Basic Essential Additional Mathematics Skills Curriculum Development Division Ministry of Education Malaysia Putrajaya 2010
First published 2010 © Curriculum Development Division, Ministry of Education Malaysia Aras 4-8, Blok E9 Pusat Pentadbiran Kerajaan Persekutuan 62604 Putrajaya Tel.: 03-88842000 Fax.: 03-88889917 Website: http://www.moe.gov.my/bpk Copyright reserved. Except for use in a review, the reproduction or utilization of this work in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, and recording is forbidden without prior written permission from the Director of the Curriculum Development Division, Ministry of Education Malaysia.
TABLE OF CONTENTS Preface i Acknowledgement ii Introduction iii Objective iii Module Layout iii BEAMS Module: Unit 1: Negative Numbers Unit 2: Fractions Unit 3: Algebraic Expressions and Algebraic Formulae Unit 4: Linear Equations Unit 5: Indices Unit 6: Coordinates and Graphs of Functions Unit 7: Linear Inequalities Unit 8: Trigonometry Panel of Contributors
ACKNOWLEDGEMENT The Curriculum Development Division, Ministry of Education wishes to express our deepest gratitude and appreciation to all panel of contributors for their expert views and opinions, dedication, and continuous support in the development of this module. ii
Additional Mathematics is an elective subject taught at the upper secondary level. This subject demands a higher level of mathematical thinking and skills compared to that required by the more general Mathematics KBSM. A sound foundation in mathematics is deemed crucial for pupils not only to be able to grasp important concepts taught in Additional Mathematics classes, but also in preparing them for tertiary education and life in general. This Basic Essential Additional Mathematics Skills (BEAMS) Module is one of the continuous efforts initiated by the Curriculum Development Division, Ministry of Education, to ensure optimal development of mathematical skills amongst pupils at large. By the acronym BEAMS itself, it is hoped that this module will serve as a concrete essential support that will fruitfully diminish mathematics anxiety amongst pupils. Having gone through the BEAMS Module, it is hoped that fears induced by inadequate basic mathematical skills will vanish, and pupils will learn mathematics with the due excitement and enjoyment. INTRODUCTION OBJECTIVE The main objective of this module is to help pupils develop a solid essential mathematics foundation and hence, be able to apply confidently their mathematical skills, specifically in school and more significantly in real-life situations. iii MODULE LAYOUT This module encompasses all mathematical skills and knowledge taught in the lower secondary level and is divided into eight units as follows: Unit 1: Negative Numbers Unit 2: Fractions Unit 3: Algebraic Expressions and Algebraic Formulae Unit 4: Linear Equations Unit 5: Indices Unit 6: Coordinates and Graphs of Functions Unit 7: Linear Inequalities Unit 8: Trigonometry
Each unit stands alone and can be used as a comprehensive revision of a particular topic. Most of the units follow as much as possible the following layout: Module Overview Objectives Teaching and Learning Strategies Lesson Notes Examples Test Yourself Answers The “Lesson Notes”, “Examples” and “Test Yourself” in each unit can be used as supplementary or reinforcement handouts to help pupils recall and understand the basic concepts and skills needed in each topic. Teachers are advised to study the whole unit prior to classroom teaching so as to familiarize with its content. By completely examining the unit, teachers should be able to select any part in the unit that best fit the needs of their pupils. It is reminded that each unit in this module is by no means a complete lesson, rather as a supporting material that should be ingeniously integrated into the Additional Mathematics teaching and learning processes. At the outset, this module is aimed at furnishing pupils with the basic mathematics foundation prior to the learning of Additional Mathematics, however the usage could be broadened. This module can also be benefited by all pupils, especially those who are preparing for the Penilaian Menengah Rendah (PMR) Examination. iv
Advisors: Haji Ali bin Ab. Ghani AMN Director Curriculum Development Division Dr. Lee Boon Hua Deputy Director (Humanities) Curriculum Development Division Mohd. Zanal bin Dirin Deputy Director (Science and Technology) Curriculum Development Division Editorial Advisor: Aziz bin Saad Principal Assistant Director (Head of Science and Mathematics Sector) Curriculum Development Division Editors: Dr. Rusilawati binti Othman Assistant Director (Head of Secondary Mathematics Unit) Curriculum Development Division Aszunarni binti Ayob Assistant Director Curriculum Development Division Rosita binti Mat Zain Assistant Director Curriculum Development Division PANEL OF CONTRIBUTORS
Abdul Rahim bin Bujang SM Tun Fatimah, Johor Ali Akbar bin Asri SM Sains, Labuan Amrah bin Bahari SMK Dato’ Sheikh Ahmad, Arau, Perlis Aziyah binti Paimin SMK Kompleks KLIA, , Negeri Sembilan Bashirah binti Seleman SMK Sultan Abdul Halim, Jitra, Kedah Bibi Kismete binti Kabul Khan SMK Jelapang Jaya, Ipoh, Perak Che Rokiah binti Md. Isa SMK Dato’ Wan Mohd. Saman, Kedah Cheong Nyok Tai SMK Perempuan, Kota Kinabalu, Sabah Ding Hong Eng SM Sains Alam Shah, Kuala Lumpur Esah binti Daud SMK Seri Budiman, Kuala Terengganu Haspiah binti Basiran SMK Tun Perak, Jasin, Melaka Noorliah binti Ahmat SM Teknik, Kuala Lumpur Ali Akbar bin Asri Nor A’idah binti Johari SM Sains, Labuan SMK Teknik Setapak, Selangor Amrah bin Bahari Nor Dalina binti Idris SMK Dato’ Sheikh Ahmad, Arau, Perlis SMK Syed Alwi, Kangar, Perlis Hon May Wan SMK Tasek Damai, Ipoh, Perak Horsiah binti Ahmad SMK Tun Perak, Jasin, Melaka Kalaimathi a/p Rajagopal SMK Sungai Layar, Sungai Petani, Kedah Kho Choong Quan SMK Ulu Kinta, Ipoh, Perak Lau Choi Fong SMK Hulu Klang, Selangor Loh Peh Choo SMK Bandar Baru Sungai Buloh, Selangor Mohd. Misbah bin Ramli SMK Tunku Sulong, Gurun, Kedah Noor Aida binti Mohd. Zin SMK Tinggi Kajang, Kajang, Selangor Noor Ishak bin Mohd. Salleh SMK Laksamana, Kota Tinggi, Johor Noorliah binti Ahmat SM Teknik, Kuala Lumpur Nor A’idah binti Johari SMK Teknik Setapak, Selangor Writers:
Layout and Illustration: Aszunarni binti Ayob Mohd. Lufti bin Mahpudz Assistant Director Assistant Director Curriculum Development Division Curriculum Development Division Writers: Nor Dalina binti Idris SMK Syed Alwi, Kangar, Perlis Norizatun binti Abdul Samid SMK Sultan Badlishah, Kulim, Kedah Pahimi bin Wan Salleh Maktab Sultan Ismail, Kelantan Rauziah binti Mohd. Ayob SMK Bandar Baru Salak Tinggi, Selangor Rohaya binti Shaari SMK Tinggi Bukit Merajam, Pulau Pinang Roziah binti Hj. Zakaria SMK Taman Inderawasih, Pulau Pinang Shakiroh binti Awang SM Teknik Tuanku Jaafar, Negeri Sembilan Sharina binti Mohd. Zulkifli SMK Agama, Arau, Perlis Sim Kwang Yaw SMK Petra, Kuching, Sarawak Suhaimi bin Mohd. Tabiee SMK Datuk Haji Abdul Kadir, Pulau Pinang Suraiya binti Abdul Halim SMK Pokok Sena, Pulau Pinang Tan Lee Fang SMK Perlis, Perlis Tempawan binti Abdul Aziz SMK Mahsuri, Langkawi, Kedah Turasima binti Marjuki SMKA Simpang Lima, Selangor Wan Azlilah binti Wan Nawi SMK Putrajaya Presint 9(1), WP Putrajaya Zainah binti Kebi SMK Pandan, Kuantan, Pahang Zaleha binti Tomijan SMK Ayer Puteh Dalam, Pendang, Kedah Zariah binti Hassan SMK Dato’ Onn, Butterworth, Pulau Pinang
Unit 1: Negative Numbers UNIT 1 NEGATIVE NUMBERS B a s i c E s s e n t i a l A d d i t i o n a l M a t h e m a t i c s S k i l l s Curriculum Development Division Ministry of Education Malaysia
TABLE OF CONTENTS Module Overview 1 Part A: Addition and Subtraction of Integers Using Number Lines 2 1.0 Representing Integers on a Number Line 3 2.0 Addition and Subtraction of Positive Integers 3 3.0 Addition and Subtraction of Negative Integers 8 Part B: Addition and Subtraction of Integers Using the Sign Model 15 Part C: Further Practice on Addition and Subtraction of Integers 19 Part D: Addition and Subtraction of Integers Including the Use of Brackets 25 Part E: Multiplication of Integers 33 Part F: Multiplication of Integers Using the Accept-Reject Model 37 Part G: Division of Integers 40 Part H: Division of Integers Using the Accept-Reject Model 44 Part I: Combined Operations Involving Integers 49 Answers 52
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 1 Curriculum Development Division Ministry of Education Malaysia MODULE OVERVIEW 1. Negative Numbers is the very basic topic which must be mastered by every pupil. 2. The concept of negative numbers is widely used in many Additional Mathematics topics, for example: (a) Functions (b) Quadratic Equations (c) Quadratic Functions (d) Coordinate Geometry (e) Differentiation (f) Trigonometry Thus, pupils must master negative numbers in order to cope with topics in Additional Mathematics. 3. The aim of this module is to reinforce pupils‟ understanding on the concept of negative numbers. 4. This module is designed to enhance the pupils‟ skills in using the concept of number line; using the arithmetic operations involving negative numbers; solving problems involving addition, subtraction, multiplication and division of negative numbers; and applying the order of operations to solve problems. 5. It is hoped that this module will enhance pupils‟ understanding on negative numbers using the Sign Model and the Accept-Reject Model. 6. This module consists of nine parts and each part consists of learning objectives which can be taught separately. Teachers may use any parts of the module as and when it is required.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 2 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES The concept of negative numbers can be confusing and difficult for pupils to grasp. Pupils face difficulty when dealing with operations involving positive and negative integers. Strategy: Teacher should ensure that pupils understand the concept of positive and negative integers using number lines. Pupils are also expected to be able to perform computations involving addition and subtraction of integers with the use of the number line. PART A: ADDITION AND SUBTRACTION OF INTEGERS USING NUMBER LINES LEARNING OBJECTIVE Upon completion of Part A, pupils will be able to perform computations involving combined operations of addition and subtraction of integers using a number lines.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 3 Curriculum Development Division Ministry of Education Malaysia PART A: ADDITION AND SUBTRACTION OF INTEGERS USING NUMBER LINES 1.0 Representing Integers on a Number Line Positive whole numbers, negative numbers and zero are all integers. Integers can be represented on a number line. Note: i) –3 is the opposite of +3 ii) – (–2) becomes the opposite of negative 2, that is, positive 2. 2.0 Addition and Subtraction of Positive Integers –3 –2 –1 0 1 2 3 4 LESSON NOTES Rules for Adding and Subtracting Positive Integers When adding a positive integer, you move to the right on a number line. When subtracting a positive integer, you move to the left on a number line. –3 –2 –1 0 1 2 3 4 –3 –2 –1 0 1 2 3 4 Positive integers may have a plus sign in front of them, like +3, or no sign in front, like 3.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 4 Curriculum Development Division Ministry of Education Malaysia (i) 2 + 3 Alternative Method: EXAMPLES Adding a positive integer: Start by drawing an arrow from 0 to 2, and then, draw an arrow of 3 units to the right: 2 + 3 = 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Start with 2 Add a positive 3 Adding a positive integer: Start at 2 and move 3 units to the right: 2 + 3 = 5 Make sure you start from the position of the first integer. –5 –4 –3 –2 –1 0 1 2 3 4 5 6
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 5 Curriculum Development Division Ministry of Education Malaysia (ii) –2 + 5 Alternative Method: Adding a positive integer: Start by drawing an arrow from 0 to –2, and then, draw an arrow of 5 units to the right: –2 + 5 = 3 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Add a positive 5 Make sure you start from the position of the first integer. –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Adding a positive integer: Start at –2 and move 5 units to the right: –2 + 5 = 3
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 6 Curriculum Development Division Ministry of Education Malaysia (iii) 2 – 5 = –3 Alternative Method: –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Subtracting a positive integer: Start by drawing an arrow from 0 to 2, and then, draw an arrow of 5 units to the left: 2 – 5 = –3 Subtract a positive 5 Subtracting a positive integer: Start at 2 and move 5 units to the left: 2 – 5 = –3 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Make sure you start from the position of the first integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 7 Curriculum Development Division Ministry of Education Malaysia (iv) –3 – 2 = –5 Alternative Method: Subtracting a positive integer: Start by drawing an arrow from 0 to –3, and then, draw an arrow of 2 units to the left: –3 – 2 = –5 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Subtract a positive 2 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Subtracting a positive integer: Start at –3 and move 2 units to the left: –3 – 2 = –5 Make sure you start from the position of the first integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 8 Curriculum Development Division Ministry of Education Malaysia 3.0 Addition and Subtraction of Negative Integers Consider the following operations: 4 – 1 = 3 4 – 2 = 2 4 – 3 = 1 4 – 4 = 0 4 – 5 = –1 4 – 6 = –2 Note that subtracting an integer gives the same result as adding its opposite. Adding or subtracting a negative integer goes in the opposite direction to adding or subtracting a positive integer. –3 –2 –1 0 1 2 3 4 –3 –2 –1 0 1 2 3 4 –3 –2 –1 0 1 2 3 4 –3 –2 –1 0 1 2 3 4 4 + (–5) = –1 –3 –2 –1 0 1 2 3 4 –3 –2 –1 0 1 2 3 4 4 + (–6) = –2 4 + (–1) = 3 4 + (–2) = 2 4 + (–3) = 1 4 + (–4) = 0
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 9 Curriculum Development Division Ministry of Education Malaysia Rules for Adding and Subtracting Negative Integers When adding a negative integer, you move to the left on a number line. When subtracting a negative integer, you move to the right on a number line. –3 –2 –1 0 1 2 3 4 –3 –2 –1 0 1 2 3 4
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 10 Curriculum Development Division Ministry of Education Malaysia (i) –2 + (–1) = –3 Alternative Method: –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Adding a negative integer: Start at –2 and move 1 unit to the left: –2 + (–1) = –3 EXAMPLES –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Adding a negative integer: Start by drawing an arrow from 0 to –2, and then, draw an arrow of 1 unit to the left: –2 + (–1) = –3 Add a negative 1 Make sure you start from the position of the first integer. This operation of –2 + (–1) = –3 is the same as –2 –1 = –3.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 11 Curriculum Development Division Ministry of Education Malaysia (ii) 1 + (–3) = –2 Alternative Method: –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Adding a negative integer: Start at 1 and move 3 units to the left: 1 + (–3) = –2 Add a negative 3 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Adding a negative integer: Start by drawing an arrow from 0 to 1, then, draw an arrow of 3 units to the left: 1 + (–3) = –2 Make sure you start from the position of the first integer. This operation of 1 + (–3) = –2 is the same as 1 – 3 = –2
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 12 Curriculum Development Division Ministry of Education Malaysia (iii) 3 – (–3) = 6 Alternative Method: –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Subtracting a negative integer: Start at 3 and move 3 units to the right: 3 – (–3) = 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Subtracting a negative integer: Start by drawing an arrow from 0 to 3, and then, draw an arrow of 3 units to the right: 3 – (–3) = 6 Subtract a negative 3 This operation of 3 – (–3) = 6 is the same as 3 + 3 = 6 Make sure you start from the position of the first integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 13 Curriculum Development Division Ministry of Education Malaysia (iv) –5 – (–8) = 3 Alternative Method: –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Subtracting a negative integer: Start at –5 and move 8 units to the right: –5 – (–8) = 3 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 Subtract a negative 8 This operation of –5 – (–8) = 3 is the same as –5 + 8 = 3 3 + 3 = 6 Subtracting a negative integer: Start by drawing an arrow from 0 to –5, and then, draw an arrow of 8 units to the right: –5 – (–8) = 3
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 14 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. –2 + 4 2. 3 + (–6) 3. 2 – (–4) 4. 3 – 5 + (–2) 5. –5 + 8 + (–5) –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 15 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES This part emphasises the first alternative method which include activities and mathematical games that can help pupils understand further and master the operations of positive and negative integers. Strategy: Teacher should ensure that pupils are able to perform computations involving addition and subtraction of integers using the Sign Model. PART B: ADDITION AND SUBTRACTION OF INTEGERS USING THE SIGN MODEL LEARNING OBJECTIVE Upon completion of Part B, pupils will be able to perform computations involving combined operations of addition and subtraction of integers using the Sign Model.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 16 Curriculum Development Division Ministry of Education Malaysia PART B: ADDITION AND SUBTRACTION OF INTEGERS USING THE SIGN MODEL In order to help pupils have a better understanding of positive and negative integers, we have designed the Sign Model. Example 1 What is the value of 3 – 5? NUMBER SIGN 3 + + + –5 – – – – – WORKINGS i. Pair up the opposite signs. ii. The number of the unpaired signs is the answer. Answer –2 + + + LESSON NOTES EXAMPLES The Sign Model This model uses the „+‟ and „–‟ signs. A positive number is represented by „+‟ sign. A negative number is represented by „–‟ sign.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 17 Curriculum Development Division Ministry of Education Malaysia Example 2 What is the value of 35 ? 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 35 ? NUMBER SIGN –3 – – – +5 + + + + + WORKINGS i. Pair up the opposite signs. ii. The number of unpaired signs is the answer. Answer 2 _ + + + _ + _ +
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 18 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. –4 + 8 2. –8 – 4 3. 12 – 7 4. –5 – 5 5. 5 – 7 – 4 6. –7 + 4 – 3 7. 4 + 3 – 7 8. 6 – 2 + 8 9. –3 + 4 + 6 TEST YOURSELF B
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 19 Curriculum Development Division Ministry of Education Malaysia PART C: FURTHER PRACTICE ON ADDITION AND SUBTRACTION OF INTEGERS TEACHING AND LEARNING STRATEGIES This part emphasises addition and subtraction of large positive and negative integers. Strategy: Teacher should ensure the pupils are able to perform computation involving addition and subtraction of large integers. LEARNING OBJECTIVE Upon completion of Part C, pupils will be able to perform computations involving addition and subtraction of large integers.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 20 Curriculum Development Division Ministry of Education Malaysia PART C: FURTHER PRACTICE ON ADDITION AND SUBTRACTION OF INTEGERS In Part A and Part B, the method of counting off the answer on a number line and the Sign Model were used to perform computations involving addition and subtraction of small integers. However, these methods are not suitable if we are dealing with large integers. We can use the following Table Model in order to perform computations involving addition and subtraction of large integers. LESSON NOTES Steps for Adding and Subtracting Integers 1. Draw a table that has a column for + and a column for –. 2. Write down all the numbers accordingly in the column. 3. If the operation involves numbers with the same signs, simply add the numbers and then put the respective sign in the answer. (Note that we normally do not put positive sign in front of a positive number) 4. If the operation involves numbers with different signs, always subtract the smaller number from the larger number and then put the sign of the larger number in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 21 Curriculum Development Division Ministry of Education Malaysia Examples: i) 34 + 37 = + – 34 37 +71 ii) 65 – 20 = + – 65 20 +45 iii) –73 + 22 = + – 22 73 –51 iv) 228 – 338 = + – 228 338 –110 Subtract the smaller number from the larger number and put the sign of the larger number in the answer. We can just write the answer as 45 instead of +45. Subtract the smaller number from the larger number and put the sign of the larger number in the answer. Subtract the smaller number from the larger number and put the sign of the larger number in the answer. Add the numbers and then put the positive sign in the answer. We can just write the answer as 71 instead of +71.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 22 Curriculum Development Division Ministry of Education Malaysia v) –428 – 316 = + – 428 316 –744 vi) –863 – 127 + 225 = + – 225 863 127 225 990 –765 vii) 234 – 675 – 567 = + – 234 675 567 234 1242 –1008 Add the numbers and then put the negative sign in the answer. Add the two numbers in the „–‟ column and bring down the number in the „+‟ column. Subtract the smaller number from the larger number in the third row and put the sign of the larger number in the answer. Add the two numbers in the „–‟ column and bring down the number in the „+‟ column. Subtract the smaller number from the larger number in the third row and put the sign of the larger number in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 23 Curriculum Development Division Ministry of Education Malaysia viii) –482 + 236 – 718 = + – 236 482 718 236 1200 –964 ix) –765 – 984 + 432 = + – 432 765 984 432 1749 –1317 x) –1782 + 436 + 652 = + – 436 652 1782 1088 1782 –694 Add the two numbers in the „–‟ column and bring down the number in the „+‟ column. Subtract the smaller number from the larger number in the third row and put the sign of the larger number in the answer. Add the two numbers in the „–‟ column and bring down the number in the „+‟ column. Subtract the smaller number from the larger number in the third row and put the sign of the larger number in the answer. Add the two numbers in the „+‟ column and bring down the number in the „–‟ column. Subtract the smaller number from the larger number in the third row and put the sign of the larger number in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 24 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. 47 – 89 2. –54 – 48 3. 33 – 125 4. –352 – 556 5. 345 – 437 – 456 6. –237 + 564 – 318 7. –431 + 366 – 778 8. –652 – 517 + 887 9. –233 + 408 – 689 TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 25 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES This part emphasises the second alternative method which include activities to enhance pupils‟ understanding and mastery of the addition and subtraction of integers, including the use of brackets. Strategy: Teacher should ensure that pupils understand the concept of addition and subtraction of integers, including the use of brackets, using the Accept-Reject Model. PART D: ADDITION AND SUBTRACTION OF INTEGERS INCLUDING THE USE OF BRACKETS LEARNING OBJECTIVE Upon completion of Part D, pupils will be able to perform computations involving combined operations of addition and subtraction of integers, including the use of brackets, using the Accept-Reject Model.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 26 Curriculum Development Division Ministry of Education Malaysia PART D: ADDITION AND SUBTRACTION OF INTEGERS INCLUDING THE USE OF BRACKETS To Accept or To Reject? Answer + ( 5 ) Accept +5 +5 – ( 2 ) Reject +2 –2 + (–4) Accept –4 –4 – (–8) Reject –8 +8 LESSON NOTES The Accept - Reject Model „+‟ sign means to accept. „–‟ sign means to reject.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 27 Curriculum Development Division Ministry of Education Malaysia i) 5 + (–1) = Number To Accept or To Reject? Answer 5 + (–1) Accept 5 Accept –1 +5 –1 + + + + + – 5 + (–1) = 4 We can also solve this question by using the Table Model as follows: 5 + (–1) = 5 – 1 + – 5 1 +4 EXAMPLES This operation of 5 + (–1) = 4 is the same as 5 – 1 = 4 Subtract the smaller number from the larger number and put the sign of the larger number in the answer. We can just write the answer as 4 instead of +4.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 28 Curriculum Development Division Ministry of Education Malaysia ii) –6 + (–3) = Number To Accept or To Reject? Answer –6 + (–3) Reject 6 Accept –3 –6 –3 – – – – – – – – – –6 + (–3) = –9 We can also solve this question by using the Table Model as follows: –6 + (–3) = –6 – 3 = + – 6 3 –9 This operation of –6 + (–3) = –9 is the same as –6 –3 = –9 Add the numbers and then put the negative sign in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 29 Curriculum Development Division Ministry of Education Malaysia iii) –7 – (–4) = Number To Accept or To Reject? Answer –7 – (–4) Reject 7 Reject –4 –7 +4 – – – – – – – + + + + –7 – (–4) = –3 We can also solve this question by using the Table Model as follows: –7 – (–4) = –7 + 4 = + – 4 7 –3 This operation of –7 – (–4) = –3 is the same as –7 + 4 = –3 Subtract the smaller number from the larger number and put the sign of the larger number in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 30 Curriculum Development Division Ministry of Education Malaysia iv) –5 – (3) = Number To Accept or To Reject? Answer –5 – (3) Reject 5 Reject 3 –5 –3 – – – – – – – – – 5 – (3) = –8 We can also solve this question by using the Table Model as follows: –5 – (3) = –5 – 3 = + – 5 3 –8 This operation of –5 – (3) = –8 is the same as –5 – 3 = –8 Add the numbers and then put the negative sign in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 31 Curriculum Development Division Ministry of Education Malaysia v) –35 + (–57) = –35 – 57 = Using the Table Model: + – 35 57 –92 vi) –123 – (–62) = –123 + 62 = Using the Table Model: + – 62 123 –61 This operation of –35 + (–57) is the same as –35 – 57 Add the numbers and then put the negative sign in the answer. Subtract the smaller number from the larger number and put the sign of the larger number in the answer. This operation of –123 – (–62) is the same as –123 + 62
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 32 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. –4 + (–8) 2. 8 – (–4) 3. –12 + (–7) 4. –5 + (–5) 5. 5 – (–7) + (–4) 6. 7 + (–4) – (3) 7. 4 + (–3) – (–7) 8. –6 – (2) + (8) 9. –3 + (–4) + (6) 10. –44 + (–81) 11. 118 – (–43) 12. –125 + (–77) 13. –125 + (–239) 14. 125 – (–347) + (–234) 15. 237 + (–465) – (378) 16. 412 + (–334) – (–712) 17. –612 – (245) + (876) 18. –319 + (–412) + (606) TEST YOURSELF D
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 33 Curriculum Development Division Ministry of Education Malaysia PART E: MULTIPLICATION OF INTEGERS TEACHING AND LEARNING STRATEGIES This part emphasises the multiplication rules of integers. Strategy: Teacher should ensure that pupils understand the multiplication rules to perform computations involving multiplication of integers. LEARNING OBJECTIVE Upon completion of Part E, pupils will be able to perform computations involving multiplication of integers.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 34 Curriculum Development Division Ministry of Education Malaysia PART E: MULTIPLICATION OF INTEGERS Consider the following pattern: 3 × 3 = 9 32 6 31 3 30 0 The result is reduced by 3 in 3(1) 3 every step. 3(2) 6 3(3) 9 (3)3 9 (3)2 6 (3)1 3 (3) 0 0 The result is increased by 3 in (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. positive × positive = positive (+) × (+) = (+) positive × negative = negative (+) × (–) = (–) negative × positive = negative (–) × (+) = (–) negative × negative = positive (–) × (–) = (+) LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 35 Curriculum Development Division Ministry of Education Malaysia 1. When multiplying two integers of the same signs, the answer is positive integer. (a) 4 × 3 = 12 (b) –8 × –6 = 48 2. When multiplying two integers of the different signs, the answer is negative integer. (a) –4 × (3) = –12 (b) 8 × (–6) = –48 3. When any integer is multiplied by zero, the answer is always zero. (a) (4) × 0 = 0 (b) (–8) × 0 = 0 (c) 0 × (5) = 0 (d) 0 × (–7) = 0 EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 36 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. –4 × (–8) 2. 8 × (–4) 3. –12 × (–7) 4. –5 × (–5) 5. 5 × (–7) × (–4) 6. 7 × (–4) × (3) 7. 4 × (–3) × (–7) 8. (–6) × (2) × (8) 9. (–3) × (–4) × (6) TEST YOURSELF E
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 37 Curriculum Development Division Ministry of Education Malaysia PART F: MULTIPLICATION OF INTEGERS USING THE ACCEPT-REJECT MODEL TEACHING AND LEARNING STRATEGIES This part emphasises the second alternative method which include activities to enhance the pupils‟ understanding and mastery of the multiplication of integers. Strategy: Teacher should ensure that pupils understand the multiplication rules of integers using the Accept-Reject Model. Pupils can then perform computations involving multiplication of integers. LEARNING OBJECTIVE Upon completion of Part F, pupils will be able to perform computations involving multiplication of integers using the Accept-Reject Model.