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Basic Essential Additional Mathematics Skills Curriculum Development Division Ministry of Education Malaysia Putrajaya 2010 First published 2010 © Curriculum Development Division, Ministry of Education Malaysia Aras 4-8, Blok E9 Pusat Pentadbiran Kerajaan Persekutuan 62604 Putrajaya Tel.: 03-88842000 Fax.: 03-88889917 Website: http://www.moe.gov.my/bpk Copyright reserved. Except for use in a review, the reproduction or utilization of this work in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, and recording is forbidden without prior written permission from the Director of the Curriculum Development Division, Ministry of Education Malaysia. TABLE OF CONTENTS Preface i Acknowledgement ii Introduction iii Objective iii Module Layout iii BEAMS Module: Unit 1: Negative Numbers Unit 2: Fractions Unit 3: Algebraic Expressions and Algebraic Formulae Unit 4: Linear Equations Unit 5: Indices Unit 6: Coordinates and Graphs of Functions Unit 7: Linear Inequalities Unit 8: Trigonometry Panel of Contributors ACKNOWLEDGEMENT The Curriculum Development Division, Ministry of Education wishes to express our deepest gratitude and appreciation to all panel of contributors for their expert views and opinions, dedication, and continuous support in the development of this module. ii 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 33 answer in its simplest form. = 6 2 43 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 m2 2 Multiply the first fraction with the second denominator Multiply the second fraction with the first denominator 2 ( m2) Multiply the first fraction by the denominator m m = of the second fraction. m2 2 2 ( m2) Multiply the second fraction by the denominator of the first fraction. 2m mm 2 = Remember to use 2m 2 2m 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 5n 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 x4 x 1 x 20. x2 x2 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 4n 21. 22. 2 4 3n 9n 2 r 5 2r 2 p3 p2 23. 24. 5 15 r p2 2p 2n 3 4n 3 3m n n 3 25. 26. 5n 2 10n mn n 5m mn m3 nm 27. 28. 5m mn 3m mn 3 5n 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: 23 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. n9 n 4m = 2n 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! 63 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 n5 25. 26. 27. 10 n 2 m 5n n3 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. 43 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) 32k 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) 32k 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 x2 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 12 2e 1 8. m n 2 m2m n 9. f g f g g 2 f g 10 . h i h i 2ih 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 2x 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 3x 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 xy Group the like terms 3 x x y2 Simplify the terms. 2 x y2 y2 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: xa 1 x a) Given that 2 , express x in terms b) Given that y , express x in terms xa 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 1x 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 1x 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) x5 x5 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 BA BA AB AB C) C N) C BA BA 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 62 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) x25 Subtract 2 from both Alternative Method: sides of the equation. x+2–2=5–2 x25 x 52 x=5–2 Simplify the LHS. x3 x=3 Simplify the RHS. (ii) x35 Add 3 to both sides of Alternative Method: the equation. x–3+3=5+3 x 35 x=5+3 Simplify the LHS. x 53 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 m4 m=4 Simplify the RHS. m (ii) 4 3 Multiply both sides of Alternative Method: m the equation by 3. 3 43 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 x2 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 x7 2 Simplify both sides of x7 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 bc 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 bc a EXAMPLES x Solve the equation 4 1. 3 Solution: Method 1 x 4 1 3 x 44 = 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. 35 2. 2 1 3. 27 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 x3 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 x3 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 x7 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 mn 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) 27 234 (ii ) 7 7 2 7 (7 7 ) 73 7 12 (iii ) ( y 1) 2 ( y 1)3 [( y 1)( y 1)] [( y 1)( y 1)( y 1)] ( y 1)5 ( y 1) 23 am an amn 4.0 Simplifying Multiplication of Numbers, Expressed In Index Notation with the Same Base (i) 6 3 6 4 6 6 3 41 68 (ii ) (5) 3 (5) 8 (5) 38 (5)11 5 15 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 w911 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 31 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 3 348 2 3 312 2 3 Sum up the indices with the same (ii ) 53 5 7 714 7 3 537 7143 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 52 n 55 m 7 n10 (ii) 3t 6 2s 3 5r 2 30t 6 s 3 r 2 2 4 1 4 13 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: mn 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 mn 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 53 (b) What is 20? (c) What is a0? 1 1 555555555 (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 mm a 0 m am am am 1 am am an amn 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 73 4k 4 5k 8h 3 8 8 (iii) 2 h 32 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 83 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 53 (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 q125 (b) 4 y9 8 y7 q7 (c) 35m10 (d) 214 b11 15m8 28 b8 3. Simplify the following. (a) 36m9 n 5 9 94 51 (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 25 35 x10 (ii ) (e 2 f 3 g 4 ) 5 e 25 f 35 g 45 e10 f 15 g 20 4 4 1 1 (iii ) a 3b a 34 b14 5 5 a12b 4 54 a12b 4 625 1 12 4 a b 625 5 2m 4 (2) 5 m 45 (iv ) n3 n 35 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 35 p 6 q 7 ( v) 12 p 3 q 2 12 p 3q 2 32 p1563 q 72 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 77 (ii ) 7 2 75 77777 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 1 101 1 10 10 1 1 102 2 100 10 1 10n 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 53 [(1) 2 ] 3 215 32768 (c) 2 (d) 3 23 3 2 2 7 5 (e) 32 (f) 3 4 5 23 2 2. (a) Simplify the following. (i) 2 6 32 4 2 64 3 24 (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 (215 )( x 35 ) (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. 107 102 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 29 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 25 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 4m (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) x3 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) x2 (c) x 2 (d) x3 ______________________________________________________________________________ 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 x9 Simplify. (2) Solve p 3 2. Solution: Add 3 to both sides of the p3 2 inequality. p 3 3 2 3 p5 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) m5 0 (b) x26 (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 p2 (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 x3 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. p2 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) p7 (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