  5  2x  7                   Subtract 2x from both sides of the equation.
   –4x – 2x – 5 = 2x – 2x + 7
        6x  5  7                           Simplify both sides of the equation.
     6x  5  5  7  5
                                              Add 5 to both sides of the equation.
            6 x  12
           6 x 12
                                             Simplify both sides of the equation.
           6      6
               x  2                       Divide both sides of the equation by –6.



Method 2

         4x  5  2x  7

    – 4x – 5 + 5 = 2x + 7 + 5                   Add 5 to both sides of the equation.

            – 4x = 2x + 12                      Simplify both sides of the equation.

      – 4x – 2x = 2x – 2x + 12
                                           Subtract 2x from both sides of the equation.
            – 6x = 12
                                               Simplify both sides of the equation.
             6 x 12
                  
            6 6                            Divide both sides of the equation by – 6.
                x  2


        Curriculum Development Division
        Ministry of Education Malaysia                                                    19
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



(ii) 3(n – 2) – 2(n – 1) = 2 (n + 5)
                                                        Expand both sides of the equation.
        3n – 6 – 2n + 2 = 2n + 10
                                                                 Simplify the LHS.
                  n – 4 = 2n + 10

             n – 2n – 4 = 2n – 2n + 10             Subtract 2n from both sides of the equation.

                – n – 4 = 10

             – n – 4 + 4 = 10 + 4                       Add 4 to both sides of the equation.

                    – n = 14

                     n 14                           Divide both sides of the equation by – 1.
                        
                    1 1
                      n  14




                                          Alternative Method:

                                       3(n  2)  2(n  1)  2(n  5)
                                          3n  6  2n  2  2n  10
                                                     n  4  2n  10
                                                        n  14
                                                         n  14




        Curriculum Development Division
        Ministry of Education Malaysia                                                            20
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



                2x  3 x  1
(iii)                        3
                   3      2
              2x  3 x  1 
            6               6(3)                  Multiply both sides of the equation by the
              3         2                                            LCM.
          2x  3   x  1 
        6          6         6(3)
          3   2 
          2(2 x  3)  3( x  1)  18                            Expand the brackets.
               4 x  6  3 x  3  18
                         7 x  3  18                                Simplify LHS.

                     7 x  3  3  18  3
                                                         Add 3 to both sides of the equation.
                             7 x  21
                            7 x 21                  Divide both sides of the equation by 7.
                               
                             7   7
                              x3




                                            Alternative Method:

                                            2x  3 x  1
                                                            3
                                               3         2
                                         2x  3 x  1 
                                       6                   3 6
                                         3            2 
                                      2(2 x  3)  3( x  1)  18
                                           4 x  6  3 x  3  18
                                                     7 x  3  18
                                                          7 x  18  3
                                                          7 x  21
                                                            21
                                                          x
                                                             7
                                                          x3




        Curriculum Development Division
        Ministry of Education Malaysia                                                             21
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations




                                      TEST YOURSELF F

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)                              3x 9
                                                4.         
                                                          4 2




     x 2 5                                               x x
5.                                            6.         2
     2 3 6                                               3 5




     y      13 y                                         x  2 x 1 9
7.     5                                      8.                
     2       6                                             3     4   2




     2 x  5 3x  4                                      2x  7     x7
9.                 0                          10.             4
        6      8                                           9         12




        Curriculum Development Division
        Ministry of Education Malaysia                                                     22
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations



                                               ANSWERS


TEST YOURSELF B:

1.    x=5                       2.    x=6                3.     x=9
4.    x = –2                    5.    x = –7             6.     x = –3
7.    x = 48                    8.    x = –45            9.     x = –50
10.   x = –111                  11.   x = 57             12.    x = –87
13.   x = –107                  14.   x = 247            15.    x = –115


TEST YOURSELF C:

1.    p=3                       2.    k=–4               3. h = –6

4.    l=8                       5.    j=–9               6. n = 12

7.    v = 12                    8.    y=–6               9.    z=8

10.   m=8                       11. r = 20               12. w = – 56

13.   t = – 64                  14. s = 108              15. u = 30


TEST YOURSELF D:

1.    m=2                       2.    p=4                3.    k=2

                                           3                      3
4.    m=3                       5. y                    6. p 
                                           2                      4

7.    p=2                       8. k = 2                 9.    x = –1

10. p = −1                      11. p = 3                12. m = 1


TEST YOURSELF E:

1.    m=4                       10. b = 9                11. k = 15

4.    h=4                       5.    h = 10             6.    m = 12

7.    h = 12                    8.    k = −12            9.    h=5

10. m = −2                      11. m = −8               12. h = −2


         Curriculum Development Division
         Ministry of Education Malaysia                                    23
Basic Essentials Additional Mathematics (BEAMS) Module
UNIT 4: Linear Equations


TEST YOURSELF F:
                                                                   1
1.   x=−2                  2.    x = − 17                3. n         4.   x=6
                                                                  14

5.   x=3                   6.    x = 15                  7.   y=3      8.   x=7

9.   x = −8                10.   x = 19




        Curriculum Development Division
        Ministry of Education Malaysia                                            24
       Basic Essential

Additional Mathematics Skills




             UNIT 5

             INDICES

 Unit 1:
 Negative Numbers




           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
                                         m n
                 Verifying a  a  a
                               m    n
          3.0                                                                         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

                                     mn
                 Verifying a  a  a
                            m   n
          1.0                                                                         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


                  Verifying (a )  a
                                m n         mn
           1.0                                                                     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
                                       1
                              a n 
           4.0    Verifying            an                                          15
                               1
           5.0    Verifying   an   na
                                                                                   16


Activity                                                                           20



Answers                                                                            22
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




                                      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:
                                            PART 1
                             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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                        1
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




                                                PART A:
                                                INDICES I



                                     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 a m  a n  a m n ;

               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.




                         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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                      2
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



                                          LESSON NOTES A




1.0   Expressing Repeated Multiplication As an and Vice Versa

           (i)      32  3  3                                      32 is read as
                                                             ‘three to the power of 2’
                       2 factors of 3
                                                                          or
                                                           ‘three to the second power’.

           (ii)     (4)3  (4)(4)(4)                                     index
                                                                      32
                              3 factors of (4)
                                                                    base

           (iii)      r3  r  r  r
                           3 factors of r
                                                            (a) What is 24?
                                                            (b) What is (−1)3?
                                                            (c) What is an?
            (iv)     (6  m) 2  (6  m)( 6  m)

                                   2 factors of (6+m)



2.0   Finding the Value of an

            (i )    25  2  2  2  2  2
                        32


            (ii )   (  5)3  ( 5)(5)(5)
                              125

                          4
                     2    24
            (iii)       4
                     3    3
                            2 2 2 2 
                                      
                            3 3 3 3 
                            16
                         
                             81




        Curriculum Development Division
        Ministry of Education Malaysia                                                    3
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices


                                       m n
      Verifying a  a  a
                      m      n
3.0

           (i)       23  24  (2  2  2)  (2  2  2  2)
                                  27          234


           (ii )      7  7 2  7  (7  7 )
                                  73          7 12


           (iii )     ( y  1) 2 ( y  1)3  [( y  1)( y  1)] [( y  1)( y  1)( y  1)]
                                               ( y  1)5    ( y  1) 23



                                                       am  an  amn




4.0   Simplifying Multiplication of Numbers, Expressed In Index Notation with the Same
      Base

           (i)      6 3  6 4  6  6 3 41
                                      68


           (ii ) (5) 3  (5) 8  (5) 38
                                       (5)11

                                 5            15
                     1 1 1
           (iii )         
                     3 3 3
                                              6
                                       1
                                       
                                       3




        Curriculum Development Division
        Ministry of Education Malaysia                                                        4
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



5.0   Simplifying Multiplication of Algebraic Terms, Expressed In Index Notation with the
      Same Base

          (i)      p 2  p 4  p 2 4  p 6
                                                                                  (ab) 5  a 5 b 5
                                                                                  Conversely,
          (ii ) 2 w9  3w11  w 20  6 w911 20  6 w 40                         a 5 b 5  (ab) 5


          (iii ) (ab) 3  (ab) 2  ab
                                           3 2
                                                   (ab) 5                              4
                                                                                  s
                                                                                               4
                                                                                         s
                                                                                     4
                                                                                  t    t
                        3                  31              4
                s s     s                     s                           Conversely,
          (iv )                             
                t t     t                     t
                                                                                               4
                                                                                   s4  s 
                                                                                      
                                                                                   t4  t 




6.0   Simplifying Multiplication of Numbers, Expressed In Index Notation with Different
      Bases

                                                                                                 Note:
          (i) 34  38  2 3  348  2 3  312  2 3                                     Sum up the indices
                                                                                          with the same
          (ii ) 53  5 7  714  7 3  537  7143  510  717                           base.
                                                                                         numbers with
                                                                                          different bases
                       3       2       4             3 2       4    5        4           cannot be
                 1 1 3       1                       3   1 3
          (iii )                                                      simplified.
                  2  2 5     2                       5    2 5




7.0   Simplifying Multiplication of Algebraic Terms Expressed In Index Notation with
      Different Bases

          (i) m 5  m 2  n 5  n 5  m 52  n 55  m 7 n10


          (ii) 3t 6  2s 3  5r 2  30t 6 s 3 r 2

                   2    4    1      4 13 3  4 4 3
          (iii )     p  p3  q3     p q     p q
                   3    5    2     15       15




        Curriculum Development Division
        Ministry of Education Malaysia                                                                         5
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




                                      EXAMPLES & TEST YOURSELF A


1. Find the value of each of the following.

     (a)     35  3  3  3  3  3               (b)   63 
                 243




     (c)     (4) 4                              (d)   1
                                                               5
                                                          
                                                        5



     (e)      3
                        3
                                                  (f)      1
                                                                   2

                                                     2  
              4                                          5



     (g)      74                                (h)      2
                                                                   5

                                                           
                                                           3




2. Simplify the following.

     (a)     3m 3  4m 2  12m 3 2               (b)   5b 2  3b 4  b 
                             12m 5




     (c)     2 x 2  (3x 4 )  3x 3             (d)   7 p 3  (2 p 2 )  ( p)3 




           Curriculum Development Division
           Ministry of Education Malaysia                                              6
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




3. Simplify the following.

     (a)     43  32  64  9                     (b)   (3) 2  23  2 2 
                      576




     (c)     (1)3  (7) 4  (7)3              (d)          2
                                                        1 1  4
                                                                       3           2

                                                              
                                                         3  3  5 




     (e)     2  23  52  54                    (f)              3       2           2
                                                         2 2  2 2
                                                                
                                                         3 7  3 7




4. Simplify the following.

     (a)     4 f 4  3g 2  12 f 4 g 2            (b)   (3r ) 2  2r 3  3s 2 




     (c)     (w) 3  (7w) 4  (3v) 3          (d)              2
                                                        3  1  4 
                                                                               3           2

                                                         h  k   k  
                                                        7  5  5 




           Curriculum Development Division
           Ministry of Education Malaysia                                                      7
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




                                             PART B:
                                          INDICES II



                                    LEARNING OBJECTIVES


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

                                     mn
                    verify a  a  a
                            m   n
               1.                        ;

               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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                      8
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



                                                                  LESSON NOTES B


                                                    mn
      Verifying a  a  a
                     m             n
1.0
                                            1        1       1
                                        2 2 2 2 2
               (i) 2  2 5         3
                                       / / /
                                                21 21 2 1                                        (a) What is 25 ÷ 25?
                                           2 2
                                                    2 53                                         (b) What is 20?
                                                                                                   (c) What is a0?
                                                1        1
                            555555555
               (ii) 5  5  / /
                             9         2

                                   51 51
                          5 7
                                 5 9 2
                                                                          1       1
                                               (2  p )(2  p )(2  p )
               (iii) (2  p ) 3  (2  p ) 2 
                                                 1
                                                   (2  p )(2  p ) 1
                                                                   (2  p)      ( 2  p ) 3 2

                                                                                                                Note:
                                                                                                         a  a m  a mm  a 0
                                                                                                           m


                                                                                                                  am
                                                                                                         am  am    1
                                                                                                                  am
                                                                 am  an  amn
                                                                                                                 a0  1




2. 0 Simplifying Division of Numbers, Expressed In Index Notation with the Same Base


               (i)           48  4 2  48  2
                                                     46
               (ii)              79  73  7 2  79  3 2
                                                                    74
                                  510
               (iii)                3
                                       510  3
                                  5
                                       57
                                    312
               (iv)                        312  4  5
                                  3 3
                                   4    5


                                           33

        Curriculum Development Division
        Ministry of Education Malaysia                                                                                           9
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



3.0   Simplifying Division of Algebraic Terms, Expressed In Index Notation with the Same
      Base

               (i)      n 6  n 4  n 6 4  n 2


                         20k 7
               (ii)          3
                                4k 73  4k 4
                          5k

                           8h 3    8         8
               (iii)          2
                                   h 32   h
                           3h       3         3




4.0   Simplifying Multiplication of Numbers, Expressed In Index Notation With Different
      Bases


                          REMEMBER!!!

                            Numbers with
                        different bases cannot
                             be simplified.




5.0   Simplifying Multiplication of Algebraic Terms, Expressed In Index Notation with
      Different Bases


                                        9h15
                (i) 9h15  3h 4 k 6 
                                       3h 4 k 6
                                      3h15 4     3h11    h11
                                                      3 6
                                        k6         k6     k

                        48 p 8 q 6     4 83 6  2
                (ii )        3 2
                                        p q
                        60 p q         5
                                     4
                                    p5q 4
                                     5



        Curriculum Development Division
        Ministry of Education Malaysia                                                10
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




                                        EXAMPLES & TEST YOURSELF B



1. Find the value of each of the following.

     (a)     12 5  12 3  12 53                  (b)   910  93  9 
                          12   2


                          144
     (c)     8 9                                   (d)   2
                                                               18
                                                              2
                                                                         12
                                                            
             83                                          3  3

     (e)     (5) 20                               (f)   318  310
                                                                  
             (5)18                                        324



2. Simplify the following.

     (a)     q12  q 5  q125                     (b)   4 y9  8 y7 
                          q7

     (c)     35m10                                 (d)   214 b11
                                                                
             15m8                                         28 b8




3. Simplify the following.

     (a)           36m9 n 5 9 94 51              (b)   64c16d 13
                            m n                                   
                    8m 4 n  2                            12c 6 d 7
                            9
                            m5 n 4
                            2


     (c)     4 f 6  6 fg 9                        (d)   8u 9  7v8  3u 4
                                                                          
               12 f 4 g 3                                     12u 6v5




           Curriculum Development Division
           Ministry of Education Malaysia                                      11
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




                                            PART C:
                                          INDICES III


                                      LEARNING OBJECTIVES


                  Upon completion of Part C of the module, pupils will be able to:

                       derive (a )  a ;
                                m n   mn
                  1.

                  2.   simplify
                       (a) numbers;
                       (b) algebraic terms, expressed in index notation raised to a power;

                               n     1
                  3.   verify a         ; and
                                      an

                                1
                  4.   verify a n  n a .




                           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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                         12
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



                                              LESSON NOTES C



1.0   Verifying     (a m )n  a mn

           (i)    (23 ) 2  23  23
                           23  3
                           26           2 3 2


           (ii ) (39  2 5 ) 3  (39  2 5 )(39  2 5 )(39  2 5 )
                                   39  9  9  2 5  5  5
                                   327  215  39 3  2 5 3

                              2
                   113             113     113   
           (iii )  4              4             
                   15              15      154   
                                                 
                                     113  3   
                                    4 4      
                                     15        
                                               
                                      116           113 2
                                             
                                      158           154 2



                                                               (a m ) n  a mn


2. 0 Simplifying Numbers Expressed In Index Notation Raised to a Power

            (i) (102 )6  102  6  1012


            (ii) (27  93 )5  27  5  93  5  235  915

                        5
            (iii)  43   (710 )2  43  5  710  2  415  720
                   
                   

                        3   13  3
                  613              639
            (iv)        6       
                  58     58  3    524
                      


        Curriculum Development Division
        Ministry of Education Malaysia                                           13
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



3.0   Simplifying Algebraic Terms Expressed In Index Notation Raised to a Power


          (i)     (3 x 2 ) 5  35 x 25
                            35 x10


          (ii )   (e 2 f 3 g 4 ) 5  e 25 f 35 g 45
                                      e10 f 15 g 20

                             4            4
                 1        1
          (iii )  a 3b     a 34 b14
                 5        5
                            a12b 4
                          
                             54
                            a12b 4
                          
                             625
                             1 12 4
                               a b
                            625

                                 5
                  2m 4        (2) 5 m 45
          (iv ) 
                 n3          
                             
                                   n 35
                                                                                 Note:
                                 (2) 5 m 20
                                                                      A negative number raised to
                                     n15                                an even power is positive.
                                  32m 20
                                                                      A negative number raised to
                                    n15
                                                                        an odd power is negative.
                                      m 20
                                 32 15
                                       n

                    (2 p 3 ) 5  4 p 6 q 7 2 5  4 p 35  p 6  q 7
          ( v)                                   
                        12 p 3 q 2           12          p 3q 2
                                                32 p1563 q 72
                                              
                                                       3
                                                     18 5
                                                 32 p q
                                              
                                                     3
                                                 32 18 5
                                                   p q
                                                  3




        Curriculum Development Division
        Ministry of Education Malaysia                                                               14
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




                n        1
4. 0 Verifying a 
                          an
                                         3 3 3 3
                  (i)      34  36 
                                      3 3 3 3 3 3
                                       1
                                      2  3 4  6  3 2
                                      3
                                       1
                                3 2  2
                                      3

                                        77
                  (ii )    7 2  75 
                                     77777
                                     1
                                    3  7 2 5  7 3
                                     7


                                                        1
                                               a n 
                                                        an




       Alternative Method
                                  104  10 000                         1000
                                                               Hint:         100
                                  10  1000
                                        3                                ?

                                  102  100
                                  101  10
                                  100  1
                                               1    1
                                  101            1
                                              10 10
                                                 1     1
                                  102             2
                                              100     10
                                  
                                              1
                                  10n 
                                             10n




        Curriculum Development Division
        Ministry of Education Malaysia                                              15
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices


                     1
5.0   Verifying     an     na

                           2
                   1                               1
                                                         2 
         (i)       32              32                                  31
                   
                   
                                                 2
                                    1
                                    32                                     3
                                    
                                    
                                                                                                                Take square root on both sides
                                            2
                            1                                                                                        of the equation.
                            32                                               3
                                       
                                       
                                1  1 
                                3 2  3 2                                      3
                                 
                                 
                                                  1
                                                 32                               3
                               5
                    1                                          1
                                                                   5 
         (ii)       25                                       25                                    21
                    
                    
                                                                          5
                                                                 1
                                                                 25                                 2
                                                                 
                                                                 
                                                                     5
                                                          1
                                                 5        25                                    5
                                                                                                       2
                                                          
                                                          
                                                                                                                                                       1

                      1        1              1          1          1                                             (a) What is 4 2 ?
                   5  25       2 5            2 5        2 5        2 5                     5
                                                                                                            2                                          3
                                                                                                                  (b) What is 4 2 ?
                                                                              
                                                                                        1                                                               m

                                                                                       25                  5
                                                                                                                2           (c) What               is a n   ?
                                    p
                     1                                         1
                                                                    p
         (iii )     m p                                   m   p
                                                                                      m1
                        
                        
                                                         p
                                   1
                                p m p                                        
                                                                                           p
                                                                                               m
                                      
                                      
                                                                      1
                                                                                               p
                                                                 m    p
                                                                                                  m

                                                                                                                                       Note:
                                                                                                                                   1
                                                                                                                               a   n
                                                                                                                                          n
                                                                                                                                                   a
                                                                           1
                                                                                                                                            a
                                                                                                                                   m
                                                                         a n a
                                                                           n
                                                                                                                               a   n
                                                                                                                                              n
                                                                                                                                                       m




        Curriculum Development Division
        Ministry of Education Malaysia                                                                                                                          16
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



                                                  EXAMPLES & TEST YOURSELF C


1. Find the value of each of the following.

     (a)                                                      (b)
               2  5 3
                          2       53
                                                                     [(1) 2 ] 3 
                           215  32768

     (c)                  2                                   (d)                          3
               23                                                   3  2 
               2
              7           
                                                                        
                                                                    5  
                                                                             


     (e)       32                                         (f)
                               3

                    
                                                                         
                                                                                           4
               5 
                                                                     23 2 
                                                                      
                                                                           
                                                                            



2. (a)       Simplify the following.

     (i)       2   6
                         32   
                               4
                                      2 64  3 24          (ii)   2   5 
                                                                       6 4           3 2
                                                                                           
                                    2 24  38




     (iii)
               4   4 
                    2 3        1 5
                                         
                                                              (iv)   3 2
                                                                             2

                                                                         
                                                                                       3


                                                                     4 5



     (v)        7 3
                          3          2                        (vi)               2
                                                                                      32  4 4
                                                                                                  4
                                                                                                  
                                                                5                         
                                                                                    5          
                4 7                                               12                       




           Curriculum Development Division
           Ministry of Education Malaysia                                                               17
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices


2.   (b)     Simplify the following.

     (i)
              2 x    3 5
                              (215 )( x 35 )
                                                   (ii)   x y 
                                                               4       7 6
                                                                             

                              25 x15
                              32 x15

     (iii)
              w   2
                        w12        3            (iv)
                                                          4 y     9
                                                                        8y7     7
                                                                                      




                                                          2m n 3mn 
     (v)                                  2        (vi)
               36 p 9 q 5                                                               4 4
                                                                       3 2
                                                                                                
              
               9 p8q 6  
                          




3.   Simplify the following expressions:

     (a)                                           (b)             1
              2 5         
                                1                         3
                                                                       
                                25                        4
                               1
                             
                               32


     (c)
               x 
                             4                    (d)    2st 4
               2                                                
               3y                                       6s 1t 5
                  




     (e)                              3           (f)                           2
               m 2 n 1                                  8ab 2 c 3 
                                                        3 6 
                                                           2a b                     
               2m 3 k 2          
                                                                    




           Curriculum Development Division
           Ministry of Education Malaysia                                                           18
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices


4.   Find the value of each of the following.

     (a)                1                          (b)            5
               64 3  3  64                             100 2 
                             4




     (c)        
                    3                              (d)        1           1
             81     4
                                                         3  27 
                                                              2           2




              a  (a
     (e)            1                      1       (f)                4
                10 5         3  2
                                   ) (a m ) m                 1 
                                                          3     
                                                               27 




           Curriculum Development Division
           Ministry of Education Malaysia                                     19
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



                                                   ACTIVITY


            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!

                410
       1.             
             4 2  45

             P    40                      O   43         R     417         T   413



       2.    107  102  53  5 2 

             T 10145 5                    O   105 56     N     105 55      B   10145 6



             2 2  32
       3.             
                42
                  22                          32               32              42
             D                            E               N                O
                  4                           22               42              3



       4. 2 y x  8 y x 
             9 3     2




               y7 x2                          4 y 11           y1 x 2          4y7
             M                            A              L                 K
                4                              x4               4               x2

       5.         2   5
                            32   
                                  4
                                      

             A 2 3                           2 9  36         2 20  36       2 9  38
                20 8
                                          N              T                 S


       6. m  m  n  n 
           5   2   2   4




             T    m7 n8                   U   m10n 8      L    m7 n 6      E   m10n 6



        Curriculum Development Division
        Ministry of Education Malaysia                                                    20
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices


                    3           4            2       3
             2 2  2 2
       7.            
              5 5 5 5

                            12                                  2                         6                         5
             F      2                          A       2                V       2                   E   2
                                                                                                         
                    5                                  5                        5                       5



                                5
                     72 
       8.            3 
                    4 
                     

                     710                                77                       71 0                    77 
             Y       15                        R        8
                                                         4     
                                                                           M        8                  A    15 
                    4                                                              4                      4 
                                                                                                         



             25a 9 b 5
       9.              
             5a 6 b 3

             L 15a15b 8                          I       5a 3b 8                S   5a 3b 2               T   15a 6 b 5


                        2           3        2       5
                   1 1  2  2
       10.                 
                    3 3  5  5

                            5           10                      6   7                     5       7                 6     10
                    1  2                             1  2                   1  2                  1  2
             P                               E                          I                     R      
                    3  5                             3  5                   3  5                  3  5



                 12 p 6 q 7
       11.                          
                  3 p 3q 2

                     p3q5                                                              1
             Y                                   A 4 p3q5                       R                         D    3 p9q9
                      3                                                             3 p9q9


             Congratulations! You have completed this activity.

 1       2              3                    4              5           6   7            8            9       10          11


The WONDERWORD IS: ........................................................



        Curriculum Development Division
        Ministry of Education Malaysia                                                                                         21
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




                                             ANSWERS

TEST YOURSELF A:

1.

     (a) 243                                       (b)    216

     (c) 256                                       (d)     1
                                                          3125
     (e)         27                                (f)       21
                                                         4
                 64                                          25

     (g)     2401                                 (h)    32
                                                          243


2.

     (a) 12m5                                       (b)   15b 7
     (c)      18x 9                                (d)   14 p 8



3.

     (a)     576                                    (b)   288

     (c) 823543                                     (d)    16
                                                          6075

     (e)     250 000                                (f)         256
                                                          
                                                               83 349




           Curriculum Development Division
           Ministry of Education Malaysia                               22
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




4.

     (a)     12 f 4 g 2                             (b)    54r 5 s 2


     (c)     64 827 w7 v 3                          (d)      144
                                                                   h2k 5
                                                            153125




TEST YOURSELF B:

1.


     (a) 144                                         (b)    531 441

     (c)     262 144                                 (d)    64
                                                            729
     (e)    25                                       (f)    81



2.

     (a)     q7                                      (b)    1 2
                                                              y
                                                            2

     (c)     7 2                                     (d)    64b3
               m
             3



3.

     (a)          9 5 4                              (b)    16 1 0 6
                    m n                                        c d
                  2                                          3

     (c)     2 f 3g6                                 (d)    14u 7 v 3




           Curriculum Development Division
           Ministry of Education Malaysia                                  23
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices



TEST YOURSELF C:

1.

      (a) 32768                                   (b)     1

      (c)      64                                 (d)     3
                                                               6
                                                                 729
                                                            
               2401                                       5   15625
      (e)          36     729                     (f)
               
                   5 3
                       
                          125
                                                          2 24  16 777 216


2. (a)


      (i)       2 24  3
                           8
                                                        (ii)       224  56

      (iii)        411                                  (iv)        32
                                                                   2(53 )
      (v)           7(32 )                              (vi)       36 (414 )
                
                     43                                               52



2.   (b)

      (i)       32x15                                   (ii)       x 24 y 42

      (iii)      1                                      (iv)          y1 4
                w30
                                                                      27
      (v)         p
                           2                            (vi)       162m 7 n18
               16 
                 q
                  




            Curriculum Development Division
            Ministry of Education Malaysia                                      24
Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 5: Indices




3.

     (a)      1     1                              (b)    4
                5
                  
              2     32                                    3

     (c)           y8                              (d)    1  s2   
             81                                                   
                   x4
                                                          3  t9
                                                            
                                                                   
                                                                   
     (e)      8k 6 m 3 n 3                         (f)     1  a 4c6   
                                                                      
                                                          16  b16
                                                             
                                                                       
                                                                       




4.

     (a)      4                                   (b)    100000

     (c)      1                                    (d)    9
              27

     (e)                                           (f)    1
              a5
                                                          81




ACTIVITY:

The WONDERWORD is ONEMALAYSIA




           Curriculum Development Division
           Ministry of Education Malaysia                                  25
       Basic Essential

Additional Mathematics Skills




           UNIT 6
      COORDINATES
              AND
 Unit 1:
 GRAPHS OF FUNCTIONS
 Negative Numbers




        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




                                          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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                          1
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions




                                            PART A:
                                      COORDINATES



                                     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.




                          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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                        2
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions


                                               PART A:
                                             COORDINATES


                                             LESSON NOTES



1.                                   y
                                                          ●P
                 Start from the
                    origin.
                                                            k units

                                                                      x
                                     O         h units

                                     Coordinates of P = (h, k)


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
                                                    left             –
                                   h
                                                   right             +
                                                     up              +
                                   k
                                                  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).




         Curriculum Development Division
         Ministry of Education Malaysia                                                         3
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART A1:          State the coordinates of the given points.



                               EXAMPLES                                                 TEST YOURSELF
                               EXAMPLES                                                  TEST YOURSELF

     1.                                                                    1.
                                   y                                                          y
                Start from             4                                                       4
                                                     A
                the origin,            3
                                                •            Next, move
                                                                                              3
                                                                                                                  A
                                                                                                                 •
               move 2 units
                                       2                     3 units up.                      2
               to the right.
                                       1                                                      1

                     –4 –3 –2 –1    0      1     2       3   4 x                   –4 –3 –2 –1 0     1   2   3   4 x
                                   –1                                                         –1
                                   –2                                                         –2
                                   –3                                                         –3
                                   –4                                                         –4

                  Coordinates of A = (2, 3)                                     Coordinates of A =

     2.                                                                    2.
              Start from the       y                                                          y
           origin, move 3 units     4                                                          4
                                                                                     B
                to the left.           3
                                       2
                                                                                    •         3
                                                                                              2
                          B
                          •            1                                                      1

                    –4 –3 –2 –1 0          1     2       3   4 x                   –4 –3 –2 –1 0     1   2   3   4 x
                               -1                                                             –1
                                   –2          Next, move                                     –2
                                                1 unit up.
                                   –3                                                         –3
                                   –4                                                         –4

                 Coordinates of B = (–3, 1)                                     Coordinates of B =

     3.                                                                    3.
                                   y                                                          y
                Start from             4                                                       4
                the origin,            3                                                      3
               move 2 units
                                       2                                                      2
                to the left.
                                       1                                                      1

                    –4 –3 –2 –1 0          1     2       3   4 x                   –4 –3 –2 –1 0     1   2   3   4 x
                               –1                                                             –1

                               •
                               C   –2                                                         –2
           Next, move 2
           units down.
                                   –3
                                                                                        C•    –3
                                   –4                                                         –4

                Coordinates of C = (–2, –2)                                     Coordinates of C =




          Curriculum Development Division
          Ministry of Education Malaysia                                                                               4
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART A1: State the coordinates of the given points.


                               EXAMPLES                                               TEST YOURSELF
                                                                                      TEST YOURSELF
                               EXAMPLES


     4.                                                                    4.
                                  y                                                           y
                Start from            4                                                           4
                                                              Next, move
                the origin,           3                                                           3
                                                               3 units
               move 4 units
                                      2                         down.                             2
               to the right.
                                      1                                                           1

                     –4 –3 –2 –1 0            1   2   3       4 x                  –4 –3 –2 –1 0       1   2       3    4 x
                                –1                                                            –1
                                  –2                                                          –2
                                  –3                          •                               –3
                                  –4
                                                          D
                                                                                              –4                   •D
                 Coordinates of D = (4, –3)                                     Coordinates of D =

     5.                                                                    5.
              Start from the      y                                                           y
           origin, move 3 units       4                                                           4
               to the right.          3                                                           3
                                      2                                                           2
                                      1                                                           1
                                                                                                               E
                     –4 –3 –2 –1 0            1   2   •3
                                                          E
                                                              4 x                  –4 –3 –2 –1 0       1   •
                                                                                                           2       3    4 x
                                –1                                                            –1
            Do not move           –2                                                          –2
           along the y-axis
                                  –3                                                          –3
             since y = 0.
                                  –4                                                          –4


                  Coordinates of E = (3, 0)                                     Coordinates of E =

     6.                                                                    6.
                                  y                                                          y
                                      4                                                          4
                Start from
                the origin,
                                      •
                                      3   F                                                      3
               move 3 units
                    up.
                                      2
                                      1
                                                                                                 2
                                                                                                  •F
                                                                                                 1

                    –4 –3 –2 –1 0             1   2   3       4 x                 –4 –3 –2 –1 0        1   2       3   4 x
                               –1                                                            –1
                                  –2                                                         –2
                                                   Do not move
                                  –3              along the x-axis                           –3
                                  –4                since x = 0.
                                                                                             –4


                  Coordinates of F = (0, 3)                                     Coordinates of F =




          Curriculum Development Division
          Ministry of Education Malaysia                                                                                      5
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART A1: State the coordinates of the given points.


                               EXAMPLES                                              TEST YOURSELF
                                                                                     TEST YOURSELF
                               EXAMPLES


     7.                                                                   7.
                                   y                                                          y
                Start from             4                                                          4
                the origin,            3                                                          3
               move 2 units
                                       2                                                          2
                to the left.
                                       1                                                          1
                                                                                          G
                               •
                               G
                     –4 –3 –2 –1 0         1   2       3   4 x                             •
                                                                                  –4 –3 –2 –1 0            1   2   3   4 x
                                –1                                                           –1
                                   –2                                                         –2
                                   –3                                                         –3
                                   –4                                                         –4


                 Coordinates of G = (–2, 0)                                    Coordinates of G =

     8.                                                                   8.
              Start from the       y                                                          y
           origin, move 2 units     4                                                             4
                   down.               3                                                          3
                                       2                                                          2
                                       1                                                          1

                    –4 –3 –2 –1 0          1   2       3   4 x                    –4 –3 –2 –1 0        1       2   3   4 x
                               –1                                                            –1
                                                                                                      •H
                                   –2  •H                                                     –2
                                   –3                                                         –3
                                   –4                                                         –4


                 Coordinates of H = (0, –2)                                    Coordinates of H =

     9.                                                                   9.
                                   y                                                           y
                                                   J
                Start from
                                       8
                                                       •                                        8
                                                                                                                       J
                the origin,
               move 6 units
                                       6
                                                            Next, move
                                                                                                  6
                                                                                                                       •
                                       4                                                          4
               to the right.                                 8units up.
                                       2                                                          2

                    –8 –6 –4 –2 0          2   4       6   8 x                    –8 –6 –4 –2 0            2   4   6   8 x
                               –2                                                            –2
                                   –4                                                         –4
                                   –6                                                         –6
                                   –8                                                         –8


                   Coordinates of J = (6, 8)                                   Coordinates of J =




          Curriculum Development Division
          Ministry of Education Malaysia                                                                                     6
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART A1: State the coordinates of the given points.


                                  EXAMPLES                                            TEST YOURSELF
                                   EXAMPLES                                           TEST YOURSELF

     10.                                                               10.
                                        y                                                       y

                              K
                                            8        Start from
                                                                                       K   •        8

                              •             6
                                            4
                                                     the origin,
                                                    move 6 units
                                                                                                    6
                                                                                                    4
                                                     to the left.
                                            2                                                       2

                       –8 –6 –4 –2 0            2   4    6   8 x                –8 –6 –4 –2      0      2   4   6   8 x
                                  –2                                                            –2

                   Next, move           –4                                                      –4
                   6 units up.          –6                                                      –6
                                        –8                                                      –8

                  Coordinates of K = (– 6 , 6)                               Coordinates of K =

     11.                                                               11.
                                        y                                                       y
               Start from the           20                                                      20
           origin, move 15 units
                 to the left.           15                                                      15
                                        10                                                      10
                                            5                                                       5

                      –20 –15 –10 –5 0          5   10 15    20 x               –20 –15 –10 –5 0        5   10 15 20 x
                                    –5                                                        –5

       Next, move                      –10                                                     –10
        20 units                       –15                                              •L     –15
         down.
                     L        •        –20                                                     –20

                 Coordinates of L = (–15, –20)                               Coordinates of L =

     12.                                                               12.
               Start from              y                                                        y
               the origin,                  4           Next, move 4                                4
              move 3 units                               units down.
              to the right.
                                            2                                                       2



                       –4         –2        0       2        4 x                 –4     –2          0       2       4 x


                                        –2                                                      –2
                                                                                                                    •M
                                        –4              •M                                      –4

                  Coordinates of M = (3, – 4)                                Coordinates of M =




        Curriculum Development Division
        Ministry of Education Malaysia                                                                                    7
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions




                                             ACTIVITY A1


                   Write the step by step directions involving integer coordinates that
                           will get the mouse through the maze to the cheese.

                                                          y

                                                      7
                                                      6
                                                      5
                                                      4
                                                      3
                                                      2
                                                      1
                                                                                              x
                          –6 –5 –4        –3 –2 –1    0        1   2   3   4   5   6      7
                                                     –1
                                                     –2
                                                     –3
                                                     –4
                                                     –5
                                                     –6




        Curriculum Development Division
        Ministry of Education Malaysia                                                            8
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART A2: Plot the point on the Cartesian plane given its coordinates.



                        EXAMPLES                                   TEST YOURSELF
.                        EXAMPLES                                  TEST YOURSELF


     1.              Plot point A (3, 4)                   1.     Plot point A (2, 3)
                                 y               A                          y
                                     4
                                     3
                                                 •                           4
                                                                                3
                                     2                                          2
                                     1                                          1

                  –4 –3 –2 –1 0          1   2   3   4 x        –4 –3 –2 –1 0       1   2   3   4 x
                             –1                                            –1
                                 –2                                         –2
                                 –3                                         –3
                                 –4                                         –4




     2.             Plot point B (–2, 3)                   2.    Plot point B (–3, 4)
                                 y                                          y
                                     4                                          4
                           B
                         •           3                                          3
                                     2                                          2
                                     1                                          1

                  –4 –3 –2 –1 0          1   2   3   4 x        –4 –3   –2 -1 0     1   2   3   4 x
                             –1                                              –1
                                 –2                                         –2
                                 –3                                         –3
                                 –4                                         –4




     3.           Plot point C (–1, –3)                    3.   Plot point C (–1, –2)
                                 y                                           y
                                     4                                        4
                                     3                                          3
                                     2                                          2
                                     1                                          1

                  –4 –3 –2 –1 0          1   2   3   4 x        –4 –3   –2 –1 0     1   2   3   4 x
                             –1                                              –1
                                 –2                                         –2

                             C • –3                                         –3
                                 –4                                         –4




          Curriculum Development Division
          Ministry of Education Malaysia                                                              9
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART A2: Plot the point on the Cartesian plane given the coordinates.


.                       EXAMPLES                                       TEST YOURSELF
                            EXAMPLES                                     TEST YOURSELF


     4.             Plot point D (2, – 4)                    4.    Plot point D (1, –3)
                              y                                                 y
                                  4                                              4
                                  3                                                3
                                  2                                                2
                                  1                                                1

                  –4 –3   –2 –1 0         1   2    3   4 x        –4 –3    –2 –1 0     1   2   3   4 x
                               –1                                               –1
                              –2                                               –2
                              –3                                               –3
                              –4              •D                               –4




     5.              Plot point E (1, 0)                     5.        Plot point E (2, 0)
                              y                                                y
                                  4                                                4
                                  3                                                3
                                  2                                                2
                                  1                                                1
                                          E
                  –4 –3 –2 –1 0       •   1   2    3   4 x        –4   –3 –2 –1 0      1   2   3   4 x
                             –1                                                –1
                              –2                                               –2
                              –3                                               –3
                              –4                                               –4




     6.              Plot point F (0, 4)                     6.        Plot point F (0, 3)
                              y                                                 y
                                  •
                                  4
                                      F                                          4
                                  3                                                3
                                  2                                                2
                                  1                                                1

                  –4 –3 –2 –1 0           1   2    3   4 x        –4 –3 –2 –1 0        1   2   3   4 x
                             –1                                              –1
                              –2                                               –2
                              –3                                               –3
                              –4                                               –4




          Curriculum Development Division
          Ministry of Education Malaysia                                                                 10
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART A2: Plot the point on the Cartesian plane given the coordinates.



                        EXAMPLES                                  TEST YOURSELF
                         EXAMPLES                                 TEST YOURSELF


     7.             Plot point G (–2, 0)                 7.     Plot point G (– 4,0)
                              y                                             y
                                  4                                          4
                                  3                                            3
                                  2                                            2
                                  1                                            1
                        G
                         •
                   –4 –3 –2 –1 0       1   2   3   4 x         –4 –3 –2 –1 0       1   2   3   4 x
                              –1                                          –1
                              –2                                           –2
                              –3                                           –3
                              –4                                           –4




     8.             Plot point H (0, – 4)                8.     Plot point H (0, –2)
                              y                                             y
                                  4                                          4
                                  3                                            3
                                  2                                            2
                                  1                                            1

                   –4 –3 –2 –1 0       1   2   3   4 x         –4 –3 –2 –1 0       1   2   3   4 x
                              –1                                          –1
                              –2                                           –2
                              –3                                           –3
                              –4  •H                                       –4




     9.              Plot point J (6, 4)                 9.      Plot point J (8, 6)
                              y                                            y
                                  8                                            8
                                  6                                            6
                                               J
                                  4
                                               •                               4
                                  2                                            2

                  –8 –6 –4   –2 0      2   4   6   8 x         –8 –6 –4   –2 0     2   4   6   8 x
                               –2                                           –2
                              –4                                           –4
                              –6                                           –6
                              –8                                           –8



.


          Curriculum Development Division
          Ministry of Education Malaysia                                                             11
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART A2: Plot the point on the Cartesian plane given the coordinates.



                       EXAMPLES                                      TEST YOURSELF
.                       EXAMPLES                                     TEST YOURSELF


     10.          Plot point K (– 4, 6)              10.            Plot point K (– 6, 2)
                                y                                              y
                                    8                                              8
                        K
                            •
                                    4                                              4



                –8      –4          0   4     8 x              -8       -4         0   4    8 x


                                 –4                                            –4


                                 –8                                            –8




     11.        Plot point L (–15, –10)              11.       Plot point L (–20, –5)
                                y                                              y
                                29                                             20


                                 10                                            10



                –20     –10         0   10    20 x             –20      –10        0   10   20 x


                       •L       –10                                           –10


                                –20                                           –20




     12.         Plot point M (30, –15)              12.       Plot point M (10, –25)
                                 y                                             y
                                 20                                            20


                                 10                                            10



                 –40    –20         0   20    40 x             –40      –20        0   20   40 x


                                –10                                           –10

                                             •M
                                –20                                           –20




        Curriculum Development Division
        Ministry of Education Malaysia                                                             12
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions



                                                 ACTIVITY A2




                                             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.


1.                                           Enjoy yourself !
       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
                                                       y


                                                   4

                                                   2

                                                                             x
                                   –4       –2     0          2        4
                                                       ,
                                                  –2


                                                  –4




        Curriculum Development Division
        Ministry of Education Malaysia                                                        13
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions




                                              PART B:
                               GRAPHS OF FUNCTIONS



                                          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.




                            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.



        Curriculum Development Division
        Ministry of Education Malaysia                                                            14
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions


                                              PART B:
                                        GRAPHS OF FUNCTIONS


                                          LESSON NOTES




1.     For a standard graph paper, 2 cm is represented by 10 small squares.




              2 cm



                        2 cm


2.     Some common scales used are as follows:

                            Scale                          Note

                                          10 small squares represent 10 units
                     2 cm to 10 units
                                          1 small square represents 1 unit


                                          10 small squares represent 5 units
                     2 cm to 5 units
                                          1 small square represents 0.5 unit


                                          10 small squares represent 2 units
                     2 cm to 2 units
                                          1 small square represents 0.2 unit


                                          10 small squares represent 1 unit
                     2 cm to 1 unit
                                          1 small square represents 0.1 unit


                                          10 small squares represent 0.1 unit
                     2 cm to 0.1 unit
                                          1 small square represents 0.01 unit




        Curriculum Development Division
        Ministry of Education Malaysia                                          15
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B1:                 Mark numbers on the x-axis and y-axis based on the scales given.


                               EXAMPLES                                                 TEST YOURSELF




1.     Mark – 4. 7, 16 and 27on the x-axis.                                1.   Mark – 6 4, 15 and 26 on the x-axis.
       Scale: 2 cm to 10 units.                                                 Scale: 2 cm to 10 units.
       [ 1 small square represents 1 unit ]                                     [ 1 small square represents 1 unit ]




                                                                       x                                                        x
     –10       –4    0         7   10     16     20          27 30



2.     Mark –7, –2, 3 and 8on the x-axis.                                  2.   Mark –8, –3, 2 and 6, on the x-axis.
       Scale: 2 cm to 5 units.                                                  Scale: 2 cm to 5 units.
       [ 1 small square represents 0.5 unit ]                                   [ 1 small square represents 0.5 unit ]




                                                                       x                                                        x
     –10       –7   –5       –2    0         3   5          8     10



3.     Mark –3.4, – 0.8, 1 and 2.6, on the x-axis.                         3.   Mark –3.2, –1, 1.2 and 2.8 on the x-axis.
       Scale: 2 cm to 2 units.                                                  Scale: 2 cm to 2 units.
       [ 1 small square represents 0.2 unit ]                                   [ 1 small square represents 0.2 unit ]




                                                                       x                                                        x
     –4 –3.4        –2      –0.8   0     1       2    2.6         4



4.     Mark –1.3, – 0.6, 0.5 and 1.6 on the x-axis.                        4.   Mark –1.7, – 0.7, 0.7 and 1.5 on the x-axis.
       Scale: 2 cm to 1 unit.                                                   Scale: 2 cm to 1 unit.
       [ 1 small square represents 0.1 unit ]                                   [ 1 small square represents 0.1 unit ]




                                                                       x                                                        x
     –2        –1.3 – 1 –0.6       0    0.5      1          1.6   2




           Curriculum Development Division
           Ministry of Education Malaysia                                                                                      16
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B1:             Mark numbers on the x-axis and y-axis based on the scales given.


                               EXAMPLES                                      TEST YOURSELF




5.     Mark – 0.15, – 0.04, 0.03 and 0.17 on the                5.   Mark – 0.17, – 0.06, 0.04 and 0.13 on the
       x-axis.                                                       x-axis.

       Scale: 2 cm to 0.1 unit                                       Scale: 2 cm to 0.1 unit
       [ 1 small square represents 0.01 unit ]                       [ 1 small square represents 0.01 unit ]




                                                            x                                                     x
     –0.2   –0.15 –0.1    –0.04   0 0.03   0.1   0.17 0.2




6.     Mark –13, –8, 2 and 14 on the y-axis.                    6.   Mark –16, – 4, 5 and 15 on the y-axis.

       Scale: 2 cm to 10 units                                       Scale: 2 cm to 10 units
       [ 1 small square represents 1 unit ]                          [ 1 small square represents 1 unit ]
                       y                                                             y
                          20


                         14

                         10



                          2
                          0



                          –8
                         –10
                         –13



                         –20




             Curriculum Development Division
             Ministry of Education Malaysia                                                                      17
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B1:        Mark numbers on the x-axis and y-axis based on the scales given.


                         EXAMPLES                                    TEST YOURSELF


7.   Mark –9, –3, 1 and 7 on the y-axis.               7.   Mark –7, – 4, 2 and 6 on the y-axis.

     Scale: 2 cm to 5 units.                                Scale: 2 cm to 5 units.
     [ 1 small square represents 0.5 unit ]                 [ 1 small square represents 0.5 unit ]
                     y                                                      y
                   10


                    7

                    5



                    1
                    0


                   –3

                   –5




                   –9
                  –10



8.   Mark –3.2, – 0.6, 1.4 and 2.4 on the y-axis.      8.   Mark –3.4, –1.4, 0.8 and 2.8 on the y-axis.

     Scale: 2 cm to 2 units.                                Scale: 2 cm to 2 units.
     [ 1 small square represents 0.2 unit ]                 [ 1 small square represents 0.2 unit ]
                     y                                                      y
                    4




                   2.4
                    2
                   1.4



                    0
                  –0.6



                   –2


                  –3.2

                   –4




         Curriculum Development Division
         Ministry of Education Malaysia                                                                   18
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B1:        Mark numbers on the x-axis and y-axis based on the scales given.


                            EXAMPLES                                TEST YOURSELF



9.   Mark –1.6, – 0.4, 0.4 and 1.5 on the y-axis.      9.   Mark –1.5, – 0.8, 0.3 and 1.7 on the y-axis.

     Scale: 2 cm to 1 unit.                                 Scale: 2 cm to 1 unit.
     [ 1 small square represents 0.1 unit ]                 [ 1 small square represents 0.1 unit ]
                     y                                                      y
                        2


                   1.5


                        1


                   0.4

                     0

                  – 0.4


                    –1


                  –1.6

                    –2


10. Mark – 0.17, – 0.06, 0.08 and 0.16 on the          10. Mark – 0.18, – 0.03, 0.05 and 0.14 on the
    y-axis.                                                y-axis.

     Scale: 2 cm to 0.1 unit.                               Scale: 2 cm to 0.1 units.
     [ 1 small square represents 0.01 unit ]                [ 1 small square represents 0.01 unit ]
                     y                                                      y
                   0.2

                  0.16


                  0.1


                  0.08

                     0


                 – 0.06

                 –0.1



                 – 0.17
                 –0.2




         Curriculum Development Division
         Ministry of Education Malaysia                                                                    19
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B2:          Draw graph of a function given a table for values of x and y.



                           EXAMPLES                                           TEST YOURSELF


1.   The table shows some values of two variables, x and y,   1.   The table shows some values of two variables, x and y,
     of a function.                                                of a function.

        x        –2       –1         0        1        2              x        –3       –2        –1        0        1
        y        –2        0         2        4        6              y        –2        0         2        4        6
     By using a scale of 2 cm to 1 unit on the x-axis and          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          2 cm to 2 units on the y-axis, draw the graph of the
     function.                                                     function.
                            y

                            6               

                            4        

                            2

                       
               –2     –1     0       1      2     x
                          –2




2.   The table shows some values of two variables, x and y,   2.   The table shows some values of two variables, x and y,
     of a function.                                                of a function.

        x        –2       –1         0        1        2              x        –2       –1         0        1        2
        y        5         3         1       –1       –3              y        7         5         3        1       –1
     By using a scale of 2 cm to 1 unit on the x-axis and          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          2 cm to 2 units on the y-axis, draw the graph of the
     function.                                                     function.
                            y

                            6
                
                            4
                       
                            2
                                

               –2     –1                          x
                             0       
                                     1      2
                           –2
                                            




         Curriculum Development Division
         Ministry of Education Malaysia                                                                                   20
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B2:          Draw graph of a function given a table for values of x and y.


                          EXAMPLES                                            TEST YOURSELF




3.   The table shows some values of two variables, x and y,    3.   The table shows some values of two variables, x and y,
     of a function.                                                 of a function.

        x     –4      –3     –2      –1     0      1      2            x      –1     0       1       2     3      4      5
        y      15     5      –1      –3     –1     5      15           y      19     4      –5      –8     –5     4      19
     By using a scale of 2 cm to 1 unit on the x-axis and           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           2 cm to 5 units on the y-axis, draw the graph of the
     function.                                                      function.
                                    y
                                  15               

                                   10

                                   5        


         –4             
                       –2     –1        0                x
                –3                          1      2
                                   –5




4.   The table shows some values of two variables, x and y,    4.   The table shows some values of two variables, x and y,
     of a function.                                                 of a function.

        x      –2     –1      0       1     2       3      4           x      –2     –1       0       1      2      3
        y      –7     –2      1       2     1      –2     –7           y      –8     –4      –2      –2     –4     –8
     By using a scale of 2 cm to 1 unit on the x-axis and           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           2 cm to 2 units on the y-axis, draw the graph of the
     function.                                                      function.
                      y
                      2        
                                       

         –2     –1 0           1         2   3      4     x
                  –2                        

                     –4

                     –6
                                                   




         Curriculum Development Division
         Ministry of Education Malaysia                                                                                    21
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B2:          Draw graph of a function given a table for values of x and y.



                             EXAMPLES                                             TEST YOURSELF



5.   The table shows some values of two variables, x and y,     5.   The table shows some values of two variables, x and y,
     of a function.                                                  of a function.

            x      –2      –1       0      1       2                        x      –2      –1       0      1       2
            y      –7      –1       1      3      11                        y      –6       2       4      6      16
     By using a scale of 2 cm to 1 unit on the x-axis and            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            2 cm to 5 units on the y-axis, draw the graph of the
     function.                                                       function.
                             y
                            15
                                             
                            10

                             5
                                      
                                 
                                                  x
                –2     –1             1      2
                              0
                            –5
                 




6.   The table shows some values of two variables, x and y,     6.   The table shows some values of two variables, x and y,
     of a function.                                                  of a function.

         x     –3     –2      –1      0     1      2       3            x      –3    –2      –1      0     1      2       3
         y     22      5      0       1     2     –3     –20            y      21     4      –1      0     1     –4     –21
      By using a scale of 2 cm to 1 unit on the x-axis and           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          2 cm to 10 units on the y-axis, draw the graph of the
     function.                                                       function.
                             y
           
                            20

                            10
                  
                                    
          –3     –2    –1   0         1      
                                             2         3    x
                         –10

                          –20                          




          Curriculum Development Division
          Ministry of Education Malaysia                                                                                    22
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

                                                         ACTIVITY B1

                         Each table below shows the values of x and y for a certain function.

                    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

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.
                                                                 y




                                                                                                             x
                                                                0
        Curriculum Development Division
        Ministry of Education Malaysia                                                                                23
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B3:                  State the values of x and y on the axes.



                                      EXAMPLES                                                    TEST YOURSELF




1.     State the values of a, b, c and d on the x-axis                           1.     State the values of a, b, c and d on the x-axis
       below.                                                                           below.




                                                                             x                                                                    x
     –20        d       –10       c    0           a   10   b           20            –20 d       –10         c   0   a       10        b   20

       Scale: 2 cm to 10 units.
       [ 1 small square represents 1 unit ]

               a = 7, b = 13, c = – 4, d = –14


2.     State the values of a, b, c and d on the x-axis                           2.     State the values of a, b, c and d on the x-axis
       below.                                                                           below.




                                                                             x                                                                    x
     –10   d             –5   c        0       a       5        b       10            –10     d   –5      c       0       a   5     b       10

       Scale: 2 cm to 5 units.
       [ 1 small square represents 0.5 unit ]

               a = 2, b = 7.5, c = –3, d = –8.5


3.     State the values of a, b, c and d on the x-axis                           3.     State the values of a, b, c and d on the x-axis
       below.                                                                           below.




                                                                             x                                                                    x
     –4             d    –2   c        0   a           2            b   4             – 4d         –2 c           0       a   2 b           4

       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



               Curriculum Development Division
               Ministry of Education Malaysia                                                                                                    24
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B3:               State the values of x and y on the axes.

                             EXAMPLES                                              TEST YOURSELF



4.     State the values of a, b, c and d on the x-axis             4.    State the values of a, b, c and d on the x-axis
       below.                                                            below.




                                                               x                                                                   x
     –2      d       –1      c   0       a 1         b   2              –2    d    –1    c     0     a     1               b 2
        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


5.     State the values of a, b, c and d on the x-axis             5.    State the values of a, b, c and d on the x-axis
       below.                                                            below.




                                                               x                                                                   x
     –0.2    d      –0.1     c   0   a    0.1        b   0.2        – 0.2 d       –0.1       c 0         a 0.1         b    0.2
       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

6.     State the values of a, b, c and d on the y-axis             6.    State the values of a, b, c and d on the y-axis
                                            y                                                                y
       below.                                                            below.
        Scale: 2 cm to 10 units.          20                                                                      20

       [ 1 small square                    b
                                                                                                                  b
         represents 1 unit ]
                                                10                                                               10
                 a = 3, b = 17
                 c = – 6, d = –15                                                                                 a
                                                 a
                                                 0                                                                0
                                                                                                                  c
                                                 c

                                                –10                                                              –10


                                                 d
                                                                                                                   d
                                                –20                                                              –20




             Curriculum Development Division
             Ministry of Education Malaysia                                                                                       25
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B3:        State the values of x and y on the axes.



                       EXAMPLES                                      TEST YOURSELF




7.   State the values of a, b, c and d on the y-axis   7.   State the values of a, b, c and d on the y-axis
     below.                              y                  below.                              y
                                       10                                                     10
     Scale: 2 cm to 5 units.            b

     [ 1 small square                                                                          b

       represents 0.5 unit ]
                                           5                                                  5
                                           a
         a = 4, b = 9.5
                                                                                              a
         c = –2, d = –7.5
                                           0                                                  0
                                                                                               c
                                           c


                                       –5                                                    –5


                                       d
                                                                                              d
                                      –10                                                    –10




8.   State the values of a, b, c and d on the y-axis   8.   State the values of a, b, c and d on the y-axis
     below.                              y                  below.                              y
                                           4                                                  4
     Scale: 2 cm to 2 units.                                                                  b
     [ 1 small square                      b
       represents 0.2 unit ]
                                       2                                                      2
        a =    0.8, b = 3.2                                                                    a
                                           a
         c = –1.2, d = –2.6
                                       0                                                      0

                                                                                              c
                                           c

                                       –2                                                    –2
                                           d

                                                                                              d
                                       –4                                                    –4




         Curriculum Development Division
         Ministry of Education Malaysia                                                                  26
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B3:        State the values of x and y on the axes.



                        EXAMPLES                                      TEST YOURSELF




9.   State the values of a, b, c and d on the y-axis   9.   State the values of a, b, c and d on the y-axis
     below.                              y                  below.                              y
                                           2                                                      2
     Scale: 2 cm to 1 unit.
                                                                                               b
     [ 1 small square
       represents 0.1 unit ]               b
                                           1                                                   1
                                        a
         a = 0.7, b = 1.2                                                                      a


         c = – 0.6, d = –1.4            0                                                      0

                                                                                              c
                                       c

                                       –1                                                     –1

                                           d
                                                                                               d

                                      –2                                                      –2




10. State the values of a, b, c and d on the y-axis    10. State the values of a, b, c and d on the y-axis
    below.                              y                  below.                              y
                                      0.2                                                    0.2
     Scale: 2 cm to 0.1 unit.
                                       b
     [ 1 small square                                                                          b
       represents 0.01 unit ]
                                      0.1                                                    0.1

                                                                                               a
         a = 0.03, b = 0.07
                                        a

         c = – 0.04, d = – 0.18            0                                                   0

                                       c
                                                                                               c
                                     –0.1                                                   –0.1


                                                                                              d
                                        d
                                     –0.2                                                   –0.2




         Curriculum Development Division
         Ministry of Education Malaysia                                                                  27
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B4:           State the value of y given the value x from the graph and vice versa.



                             EXAMPLES                                             TEST YOURSELF




1.   Based on the graph below, find the value of y                   1.   Based on the graph below, find the value of y
     when (a) x = 1.5                                                     when (a) x = 0.6
             (b) x = –2.8                                                         (b) x = –1.7
                              y                                                                   y
                                   7
                              6                                                                   6

                              4                                                                   4

                              2                                                                   2
          – 2.8

                  –2    –1     0               1     1.5
                                                           2     x                 –2      –1      0             1     2     x
                                       – 1.6
                             –2                                                                  –2



              (a)      7               (b)      –1.6                              (a)                      (b)



2.   Based on the graph below, find the value of y                   2.   Based on the graph below, find the value of y
     when ( a ) x = 0.14                                                  when ( a ) x = 0.07
             ( b ) x = – 0.26                                                     ( b ) x = – 0.18
                              y                                                                   y
                                        11.5
                             10                                                                  10


                              5                                                                   5
                              1.5
          – 0.26                                   0.14          x                                                           x
              – 0. 2 –0.1          0           0.1         0.2                    –0. 2   –0.1         0         0.1   0.2
                             –5                                                                  –5


                             –10                                                                 –10




             (a)       1.5              (b)        11.5                          (a)                       (b)




        Curriculum Development Division
        Ministry of Education Malaysia                                                                                           28
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B4:           State the value of y given the value x from the graph and vice versa.


                                EXAMPLES                                                 TEST YOURSELF




3.   Based on the graph below, find the value of y                           3.   Based on the graph below, find the value of y
     when ( a ) x = 0.6                                                           when ( a ) x = 1.2
             ( b ) x = –2.7                                                               ( b ) x = –1.8

                                               y                                                           y
                                             15                                                           15
                                               11
                                             10                                                           10

                                               5                                                           5
                   – 2.7

        –4    –3            –2          –1         0       0.6
                                                                 1   2   x          –4    –3    –2   –1    0       1   2    x
                                                       – 3.5
                                             –5                                                           –5



             (a)            11               (b)           –3.5                          (a)               (b)



4.   Based on the graph below, find the value of y                           4.   Based on the graph below, find the value of y
     when (a) x = 1.4                                                             when (a) x = 2.7
             (b) x = –1.5                                                                 (b) x = –2.1

                        y                                                                       y
                            3
                        2                                                                       2
          – 1.5

        –2        –1        0           1
                                             1.4
                                                       2         3   4   x          –2    –1    0     1        2   3   4    x
                       –2                                                                      –2

                       –4                                                                      –4
                                – 5.8
                       –6                                                                      –6


             (a)        3               (b)            –5.8                              (a)              (b)




        Curriculum Development Division
        Ministry of Education Malaysia                                                                                      29
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions



PART B4:             State the value of y given the value x from the graph and vice versa.


                                EXAMPLES                                                  TEST YOURSELF




5.   Based on the graph below, find the value of y                      5.   Based on the graph below, find the value of y
     when (a) x = 1.7                                                        when (a) x = 1.2
             (b) x = –1.3                                                            (b) x = –1.9

                                 y                                                                 y
                                15                                                                15

                                10                                                                10
                                 5.5
                                 5                                                                 5
                       – 1.3

                –2        –1         0           1    1.7
                                                            2   x                    –2      –1        0         1   2   x
                                         – 3.5
                                –5                                                                –5


              (a)       5.5               (b)        –3.5                           (a)                    (b)


6.   Based on the graph below, find the value of y                      6.   Based on the graph below, find the value of y
     when (a) x = 1.6                                                        when (a) x = 2.8
             (b) x = –2.3                                                            (b) x = –2.6

                                 y                                                                 y
                                         25
                                20                                                                20

                                10                                                                10
                                                     1.6

        –3   – 2.3
                –2        –1 0                   1         2    3   x          –3    –2     –1 0                 1   2   3   x
                              –9
                            –10                                                               –10

                               –20                                                            –20


              (a)       –9                (b)        25                             (a)                    (b)




        Curriculum Development Division
        Ministry of Education Malaysia                                                                                       30
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B4:          State the value of y given the value x from the graph and vice versa.



                            EXAMPLES                                              TEST YOURSELF




7.   Based on the graph below, find the value of x                   7.   Based on the graph below, find the value of x
     when (a) y = 5.4                                                     when (a) y = 2.8
             (b) y = –1.6                                                         (b) y = –2.4
                                  y                                                                y

                                  6                                                                6
                                  5.4

                                  4                                                                4

                                  2                                                                2
          – 2.8

                  –2       –1     0              1
                                                     1.4
                                                            2    x                 –2      –1      0              1     2    x
                                – 1.6
                                –2                                                               –2



              (a)      1.4               (b)     –2.8                             (a)                      (b)



8.   Based on the graph below, find the value of x                   8.   Based on the graph below, find the value of x
     when ( a ) y = 4                                                     when ( a ) y = 6.5
             ( b ) y = –7.5                                                       ( b ) y = –7
                       y                                                                    y
                                10                                                               10

                                 5                                                                5
                                     4
                                          0.08
                  – 0.07                                         x                                                           x
             –0. 2     –0.1          0         0.1         0.2                    –0. 2   –0.1         0         0.1   0.2
                                –5                                                               –5
                                – 7.5
                              –10                                                                –10



             (a)       – 0.07            (b)         0.08                        (a)                   (b)




        Curriculum Development Division
        Ministry of Education Malaysia                                                                                           31
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

PART B4:             State the value of y given the value x from the graph and vice versa.



                               EXAMPLES                                                    TEST YOURSELF




9. Based on the graph below, find the values of x                            9. Based on the graph below, find the values of x
   when (a) y = 8.5                                                             when (a) y = 3.5
           (b) y = 0                                                                    (b) y = 0

                                          y                                                                 y
                                         15                                                               15

                                         10                                                               10
                                          8.5
                                          5                                                                 5

             – 3.1                         0                       2.1   x                                   0               x
        –4      –3        –2     –1                    1       2                    –4    –3    –2   –1              1   2
                                         –5                                                               –5



       (a)      –3.1 , 2.1                (b)         –2 , 1                        (a)                (b)



10. Based on the graph below, find the values of x                           10. Based on the graph below, find the values of x
    when (a) y = 2.6                                                             when (a) y = 1.2
            (b) y = – 4.8                                                                (b) y = – 4.4

                        y                                                                       y
                         2.6
                        2                                                                       2
             – 1.2                                         3.9
                          0    0.6              2.1                      x                                                   x
        –2      –1                   1          2      3       4                     –2    –1 0       1          2   3   4
                      –2                                                                     –2

                      –4                                                                       –4
                       – 4.8
                      –6                                                                       –6


       (a)     0.6 , 2.1                 (b)        –1.2 , 3.9                     (a)                (b)




        Curriculum Development Division
        Ministry of Education Malaysia                                                                                       32
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions



PART B4:             State the value of y given the value x from the graph and vice versa.



                           EXAMPLES                                                      TEST YOURSELF




11. Based on the graph below, find the value of x                          11. Based on the graph below, find the value of x
    when (a) y = 14                                                            when (a) y = 11
            (b) y = –17                                                                (b) y = –23

                               y                                                                        y
                              20                                                                       20
                                14

                              10                                                                       10
             – 2.3

        –3      –2       –1 0                1         2
                                                             2.6
                                                                   3   x          –3         –2   –1 0                1         2   3   x
                           –10                                                                      –10
                              – 17
                           –20                                                                     –20


              (a)       2.6            (b)   –2.3                                       (a)                     (b)


12. Based on the graph below, find the value of x                          12. Based on the graph below, find the value of x
    when (a) y = 6.5                                                           when (a) y = 7.5
            (b) y = 0                                                                  (b ) y = 0
            (c) y = –6                                                                 (c) y = –9

                                y                                                                       y
                              15                                                                       15

                              10                                                                       10
                               6.5

                               5                                                                        5
                              – 0.8              1.3       2.3

                –2       –1        0         1         2         x                           –2   –1        0         1         2   x
                              –5                                                                       –5
                                –6



          (a) – 0.8 (b) 1.3 (c) 2.3                                                    (a)             (b)                (c)


        Curriculum Development Division
        Ministry of Education Malaysia                                                                                                  33
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

                                                   ACTIVITY B2


                    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.


          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.

              y




         60

         55

         50

         45

         40

         35

        30

         25

        20

        15

        10

          5

          0                                                                                      x
                      1        2       3       4        5      6         7    8       9



        Curriculum Development Division
        Ministry of Education Malaysia                                                               34
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions



                                              ANSWERS

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).




          Curriculum Development Division
          Ministry of Education Malaysia                                                                35
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions




PART A2:



     1.                                                     4.
                              y                                             y
                                  4                                          4
                                              A
                                  3
                                  2
                                          •                                    3
                                                                               2
                                  1                                            1

                 –4 –3 –2 –1 0        1   2       3   4 x        –4 –3 –2 –1 0         1       2       3   4 x
                            –1                                              –1
                              –2                                           –2
                                                                                           D
                              –3                                           –3      •
                              –4                                           –4



     2.                                                     5.
                      B       y                                            y
                    •             4
                                  3
                                                                               4
                                                                               3
                                  2                                            2
                                  1                                            1
                                                                                                   E
                 –4 –3 –2 –1 0        1   2       3   4 x        –4 –3 –2 –1 0         1   •   2       3   4 x
                            –1                                              –1
                              –2                                           –2
                              –3                                           –3
                              -–4                                          –4


     3.                                                     6.
                              y                                             y
                                  4                                          4
                                                                                   F
                                  3
                                                                               •
                                                                               3
                                  2                                            2
                                  1                                            1

                 –4 –3 –2 –1 0        1   2       3   4 x        –4 –3 –2 –1 0         1       2       3   4 x
                            –1                                              –1

                          •
                          C
                              –2                                           –2
                              –3                                           –3
                              –4                                           –4




          Curriculum Development Division
          Ministry of Education Malaysia                                                                         36
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions




     7.                                                   10.
                            y                                                    y
                                4                                                 8
                                3
                                2                                                4
                                                                      K
                   G
                                1
                                                                      •
                 •
                 –4 –3 –2 –1 0          1   2   3   4 x         –8        –4      0       4    8 x
                            –1
                            –2                                                   –4
                            –3
                            –4                                                   –8



     8.                                                   11.
                            y                                                    y
                                4                                                20
                                3
                                2                                                10
                                1

                 –4 –3 –2 –1 0          1   2   3   4 x         –20       –10     0       10   20 x
                            –1
                              –     H                           •L
                            -2                                                  –10
                            –3  •
                            –4                                                  –20



     9.                                                   12.
                            y                                                    y
                                8                                                20
                                                     J
                                6
                                4
                                                    •                            10
                                2

                 –8 –6 –4 –2 0          2   4   6   8 x         –40       –20     0       20   40 x
                            –2
                            –4                                                  –10
                            –6
                            –8                                                  –20
                                                                                      M
                                                                                      •




          Curriculum Development Division
          Ministry of Education Malaysia                                                              37
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions




ACTIVITY A2:



                                    YAKOMI ISLANDS
                                                       y


                                                  4
                                    A                              P        Q

                                                  2
                                                                   R        S
                                                           E   F
                                                                                x
                               –4           –2    O            2       4
                          B                   C        ,
                                                  –2           U
                                        D                                   T

                                                  –4
                                                                          RM 1 million




        Curriculum Development Division
        Ministry of Education Malaysia                                                    38
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions




PART B1:


1                                                                                 2.

                                                                              x                                                                              x
     –10        –6      0    4     10        15         20         26   30             –10         –8   –5   –3       0    2               5 6         10


3.                                                                                4.

                                                                              x                                                                              x
      –4       –3.2    –2     –1   0          1.2        2    2.8       4              –2 –1.7          –1 –0.7       0          0.7       1     1.5   2
                                                                                                                  y
5.                                                                                6.                          20


                                                                                                              15

                                                                              x
     –0.2 –0.16       –0.1 –0.06   0     0.04           0.1 0.13        0.2                                   10

                                                                                                                  5


                                                                                                                  0

                                                                                                             –4


                                                                                                             –10


                                                                                                             –16

                                                                                                             –20



7.           y                          8.          y                             9.           y                          10.           y
           10                                       4                                         2                                   0.2
                                                                                             1.7
                                                  2.8                                                                            0.14
           6
           5                                        2                                         1                                  0.1


                                                                                                                                 0.05
           2                                      0.8
                                                                                         0.3

           0                                        0                                         0                                        0
                                                                                                                                – 0.03

                                                –1.4
        –4                                                                              –0.8
        –5                                        –2                                      –1                                    – 0.1

        –7
                                                                                         –1.5
                                                –3.4
                                                                                                                                – 0.18
       –10                                        –4                                         –2                                 – 0.2




                 Curriculum Development Division
                 Ministry of Education Malaysia                                                                                                             39
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions



PART B2:


1.                                   y                        2.                       y
                                                                        
                                    6                                                 6
                                                                                  
                                    4                                                 4
                                                                                            
                                 2                                                    2
                                                                                                
                       
              –3      –2     –1                   x                     –2    –1                        x
                                      0       1                                         0       1   
                                                                                                    2
                                   –2                                                –2




3.           y                                              4.              y

             15                                                                 0                               x
                                                                   –2   –1                  1   2   3
             10                                                              –2            

              5                                                         –4                   

                                                                             –6
        –1    0        1        2         3   4       5   x
             –5                                                           –8                     
                                



5.                          y                                 6.                        y
                                                                  
                           15                                                          20

                           10                                                          10
                                                                        
                                                                                               
                            5                                                                                 x
                                                                  –3   –2      –1 0            1   
                                                                                                    2       3

                            0                                                     –10
              –2      –1                  1   2   x

                          –5                                                         –20                   




        Curriculum Development Division
        Ministry of Education Malaysia                                                                          40
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

ACTIVITY B1:
                                                         y




                                                 20
                                                                
                                                18                   
                                                                        
                                                16                            

                                                14                    

                                                 12

                                                 10
                                                                          
                                                   8                
                                                                
                                                     6
                                                            
                                                    4                 

                                                     2
                                                                
                                                                                  x
                      –4     –3      –2     –1           0       1     2   3   4




        Curriculum Development Division
        Ministry of Education Malaysia                                                 41
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 6: Coordinates and Graphs of Functions

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)


        Curriculum Development Division
        Ministry of Education Malaysia                               42
       Basic Essential

Additional Mathematics Skills




             UNIT 7
  LINEAR INEQUALITIES


 Unit 1:
 Negative Numbers




           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




                                    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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                           1
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                          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.




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                         2
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


                                              PART A:
                                       LINEAR INEQUALITY

                                 OVERALL LESSON NOTES


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


                                                                                           x
        −3         −2           −1           0            1             2              3



                    −3 < − 1                                            1<3
               −3 is less than − 1                                1 is less than 3

                        and                                             and

                     −1 > − 3                                          3>1
              −1 is greater than − 3                             3 is greater than 1




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                         3
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


3.0   Properties of Inequalities

      (a) Addition Involving Inequalities

               Arithmetic Form                       Algebraic Form


              12  8 so 12  4  8  4          If a > b, then a  c  b  c
             29    so 2  6  9  6            If a < b, then a  c  b  c




       (b) Subtraction Involving Inequalities

               Arithmetic Form                       Algebraic Form


             7 > 3 so 7  5  3  5             If a > b, then a  c  b  c
             2 < 9 so 2  6  9  6             If a < b, then a  c  b  c



       (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)          If a > b and c > 0 , then ac > bc
                         12 9                                            a b
               12 > 9 so                      If a > b and c > 0, then 
                          3 3                                            c c

                25     so 2(3)  5(3)         If a  b and c  0 , then ac  bc
                            8 12                                          a b
                8  12 so                       If a  b and c  0 , then 
                            2 2                                           c c




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                   4
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


        (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)      If a > b and c < 0, then ac < bc
                    6<7      so    6(−3) > 7(−3)      If a < b and c < 0, then ac > bc
                                   16   8                                      a b
                    16 > 8   so                      If a > b and c < 0, then 
                                   4 4                                        c c
                                   10 15                                       a b
                    10 <15 so                        If a < b and c < 0, then 
                                   5 5                                        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
                               4m

      (b)     A solution of an inequality is any value of the variable that satisfies the inequality.

              Examples:

              (i)     Consider the inequality x  3

                      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 x  3 can be written as
                      x  4, 5, 6, 7, 8,...

______________________________________________________________________________

            Curriculum Development Division
            Ministry of Education Malaysia                                                              5
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


                 A number line is normally used to represent all the solutions of an inequality.

                                    To draw a number line representing x  3 , 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.




                                                                     The open dot
                                                                    means the value
     (ii)      x>2                                                     2 is not
                                                                      included.


                                                    o
                                                                                      x
            −2       −1       0           1          2          3         4




                                                                       The solid dot
       (iii)     x3                                                  means the value
                                                                       3 is included.




                                                                                              x
       −2         −1           0              1             2             3               4




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                             6
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                          PART B:
                    POSSIBLE SOLUTIONS FOR A
                   GIVEN LINEAR INEQUALITY IN
                          ONE UNKNOWN


                                  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) x  h
                         (d) x  h

            (ii) representing a linear inequality:
                          (a) x  h
                          (b) x  h
                          (c) x  h
                          (d) x  h
                               on a number line and vice versa.




                       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.

______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                     7
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


                                                     PART B:
                                    POSSIBLE SOLUTIONS FOR
                     A GIVEN LINEAR INEQUALITY IN ONE UNKNOWN

                                                 EXAMPLES



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:



                                                                                                                                                     x
          −2         −1        0        1        2        3        4        5           6           7           8           9               10
                  The possible integers are: 5, 6, 7, …


       (2)        x  3

                  Solution:



                                                                                                                                            x
     −8         −7        −6       −5       −4       −3       −2       −1       0           1           2           3               4


             The possible integers are: – 4, − 5, −6, …


       (3)         3  x 1

                  Solution:


                                                                                                                                                 x
        −8        −7       −6       −5       −4        −3      −2       −1          0           1           2           3               4

               The possible integers are: −2, −1, 0, and 1.



______________________________________________________________________________

          Curriculum Development Division
          Ministry of Education Malaysia                                                                                        8
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                      TEST YOURSELF B


Draw a number line to represent the following inequalities:

   (a)    x>1




   (b)     x2




   (c)     x  2




   (d)     x3




______________________________________________________________________________

         Curriculum Development Division
         Ministry of Education Malaysia                                     9
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                           PART C:
                      COMPUTATIONS INVOLVING
                    ADDITION AND SUBTRACTION ON
                        LINEAR INEQUALITIES


                                  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.




                        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.




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                   10
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


                                              PART C:
           COMPUTATIONS INVOLVING ADDITION AND SUBTRACTION
                                 ON LINEAR INEQUALITIES

                                            LESSON NOTES



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

                                   2<4




                                                                       x
                                    1         2       3        4



         Adding 1 to both sides of the inequality:


                                                                           The inequality
                                                                              sign is
                              2+1<4+1
                                                                            unchanged.
                                3<5



                                                                       x
                                        2         3       4        5




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                      11
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


      (ii)     4>2                                       4>2




                                                                            x
                                             1     2       3         4


             Subtracting 3 from both sides of the inequality:


                                            4−3>2−3
                                                                                    The inequality
                                              1>−1
                                                                                       sign is
                                                                                     unchanged.


                                                                            x
                                            −1     0        1        2




                                                 EXAMPLES



(1)          Solve x  5  14 .

             Solution:
                                                       Subtract 5 from both sides
                       x  5  14                          of the inequality.
                       x  5  5  14  5
                       x9                                      Simplify.




(2)          Solve p  3  2.

             Solution:
                                                       Add 3 to both sides of the
                         p3 2
                                                              inequality.
                         p  3 3  2  3
                         p5                                    Simplify.


______________________________________________________________________________

              Curriculum Development Division
              Ministry of Education Malaysia                                                         12
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                          TEST YOURSELF C


Solve the following inequalities:


(1)     m  4  2                                 (2)     x  3.4  2.6




(3)     x  13  6                                (4)     4.5  d  6




(5)     23  m  17                                (6)     y  78  54




(7)     9  d 5                                   (8)     p  2  1




             1                                     (10)    3 x 8
(9)     m     3
             2




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                      13
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                             PART D:
                        COMPUTATIONS INVOLVING
                      DIVISION AND MULTIPLICATION
                         ON LINEAR INEQUALITIES



                                  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.




                        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)




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                    14
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities



                                              PART D1:
                                  COMPUTATIONS INVOLVING
            MULTIPLICATION AND DIVISION ON LINEAR INEQUALITIES


                                            LESSON NOTES




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

                             2<4



                                                           x
                              1        2       3      4


         Multiplying both sides of the inequality by 3:
                                                                    The inequality
                                                                       sign is
                                                                     unchanged.
              2  3<4  3
                 6 < 12


                                                                    x
                              6        8       10    12        14




______________________________________________________________________________

          Curriculum Development Division
          Ministry of Education Malaysia                                             15
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities



     (ii)    −4<2

                       −4<2


                                                                       x
                                 −4         −2         0       2

             Dividing both sides of the inequality by 2:

                                                                                    The inequality
               −4  2<2  2                                                            sign is
                    −2 <1                                                            unchanged.



                                                                                    x
                                 −2         −1         0       1               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

                                       4<6



                                                                           x
                                        3        4         5       6


               Dividing both sides of the inequality by −1:


                                      4  (−1) > 6                                 The inequality
                               (−1)                                                sign is reversed.
                                            −4>−6


                                                                               x
                            −6    −5    −4     −3
______________________________________________________________________________

             Curriculum Development Division
             Ministry of Education Malaysia                                                            16
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities



   (ii)    1 > −3


                                     1 > −3



                                                                     x
                              −3      −2       −1      0         1


           Multiply both sides of the inequality by −1:


                                                                          The inequality
             (− 1) (1) < (−1) (−3)                                       sign is reversed.
                    1  3



                                                                     x
                              −1       0        1      2         3




                                               EXAMPLES




Solve the inequality 3q  12 .

Solution:

     (i)     3q  12
                                       Divide each side of the
              3q 12                     inequality by −3.                 The inequality
                 
             3 3                                                        sign is reversed.

                q  4                     Simplify.




______________________________________________________________________________

           Curriculum Development Division
           Ministry of Education Malaysia                                                     17
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                    TEST YOURSELF D1


Solve the following inequalities:


(1) 7 p  49                                       (2) 6 x  18




(3) −5c > 15                                       (4) 200 < −40p




(5) 3d  24                                       (6)  2x  8




(7)  12  3x                                     (8) 25  5 y




(9)  2m  16                                     (10)  6b  27




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                      18
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                              PART D2:
                          PERFORM COMPUTATIONS INVOLVING
                     MULTIPLICATION OF LINEAR INEQUALITIES


                                            EXAMPLES




                         x
Solve the inequality       3.
                         2
Solution:

                                       x
                                         3.             Multiply both sides of the
                                       2                     inequality by −2.
                                       x
                                   2( )  (2)3
                                       2                          Simplify.
                                         x  6




                                   The inequality
                                  sign is reversed.




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                 19
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                       TEST YOURSELF D2


1. Solve the following inequalities:



            d                                            n
 (1)    −     3                                   (2)      8
            8                                            2




                   y                                           b
 (3)    10                                       (4) 6 
                   5                                           7




                        x                                      x
 (5)        0  12                               (6)   8      0
                        8                                      6




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                      20
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                          PART E:
                        FURTHER PRACTICE ON
                      COMPUTATIONS INVOLVING
                         LINEAR INEQUALITIES



                                  LEARNING OBJECTIVES

            Upon completion of Part E, pupils will be able perform computations
            involving linear inequalities.




                        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.




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                     21
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


                                             PART E:
                       FURTHER PRACTICE ON COMPUTATIONS
                           INVOLVING LINEAR INEQUALITIES


                                     TEST YOURSELF E1


Solve the following inequalities:


1.    (a)    m5 0




      (b)    x26




      (c)   3+m>4




2.    (a)   3m < 12




      (b)   2m > 42




      (c) 4x > 18




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                      22
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities



3.    (a)   m + 4 > 4m + 1




      (b)   14  m  6  m




      (c)    3  3m  4  m




4.    (a)    4  x  6




      (b)   15  3m  12




                  x
      (c)    3     5
                  4




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                      23
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




      (d)    5x  3  18




      (e)   1  3 p  10




             x
      (f)      3 4
             2




                 x
       (g) 3      8
                 5




             p2
      (h)        4
              3




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                      24
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                            EXAMPLES




What is the smallest integer for x if 5x  3  18 ?


                                                                    A number line can
                                                                   be used to obtain the
                                                                         answer.
Solution:


  5x  3  18

      5x  18  3
                                                                              x3
      5x  15                                                  O
         x 3                                                                                   x
                         0           1             2           3        4            5     6
       x = 4, 5, 6,…
     Therefore, the smallest integer for x is 4.




______________________________________________________________________________

         Curriculum Development Division
         Ministry of Education Malaysia                                                    25
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                            TEST YOURSELF E2


1.   If    3x  1  14, what is the smallest integer for x?




2.   What is the greatest integer for m if m  7  4m  1 ?




3.        x
     If      3  4 , find the greatest integer value of x.
          2




4.         p2
     If         4 , what is the greatest integer for p?
            3




5.                                            3 m
     What is the smallest integer for m if          9?
                                               2




______________________________________________________________________________

          Curriculum Development Division
          Ministry of Education Malaysia                                    26
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                           ACTIVITY




                                                     1




          2                       3

          4




   5                  6




                                               7           8




                                               9




                                                     10

                      11          12




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                      27
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




HORIZONTAL:
4.     1  3 is an ___________.


5.     An inequality can be represented on a number __________.

7.      2  6 is read as 2 is __________ than 6.


9.     Given 2x  1  9 , x  5 is a _____________ of the inequality.

11.      3x  12


            x  4


       The inequality sign is reversed when divided by a ____________ integer.



VERTICAL:
          x
             1
1.        2
          x  2

       The inequality sign remains unchanged when multiplied by a ___________ integer.

2.      6 x  24 equals to x  4 when both sides are _____________ by 6.


3.      x  5 equals to 3x  15 when both sides are _____________ by 3.


6.     ___________ inequalities are inequalities with the same solution(s).

8.      x  2 is represented by a ____________ dot on a number line.


10.     3x  6 is an example of ____________ inequality.


12.     5  3 is read as 5 is _____________ than 3.




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                   28
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities




                                                  ANSWERS



TEST YOURSELF B:

(a)                                                                x
       −3        −2    −1      0       1           2       3

(b)                                                                x
        −3       −2     −1     0          1        2       3



(c)                                                                x
       −3        −2    −1      0       1           2       3


                                                                       x
(d)         −3    −2     −1        0          1        2       3


TEST YOURSELF C:

(1) m  6       (2) x  6    (3) x  19         (4) d  1.5 (5) m  6
                                                            5
(6) y  24 (7) d  4          (8) p  3          (9) m       (10) x  5
                                                            2


TEST YOURSELF D1:

(1)   p7        (2) x  3    (3) c  3          (4) p  5      (5)     d  8

                                                                              9
(6) x  4       (7) x  4     (8) y  5          (9) m  8       (10) b 
                                                                              2


TEST YOURSELF D2:

(1) d  24      (2) n  16   (3) y  50         (4) b  42     (5) x  96    (6) x  48




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                         29
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 7: Linear Inequalities


TEST YOURSELF E1:

1. (a) m  5           (b) x  8  (c ) m  1
                                             9
2. (a) m  4        (b) m  21      (c ) x 
                                             2
                                   1
3. (a ) m  1 (b) m  4 (c) m 
                                   2
4. (a) x  10 (b) m  1 (c) x  8 (d) x  3 (e) p  3 (f) x  2 (g) x  25 (h) p  10



TEST YOURSELF E2:

(1) x  6      (2) m  1      (3) x  13      (4) p  9        (5) m  14


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




______________________________________________________________________________

        Curriculum Development Division
        Ministry of Education Malaysia                                                     30
       Basic Essential

Additional Mathematics Skills




             UNIT 8

     TRIGONOMETRY

 Unit 1:
 Negative Numbers




           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




                                    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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                    1
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                            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.




                         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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                  2
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                     LESSON NOTES




                                      θ




                    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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                   3
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



                                           EXAMPLES




Example 1:




                                           θ



           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:




                                       θ




              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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                       4
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                       TEST YOURSELF A


Identify the opposite, adjacent and hypotenuse sides of the following right-angled triangles.

 1.                              2.                             3.




      Opposite side =                 Opposite side =                Opposite side =
      Adjacent side =                 Adjacent side =                Adjacent side =
      Hypotenuse side =               Hypotenuse side =              Hypotenuse side =


 4.                              5.                             6.




      Opposite side =                 Opposite side =                Opposite side =
      Adjacent side =                 Adjacent side =                Adjacent side =
      Hypotenuse side =               Hypotenuse side =              Hypotenuse side =




          Curriculum Development Division
          Ministry of Education Malaysia                                                        5
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                           PART B:
                                 TRIGONOMETRY II




                                     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.




                          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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                   6
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



                                             LESSON NOTES



Definition of the Three Trigonometric Functions


                                                                    Acronym:
                      opposite side
     (i)     sin  =
                     hypotenuse side                                  SOH:
                                                           Sine – Opposite - Hypotenuse


                        adjacent side                                Acronym:
     (ii)    cos  =
                       hypotenuse side
                                                                       CAH:
                                                           Cosine – Adjacent - Hypotenuse


                       opposite side                                Acronym:
     (iii) tan  =
                       adjacent side
                                                                     TOA:
                                                          Tangent – Opposite - Adjacent




                                               θ


                                              opposite side    AB
                                  sin  =                    =
                                             hypotenuse side   AC

                                              adjacent side    BC
                                  cos  =                    =
                                             hypotenuse side   AC

                                             opposite side   AB
                                  tan  =                  =
                                             adjacent side   BC




           Curriculum Development Division
           Ministry of Education Malaysia                                                   7
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



                                          EXAMPLES



Example 1:




                                              θ



       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.


                               opposite side    AB
       Thus       sin    =                   =
                              hypotenuse side   AC

                               adjacent side    BC
                  cos  =                     =
                              hypotenuse side   AC

                              opposite side           AB
                  tan  =                         =
                              adjacent side           BC




        Curriculum Development Division
        Ministry of Education Malaysia                                                        8
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



Example 2:



                                           θ
                                                                  You have to identify the
                                                                   opposite, adjacent and
                                                                     hypotenuse sides.




       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.


                             opposite side    WU
       Thus,      sin  =                   =
                            hypotenuse side   TW

                             adjacent side    TU
                  cos  =                   =
                            hypotenuse side   TW

                            opposite side   WU
                  tan  =                 =
                            adjacent side   TU




        Curriculum Development Division
        Ministry of Education Malaysia                                                        9
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                        TEST YOURSELF B



Write the ratios of the trigonometric functions, sin , cos  and tan  , for each of the diagrams
below:

 1.                                2.             θ               3.

                                                                             θ
       θ

                                                  θ



      sin  =                           sin  =                        sin  =

      cos  =                           cos  =                        cos  =

      tan  =                           tan  =                        tan  =




 4.                                5.                             6.

                      θ                                                                 θ

                                             θ




      sin  =                           sin  =                        sin  =

      cos  =                           cos  =                        cos  =

      tan  =                           tan  =                        tan  =




           Curriculum Development Division
           Ministry of Education Malaysia                                                        10
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                          PART C:
                               TRIGONOMETRY III




                                    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.




                         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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                 11
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



                                            LESSON NOTES


                          opposite                      adjacent                        opposite
     Since sin  =                    Since cos  =                 Since tan  =
                         hypotenuse                    hypotenuse                       adjacent

                          opposite                      adjacent                        opposite
     then  = sin-1                   then  = cos-1                then  = tan-1
                         hypotenuse                    hypotenuse                       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.)


                                              EXAMPLES


Find the angle  in degrees and minutes.

 Example 1:                                       Example 2:



                                                                    θ

                     θ

                   o 2
           sin  =                                                      a    3
                   h 5                                          cos  =    =
                                                                         h    5
                = sin-1 2
                         5                                           = cos-1 3
                                                                              5
                 = 23o 34 4l
                                                                        = 53o 7 48
                 = 23o 35
                                                                        = 53o 8
 (Note that 34 41 is rounded off to 35)       (Note that 7 48 is rounded off to 8)




        Curriculum Development Division
        Ministry of Education Malaysia                                                             12
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




 Example 3:                                      Example 4:


                                                                  θ


                                  θ


           tan  = o = 7                                        cos  = a = 5
                      a       6                                           h       7

               = tan-1 7                                        = cos-1 5
                              6                                               7

                = 49o 23 55                                   = 44o 24 55

                = 49o 24                                        = 44o 25




 Example 5:                                      Example 6:



                          θ


                                                                                  θ



                                                                          o   5
                      o   4                                     tan  =     =
            sin  =     =                                                 a   6
                      h   7

                = sin-1 4                                             = tan-1 5
                                                                                  6
                              7
                                                                       = 39o 48 20
                 = 34o 50 59
                                                                        = 39o 48
                 = 34o 51




        Curriculum Development Division
        Ministry of Education Malaysia                                                  13
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



                                      TEST YOURSELF C

Find the value of  in degrees and minutes.

 1.                                              2.



                   θ
                                                                    θ




 3.                                              4.


                             θ

                                                                    θ




 5.                                              6.



                        θ


                                                                θ




        Curriculum Development Division
        Ministry of Education Malaysia                                  14
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                            PART D:
                                 TRIGONOMETRY IV




                                    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.




                         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.




        Curriculum Development Division
        Ministry of Education Malaysia                                               15
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                          LESSON NOTES




 Find the length of PR.                          Find the length of TS.




 With reference to the given angle, PR is the With reference to the given angle, TR is the
 opposite side and QR is the adjacent side.   adjacent side and TS is the hypotenuse
                                              side.
 Thus tangent ratio is used to form the
 relation of the sides.                       Thus cosine ratio is used to form the
                                              relation of the sides.
                    o   PR
             tan 50 =
                         5                                              8
                                                              cos 32o =
                                                                           TS
                 PR = 5  tan 50   o

                                                           TS  cos 32o = 8

                                                                              8
                                                                    TS =
                                                                           cos 32o




        Curriculum Development Division
        Ministry of Education Malaysia                                                       16
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                            EXAMPLES




Find the value of x in each of the following.

 Example 1:                                      Example 2:




                      3
          tan 25o =
                      x                                                         x
                                                                sin 41.27o =
                                                                                5
                         3
               x =
                      tan 25o                                           x = 5  sin 41.27o
                  = 6.434 cm                                                = 3.298 cm



 Example 3:                                      Example 4:




                            x
          cos 34o 12 =
                            6
                                                                            x
                                                                tan 63o =
                      x = 6  cos 34o 12                                   9

                          = 4.962 cm                                 x = 9  tan 63o

                                                                       = 17.66 cm




        Curriculum Development Division
        Ministry of Education Malaysia                                                       17
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                     TEST YOURSELF D


Find the value of x for each of the following.

 1.                                               2.




 3.                                               4.




                 10 cm


                                                                6 cm




 5.                                               6.
                      13 cm




        Curriculum Development Division
        Ministry of Education Malaysia                                 18
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                          PART E:
                               TRIGONOMETRY V




                                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.




                      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.




        Curriculum Development Division
        Ministry of Education Malaysia                                             19
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                      LESSON NOTES




                                                  θ




In the diagram, with reference to the angle , PR is the opposite side, OP is the adjacent side
and OR is the hypotenuse side.




                                            opposite   PR y
                                 sin                  
                                           hypotenuse OR r

                                            adjacent   OP x
                                 cos                  
                                           hypotenuse OR r

                                             opposite PR y
                                   tan               
                                             adjacent OP x




        Curriculum Development Division
        Ministry of Education Malaysia                                                        20
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                          PART F:
                               TRIGONOMETRY VI




                                      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.




                         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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                            21
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry


                                          LESSON NOTES


                   First Quadrant                                   Second Quadrant




                                                                             θ
                            θ




                                                                          y
                     y                                            sin  =     (Positive)
              sin  =  (Positive)                                         r
                     r
                                                                          x
                     x                                            cos  =      (Negative)
              cos  = (Positive)                                            r
                     r
                                                                            y
                     y                                            tan  =      (Negative)
              tan  = (Positive)                                          x
                     x
                                                         (Only sine is positive in the second
    (All trigonometric ratios are positive in the
                                                                       quadrant)
                 first quadrant)

                  Third Quadrant                                    Fourth Quadrant




                            θ                                               θ




                       y                                               y
               sin  =      (Negative)                          sin  =     (Negative)
                        r                                                r
                        x
               cos  =      (Negative)                                   x
                                                                cos  = (Positive)
                          r                                              r
                       y y                                             y
               tan  =       (Positive)                        tan  =     (Negative)
                       x x                                               x

       (Only tangent is positive in the third            (Only cosine is positive in the fourth
                     quadrant)                                         quadrant)



        Curriculum Development Division
        Ministry of Education Malaysia                                                            22
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                        Using acronym: Add Sugar To Coffee (ASTC)


  sin  is positive              cos  is positive               tan  is positive 


  sin  is negative              cos  is negative               tan  is negative 




                    S – only sin  is positive          A – All positive



                   T – only tan  is positive           C – only cos  is positive




        Curriculum Development Division
        Ministry of Education Malaysia                                                   23
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                          TEST YOURSELF F


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




        Curriculum Development Division
        Ministry of Education Malaysia                                              24
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                          PART G:
                             TRIGONOMETRY VII




                                  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




                        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.




        Curriculum Development Division
        Ministry of Education Malaysia                                                  25
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                           LESSON NOTES



                                   The Pythagoras Theorem:




                               The sum of the squares of two sides of
                                a right-angled triangle is equal to the
                                    square of the hypotenuse side.

                                            PR2 + QR2 = PQ2



(a) 3, 4, 5 or equivalent        (b) 5, 12, 13 or equivalent        (c) 8, 15, 17 or equivalent




         Curriculum Development Division
         Ministry of Education Malaysia                                                           26
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



                                          EXAMPLES



1. Write the values of sin , cos  and tan     2. Write the values of sin , cos  and tan 
   from the diagram below.                          from the diagram below.




                                                                         θ
                          θ




                                                                OB2 = (−12)2 + (−5)2
                                                                    = 144 + 25
       OA2 = (−6)2 + 82                                             = 169
           = 100
                                                                 OB = 169
         OA = 100
                                                                    = 13
            = 10
                                                                             y     5
               y 8 4                                              sin  =      
       sin  =                                                             r    13
               r 10 5
               x 6    3                                          cos  = x   12
       cos  =                                                           r  13
               r 10    5                                                   5
                y 8     4                                        tan    =    
                                                                                5
        tan  =                                                         12 12
                x 6    3




        Curriculum Development Division
        Ministry of Education Malaysia                                                           27
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                        TEST YOURSELF G



Write the value of the trigonometric ratios from the diagrams below.

 1.                            2.                                  3.
                                    y
                                                      B(5,4)

                                          B(5,12)
                 θ                         θ
                                           θ                                      θ



                                                               x

       sin  =                          sin  =                         sin  =


       cos  =                          cos  =                         cos  =


       tan  =                          tan  =                         tan  =


 4.                            5.                                  6.



                 θ                                                        θ
                                                  θ




       sin  =                          sin  =                         sin  =


       cos  =                          cos  =                         cos  =


       tan  =                          tan  =                         tan  =




        Curriculum Development Division
        Ministry of Education Malaysia                                                28
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                          PART H:
                             TRIGONOMETRY VIII




                                   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.




                        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.




        Curriculum Development Division
        Ministry of Education Malaysia                                               29
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry




                                          LESSON NOTES

(a)    y = sin x




       The domain for x can be from 0o to 360o or 0 to 2 in radians.
       Important points: (0, 0), (90o, 1), (180o, 0), (270o, −1) and (360o, 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), (90o, 0), (180o, −1), (270o, 0) and (360o, 1)
       Important features: Maximum point (0o, 1) and (360o, 1),
                             Maximum value = 1       Minimum point (180o, −1)
                             Minimum value = 1



        Curriculum Development Division
        Ministry of Education Malaysia                                           30
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



(c)    y = tan x




       Important points: (0o, 0), (180o, 0) and (360o, 0)


                                                          Is there any
                                                         maximum or
                                                        minimum point
                                                        for the tangent
                                                             graph?




        Curriculum Development Division
        Ministry of Education Malaysia                                    31
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



                                             TEST YOURSELF H



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)

                          y = cos x                                   y = sin x




     A(0,1)




           Curriculum Development Division
           Ministry of Education Malaysia                                               32
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



                                          ANSWERS



TEST YOURSELF A:


 1. Opposite side     = AB      2. Opposite side      = PQ      3. Opposite side   = YZ

    Adjacent side     = AC          Adjacent side     = QR         Adjacent side   = XZ

    Hypotenuse side = BC            Hypotenuse side = PR           Hypotenuse side = XY




 4. Opposite side      = LN     5. Opposite side      = UV      6. Opposite side   = RT

    Adjacent side      = MN         Adjacent side     = TU         Adjacent side   = ST

    Hypotenuse side = LM            Hypotenuse side = TV           Hypotenuse side = RS




TEST YOURSELF B:

            AB                              PQ                             YZ
 1. sin  =                     2. sin  =                      3. sin  =
            BC                              PR                             YX
            AC                              QR                              XZ
    cos  =                         cos  =                        cos  =
            BC                              PR                              XY
            AB                              PQ                             YZ
    tan  =                         tan  =                        tan  =
            AC                              QR                             XZ

            LN                              UV                             RT
 4. sin  =                     5. sin  =                      6. sin  =
            LM                              TV                             RS
            MN                              UT                             ST
    cos  =                         cos  =                        cos  =
            LM                              TV                             RS
            LN                              UV                             RT
    tan  =                         tan  =                        tan  =
            MN                              UT                             TS




        Curriculum Development Division
        Ministry of Education Malaysia                                                    33
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



TEST YOURSELF C:

 1. sin  = 1                                    2. cos  = 1
             3                                                  2
         = sin-1 1 = 19o 28                            = cos-1 1 = 60o
                     3                                                   2

 3. tan  = 5                                    4. cos  = 5
             3                                                  8
         = tan-1 5 = 59o 2                             = cos-1 5 = 51o 19
                      3                                               8

 5. tan  = 7.5                                  6. sin  = 6.5
             9.2                                             8.4

         = tan-1 7.5 = 39o 11                          = sin-1 6.5 = 50o 42
                      9.2                                            8.4




TEST YOURSELF D:

                 4                                                    x
 1. tan 32o =                                    2. sin 53.17o =
                 x                                                    7

          x=
                    4
                         = 6.401 cm                             x = 7  sin 53.17o = 5.603 cm
                 tan 32o

                          x                                     o
 3. cos 74o 25 =                                            1   6
                         10                      4. sin 55     =
                                                             3   x
                 x = 10  cos 74o 25                                    6
                                                                x=            o
                                                                                = 7.295 cm
                   = 2.686 cm                                        sin 55 1
                                                                            3


                  x                                                 10
 5. tan 47o =                                    6.   cos 61o =
                 13                                                  x

           x = 13  tan 47o = 13.94 cm                       x=
                                                                      10
                                                                            = 20.63 cm
                                                                    cos 61o




        Curriculum Development Division
        Ministry of Education Malaysia                                                          34
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



TEST YOURSELF F:

 1. 1ST and 2nd                 2. 1st and 3rd                  3. 2nd and 3rd




 4. 1st and 4th                 5. 3rd and 4th                  6. 2nd and 4th




TEST YOURSELF G:

              4                              12                                  4
 1.   sin  =                   2.   sin  =                     3.   sin  =
              5                              13                                  5
              3                               5                                 3
      cos  =                        cos  =                          cos  = 
              5                              13                                 5
              4                              12                                 4
      tan  =                        tan  =                          tan  = 
              3                               5                                 3



                4                               8                                5
 4.   sin  =                  5.   sin  =                    6.   sin  = 
                5                              17                               13
                3                               15                            12
      cos  =                       cos  =                         cos  =
                5                               17                            13
              4                               8                                  5
      tan  =                        tan  =                          tan  = 
              3                              15                                 12




        Curriculum Development Division
        Ministry of Education Malaysia                                               35
Basic Essentials Additional Mathematics Skills (BEAMS) Module
Unit 8: Trigonometry



TEST YOURSELF H:

1.




            y = tan x                        y = sin x          y = cos x



2. (a)    A (0, 1), B (90o, 0), C (180o, 1), D (270o, 0)

     (b) P (90o, 1), Q (180o, 0), R (270o, 1), S (360o, 0)




          Curriculum Development Division
          Ministry of Education Malaysia                                    36

				
DOCUMENT INFO
Shared By:
Stats:
views:34746
posted:1/14/2012
language:English
pages:322