Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 38 Curriculum Development Division Ministry of Education Malaysia PART F: MULTIPLICATION OF INTEGERS USING THE ACCEPT-REJECT MODEL The Accept-Reject Model In order to help pupils have a better understanding of multiplication of integers, we have designed the Accept-Reject Model. Notes: (+) × (+) : The first sign in the operation will determine whether to accept or to reject the second sign. Multiplication Rules: To Accept or to Reject Answer (2) × (3) Accept + 6 (–2) × (–3) Reject – 6 (2) × (–3) Accept – –6 (–2) × (3) Reject + –6 Sign To Accept or To Reject Answer ( + ) × ( + ) Accept + ( – ) × ( – ) Reject – ( + ) × ( – ) Accept – – ( – ) × ( + ) Reject + – LESSON NOTES EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 39 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. 3 × (–5) = 2. –4 × (–8) = 3. 6 × (5) = 4. 8 × (–6) = 5. – (–5) × 7 = 6. (–30) × (–4) = 7. 4 × 9 × (–6) = 8. (–3) × 5 × (–6) = 9. (–2) × ( –9) × (–6) = 10. –5× (–3) × (+4) = 11. 7 × (–2) × (+3) = 12. 5 × 8 × (–2) = TEST YOURSELF F
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 40 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES This part emphasises the division rules of integers. Strategy: Teacher should ensure that pupils understand the division rules of integers to perform computation involving division of integers. PART G: DIVISION OF INTEGERS LEARNING OBJECTIVE Upon completion of Part G, pupils will be able to perform computations involving division of integers.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 41 Curriculum Development Division Ministry of Education Malaysia PART G: DIVISION OF INTEGERS Consider the following pattern: 3 × 2 = 6, then 6 ÷ 2 = 3 and 6 ÷ 3 = 2 3 × (–2) = –6, then (–6) ÷ 3 = –2 and (–6) ÷ (–2) = 3 (–3) × 2 = –6, then (–6) ÷ 2 = –3 and (–6) ÷ (–3) = 2 (–3) × (–2) = 6, then 6 ÷ (–3) = –2 and 6 ÷ (–2) = –3 Rules of Division 1. Division of two integers of the same signs results in a positive integer. i.e. positive ÷ positive = positive (+) ÷ (+) = (+) negative ÷ negative = positive (–) ÷ (–) = (+) 2. Division of two integers of different signs results in a negative integer. i.e. positive ÷ negative = negative (+) ÷ (–) = (–) negative ÷ positive = negative (–) ÷ (+) = (–) 3. Division of any number by zero is undefined. LESSON NOTES Undefined means “this operation does not have a meaning and is thus not assigned an interpretation!” Source: http://www.sn0wb0ard.com
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 42 Curriculum Development Division Ministry of Education Malaysia 1. Division of two integers of the same signs results in a positive integer. (a) (12) ÷ (3) = 4 (b) (–8) ÷ (–2) = 4 2. Division of two integers of different signs results in a negative integer. (a) (–12) ÷ (3) = –4 (b) (+8) ÷ (–2) = –4 3. Division of zero by any number will always give zero as an answer. (a) 0 ÷ (5) = 0 (b) 0 ÷ (–7) = 0 EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 43 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. (–24) ÷ (–8) 2. 8 ÷ (–4) 3. (–21) ÷ (–7) 4. (–5) ÷ (–5) 5. 60 ÷ (–5) ÷ (–4) 6. 36 ÷ (–4) ÷ (3) 7. 42 ÷ (–3) ÷ (–7) 8. (–16) ÷ (2) ÷ (8) 9. (–48) ÷ (–4) ÷ (6) TEST YOURSELF G
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 44 Curriculum Development Division Ministry of Education Malaysia PART H: DIVISION OF INTEGERS USING THE ACCEPT-REJECT MODEL TEACHING AND LEARNING STRATEGIES This part emphasises the alternative method that include activities to help pupils further understand and master division of integers. Strategy: Teacher should make sure that pupils understand the division rules of integers using the Accept-Reject Model. Pupils can then perform division of integers, including the use of brackets. LEARNING OBJECTIVE Upon completion of Part H, pupils will be able to perform computations involving division of integers using the Accept-Reject Model.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 45 Curriculum Development Division Ministry of Education Malaysia PART H: DIVISION OF INTEGERS USING THE ACCEPT-REJECT MODEL In order to help pupils have a better understanding of division of integers, we have designed the Accept-Reject Model. Notes: (+) ÷ (+) : The first sign in the operation will determine whether to accept or to reject the second sign. : The sign of the numerator will determine whether to accept or to reject the sign of the denominator. Division Rules: Sign To Accept or To Reject Answer ( + ) ÷ ( + ) Accept + + ( – ) ÷ ( – ) Reject – + ( + ) ÷ ( – ) Accept – – ( – ) ÷ ( + ) Reject + – ( ) ( ) LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 46 Curriculum Development Division Ministry of Education Malaysia To Accept or To Reject Answer (6) ÷ (3) Accept + 2 (–6) ÷ (–3) Reject – 2 (+6) ÷ (–3) Accept – – 2 (–6) ÷ (3) Reject + – 2 Division [Fraction Form]: Sign To Accept or To Reject Answer ( ) ( ) Accept + + ( ) ( ) Reject – + ( ) ( ) Accept – – ( ) ( ) Reject + – EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 47 Curriculum Development Division Ministry of Education Malaysia To Accept or To Reject Answer ( 2) ( 8) Accept + 4 ( 2) ( 8) Reject – 4 ( 2) ( 8) Accept – – 4 ( 2) ( 8) Reject + – 4 EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 48 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. 18 ÷ (–6) 2. 2 12 3. 8 24 4. 5 25 5. 3 6 6. – (–35) ÷ 7 7. (–32) ÷ (–4) 8. (–45) ÷ 9 ÷ (–5) 9. ( 6) ( 30) 10. ( 5) 80 11. 12 ÷ (–3) ÷ (–2) 12. – (–6) ÷ (3) TEST YOURSELF H
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 49 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES This part emphasises the order of operations when solving combined operations involving integers. Strategy: Teacher should make sure that pupils are able to understand the order of operations or also known as the BODMAS rule. Pupils can then perform combined operations involving integers. PART I: COMBINED OPERATIONS INVOLVING INTEGERS LEARNING OBJECTIVES Upon completion of Part I, pupils will be able to: 1. perform computations involving combined operations of addition, subtraction, multiplication and division of integers to solve problems; and 2. apply the order of operations to solve the given problems.
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 50 Curriculum Development Division Ministry of Education Malaysia PART I: COMBINED OPERATIONS INVOLVING INTEGERS 1. 10 – (–4) × 3 =10 – (–12) = 10 + 12 = 22 2. (–4) × (–8 – 3 ) = (–4) × (–11 ) = 44 3. (–6) + (–3 + 8 ) ÷5 = (–6 )+ (5) ÷5 = (–6 )+ 1 = –5 LESSON NOTES EXAMPLES A standard order of operations for calculations involving +, –, ×, ÷ and brackets: Step 1: First, perform all calculations inside the brackets. Step 2: Next, perform all multiplications and divisions, working from left to right. Step 3: Lastly, perform all additions and subtractions, working from left to right. The above order of operations is also known as the BODMAS Rule and can be summarized as: Brackets power of Division Multiplication Addition Subtraction
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 51 Curriculum Development Division Ministry of Education Malaysia Solve the following. 1. 12 + (8 ÷ 2) 2. (–3 – 5) × 2 3. 4 – (16 ÷ 2) × 2 4. (– 4) × 2 + 6 × 3 5. ( –25) ÷ (35 ÷ 7) 6. (–20) – (3 + 4) × 2 7. (–12) + (–4 × –6) ÷ 3 8. 16 ÷ 4 + (–2) 9. (–18 ÷ 2) + 5 – (–4) TEST YOURSELF I
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 52 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF A: 1. 2 2. –3 3. 6 4. –4 5. –2 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 53 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF B: 1) 4 2) –12 3) 5 4) –10 5) –6 6) –6 7) 0 8) 12 9) 7 TEST YOURSELF C: 1) –42 2) –102 3) –92 4) –908 5) –548 6) 9 7) –843 8) –282 9) –514 TEST YOURSELF D: 1) –12 2) 12 3) –19 4) –10 5) 8 6) 0 7) 8 8) 0 9) –1 10) –125 11) 161 12) –202 13) –364 14) 238 15) –606 16) 790 17) 19 18) –125 TEST YOURSELF E: 1) 32 2) –32 3) 84 4) 25 5) 140 6) –84 7) 84 8) –96 9) 72
Basic Essential Additional Mathematics Skills (BEAMS) Module Unit 1: Negative Numbers 54 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF F: 1) –15 2) 32 3) 30 4) –48 5) 35 6) 120 7) –216 8) 90 9) –108 10) 60 11) –42 12) –80 TEST YOURSELF G: 1) 3 2) –2 3) 3 4) 1 5) 3 6) –3 7) 2 8) –1 9) 2 TEST YOURSELF H: 1. –3 2. –6 3. 3 4. 5 5. –2 6. 5 7. 8 8. 1 9. 5 10. –16 11. 2 12. 2 TEST YOURSELF I: 1. 16 2. –16 3. –12 4. 10 5. –5 6. –34 7. –4 8. 2 9. 0
Unit 1: Negative Numbers UNIT 2 FRACTIONS B a s i c E s s e n t i a l A d d i t i o n a l M a t h e m a t i c s S k i l l s Curriculum Development Division Ministry of Education Malaysia
TABLE OF CONTENTS Module Overview 1 Part A: Addition and Subtraction of Fractions 2 1.0 Addition and Subtraction of Fractions with the Same Denominator 5 1.1 Addition of Fractions with the Same Denominators 5 1.2 Subtraction of Fractions with The Same Denominators 6 1.3 Addition and Subtraction Involving Whole Numbers and Fractions 7 1.4 Addition or Subtraction Involving Mixed Numbers and Fractions 9 2.0 Addition and Subtraction of Fractions with Different Denominator 10 2.1 Addition and Subtraction of Fractions When the Denominator of One Fraction is A Multiple of That of the Other Fraction 11 2.2 Addition and Subtraction of Fractions When the Denominators Are Not Multiple of One Another 13 2.3 Addition or Subtraction of Mixed Numbers with Different Denominators 16 2.4 Addition or Subtraction of Algebraic Expression with Different Denominators 17 Part B: Multiplication and Division of Fractions 22 1.0 Multiplication of Fractions 24 1.1 Multiplication of Simple Fractions 28 1.2 Multiplication of Fractions with Common Factors 29 1.3 Multiplication of a Whole Number and a Fraction 29 1.4 Multiplication of Algebraic Fractions 31 2.0 Division of Fractions 33 2.1 Division of Simple Fractions 36 2.2 Division of Fractions with Common Factors 37 Answers 42
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 1 Curriculum Development Division Ministry of Education Malaysia PART 1 MODULE OVERVIEW 1. The aim of this module is to reinforce pupils’ understanding of the concept of fractions. 2. It serves as a guide for teachers in helping pupils to master the basic computation skills (addition, subtraction, multiplication and division) involving integers and fractions. 3. This module consists of two parts, and each part consists of learning objectives which can be taught separately. Teachers may use any parts of the module as and when it is required.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 2 Curriculum Development Division Ministry of Education Malaysia PART A: ADDITION AND SUBTRACTION OF FRACTIONS LEARNING OBJECTIVES Upon completion of Part A, pupils will be able to: 1. perform computations involving combination of two or more operations on integers and fractions; 2. pose and solve problems involving integers and fractions; 3. add or subtract two algebraic fractions with the same denominators; 4. add or subtract two algebraic fractions with one denominator as a multiple of the other denominator; and 5. add or subtract two algebraic fractions with denominators: (i) not having any common factor; (ii) having a common factor.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 3 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES Pupils have difficulties in adding and subtracting fractions with different denominators. Strategy: Teachers should emphasise that pupils have to find the equivalent form of the fractions with common denominators by finding the lowest common multiple (LCM) of the denominators.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 4 Curriculum Development Division Ministry of Education Malaysia numerator denominator Fraction is written in the form of: b a Examples: 3 4 , 3 2 Proper Fraction Improper Fraction Mixed Numbers The numerator is smaller than the denominator. Examples: 20 9 , 3 2 The numerator is larger than or equal to the denominator. Examples: 12 108 , 4 15 A whole number and a fraction combined. Examples: 6 5 7 1 2 , 8 Rules for Adding or Subtracting Fractions 1. When the denominators are the same, add or subtract only the numerators and keep the denominator the same in the answer. 2. When the denominators are different, find the equivalent fractions that have the same denominator. Note: Emphasise that mixed numbers and whole numbers must be converted to improper fractions before adding or subtracting fractions. LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 5 Curriculum Development Division Ministry of Education Malaysia 1.0 Addition And Subtraction of Fractions with the Same Denominator 1.1 Addition of Fractions with the Same Denominators 8 5 8 4 8 1 i) 2 1 8 4 8 3 8 1 ii) f f f 1 5 6 iii) EXAMPLES Add only the numerators and keep the denominator same. Write the fraction in its simplest form. Add only the numerators and keep the denominator the same. Add only the numerators and keep the denominator the same. 8 1 8 4 8 5
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 6 Curriculum Development Division Ministry of Education Malaysia 1.2 Subtraction of Fractions with The Same Denominators 2 1 8 4 8 1 8 5 i) 7 4 7 5 7 1 ii) n n n 3 1 2 iii) Write the fraction in its simplest form. Subtract only the numerators and keep the denominator the same. Subtract only the numerators and keep the denominator the same. Subtract only the numerators and keep the denominator the same. 8 5 8 1 2 1 8 4
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 7 Curriculum Development Division Ministry of Education Malaysia 1.3 Addition and Subtraction Involving Whole Numbers and Fractions . 8 1 i) Calculate 1 7 29 7 1 7 28 7 1 4 7 1 4 5 18 5 2 5 20 5 2 4 5 3 3 3 12 3 1 3 12 3 1 4 y y y First, convert the whole number to an improper fraction with the same denominator as that of the other fraction. Then, add or subtract only the numerators and keep the denominator the same. 1 8 1 8 1 1 8 9 + 8 8 + 8 1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 8 Curriculum Development Division Ministry of Education Malaysia n n n n n n 2 5 5 2 5 2 k k k k k k 2 3 2 3 3 2 First, convert the whole number to an improper fraction with the same denominator as that of the other fraction. Then, add or subtract only the numerators and keep the denominator the same.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 9 Curriculum Development Division Ministry of Education Malaysia 1.4 Addition or Subtraction Involving Mixed Numbers and Fractions . 8 4 8 1 i) Calculate 1 7 5 7 15 7 5 7 1 2 = 7 20 = 7 6 2 9 4 9 29 9 4 9 2 3 = 9 25 = 9 7 2 8 8 11 8 8 3 1 x x = 8 11 x First, convert the mixed number to improper fraction. Then, add or subtract only the numerators and keep the denominator the same. 8 1 1 8 4 8 5 1 8 13 + 8 9 + 8 4
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 10 Curriculum Development Division Ministry of Education Malaysia 2.0 Addition and Subtraction of Fractions with Different Denominators . 2 1 8 1 i) Calculate To make the denominators the same, multiply both the numerator and the denominator of the second fraction by 4: Now, the question can be visualized like this: ? The denominators are not the same. See how the slices are different in sizes? Before we can add the fractions, we need to make them the same, because we can't add them together like this! 8 1 8 + 4 8 5 8 4 2 1 4 4 Now, the denominators are the same. Therefore, we can add the fractions together! 8 1 2 + 1 ?
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 11 Curriculum Development Division Ministry of Education Malaysia Hint: Before adding or subtracting fractions with different denominators, we must convert each fraction to an equivalent fraction with the same denominator. 2.1 Addition and Subtraction of Fractions When the Denominator of One Fraction is A Multiple of That of the Other Fraction Multiply both the numerator and the denominator with an integer that makes the denominators the same. (i) 6 5 3 1 6 5 6 2 6 7 = 6 1 1 (ii) 4 3 12 7 12 9 12 7 12 2 6 1 Change the first fraction to an equivalent fraction with denominator 6. (Multiply both the numerator and the denominator of the first fraction by 2): 6 2 3 1 2 2 Add only the numerators and keep the denominator the same. Change the second fraction to an equivalent fraction with denominator 12. (Multiply both the numerator and the denominator of the second fraction by 3): 12 9 4 3 3 3 Subtract only the numerators and keep the denominator the same. Write the fraction in its simplest form. Convert the fraction to a mixed number.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 12 Curriculum Development Division Ministry of Education Malaysia (iii) v 5v 1 9 v 5v 9 5 5 5v 14 Change the first fraction to an equivalent fraction with denominator 5v. (Multiply both the numerator and the denominator of the first fraction by 5): v 5v 1 5 5 5 Add only the numerators and keep the denominator the same.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 13 Curriculum Development Division Ministry of Education Malaysia 2.2 Addition and Subtraction of Fractions When the Denominators Are Not Multiple of One Another Method I 4 3 6 1 (i) Find the Least Common Multiple (LCM) of the denominators. 2) 4 , 6 2) 2 , 3 3) 1 , 3 - , 1 LCM = 2 2 3 = 12 The LCM of 4 and 6 is 12. (ii) Change each fraction to an equivalent fraction using the LCM as the denominator. (Multiply both the numerator and the denominator of each fraction by a whole number that will make their denominators the same as the LCM value). = 4 3 6 1 = 12 9 12 2 = 12 11 Method II 4 3 6 1 (i) Multiply the numerator and the denominator of the first fraction with the denominator of the second fraction and vice versa. = 4 3 6 1 = 24 18 24 4 = 24 22 = 12 11 Write the fraction in its simplest form. This method is preferred but you must remember to give the answer in its simplest form. 3 2 3 2 4 4 6 6
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 14 Curriculum Development Division Ministry of Education Malaysia Multiply the first fraction with the second denominator and multiply the second fraction with the first denominator. 1. 5 1 3 2 = 5 5 3 2 + 3 3 5 1 15 3 15 10 = 15 13 2. 8 3 6 5 = 8 8 6 5 – 6 6 8 3 = 48 18 48 40 = 48 22 = 24 11 Write the fraction in its simplest form. EXAMPLES Multiply the first fraction by the denominator of the second fraction and multiply the second fraction by the denominator of the first fraction. Multiply the first fraction by the denominator of the second fraction and multiply the second fraction by the denominator of the first fraction. Add only the numerators and keep the denominator the same. Subtract only the numerators and keep the denominator the same.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 15 Curriculum Development Division Ministry of Education Malaysia 3. 7 1 3 2 g = 3 3 7 7 7 1 3 2 g = 21 3 21 14 g = 21 14g 3 4. 3 5 2g h 3 3 5 5 5 3 2 g h 15 3 15 10g h 15 10g 3h 5. c d 6 4 = c c d d c d 6 4 cd c cd 6d 4 = cd 6d 4c Multiply the first fraction by the denominator of the second fraction and multiply the second fraction by the denominator of the first fraction. Write as a single fraction. Write as a single fraction. Write as a single fraction. Multiply the first fraction by the denominator of the second fraction and multiply the second fraction by the denominator of the first fraction. Multiply the first fraction by the denominator of the second fraction and multiply the second fraction by the denominator of the first fraction.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 16 Curriculum Development Division Ministry of Education Malaysia Convert the mixed numbers to improper fractions. Convert the mixed numbers to improper fractions. 2.3 Addition or Subtraction of Mixed Numbers with Different Denominators 1. 4 3 2 2 1 2 = 4 11 2 5 = 4 11 2 5 2 2 = 4 11 4 10 = 4 21 4 1 5 2. 4 3 1 6 5 3 = 4 7 6 23 = 6 6 4 4 4 7 6 23 = 24 42 24 92 = 24 50 = 12 25 = 12 1 2 Change the first fraction to an equivalent fraction with denominator 4. (Multiply both the numerator and the denominator of the first fraction by 2) The denominators are not multiples of one another: Multiply the first fraction by the denominator of the second fraction. Multiply the second fraction by the denominator of the first fraction. Convert the mixed numbers to improper fractions. Convert the mixed numbers to improper fractions. Add only the numerators and keep the denominator the same. Change the fraction back to a mixed number. Add only the numerators and keep the denominator the same. Change the fraction back to a mixed number. Write the fraction in its simplest form.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 17 Curriculum Development Division Ministry of Education Malaysia The denominators are not multiples of one another Multiply the first fraction with the second denominator Multiply the second fraction with the first denominator The denominators are not multiples of one another Multiply the first fraction with the second denominator Multiply the second fraction with the first denominator 2.4 Addition or Subtraction of Algebraic Expression with Different Denominators 1. 2 2 m m m = ( 2) ( 2) 2 2 2 2 m m m m m = 2 2 2 2 2 2 m m m m m = 2( 2) 2 ( 2) m m m m = 2( 2) 2 2 2 m m m m = 2( 2) 2 m m 2. y y y y 1 1 = ( 1) ( 1) 1 1 y y y y y y y y = ( 1) ( 1)( 1) 2 y y y y y = ( 1) ( 1) 2 2 y y y y = ( 1) 1 2 2 y y y y = ( 1) 1 y y Remember to use brackets Write the above fractions as a single fraction. The denominators are not multiples of one another: Multiply the first fraction by the denominator of the second fraction. Multiply the second fraction by the denominator of the first fraction. Expand: m (m – 2) = m 2 – 2m Expand: (y – 1) (y + 1) = y 2 + y – y – 1 2 = y 2 – 1 Expand: – (y 2 – 1) = –y 2 + 1 Write the fractions as a single fraction. The denominators are not multiples of one another: Multiply the first fraction by the denominator of the second fraction. Multiply the second fraction by the denominator of the first fraction.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 18 Curriculum Development Division Ministry of Education Malaysia The denominators are not multiples of one another Multiply the first fraction with the second denominator Multiply the second fraction with the first denominator 3. 2 4 5 8 3 n n n = n n n n n n n 8 8 2 4 4 4 5 8 3 2 2 = 8 (4 ) 8 (5 ) 8 (4 ) 12 2 2 2 n n n n n n n = 8 (4 ) 12 8 (5 ) 2 2 n n n n n = 8 (4 ) 12 40 8 2 2 2 n n n n n = 8 (4 ) 4 40 2 2 n n n n = 4 (8 ) 4 ( 10) 2 n n n n = 2 8 10 n n Factorise and simplify the fraction by canceling out the common factors. Expand: – 8n (5 + n) = –40n – 8n 2 Subtract the like terms. Write as a single fraction. The denominators are not multiples of one another: Multiply the first fraction by the denominator of the second fraction. Multiply the second fraction by the denominator of the first fraction.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 19 Curriculum Development Division Ministry of Education Malaysia Calculate each of the following. 1. 7 1 7 2 2. 12 5 12 11 3. 14 1 7 2 4. 12 5 3 2 5. 5 4 7 2 6. 7 5 2 1 7. 3 13 2 2 8. 9 7 2 5 2 4 9. s s 2 1 10. w w 11 5 TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 20 Curriculum Development Division Ministry of Education Malaysia 11. a 2a 2 1 12. f 3 f 2 5 13. a b 2 4 14. p q 1 5 15. m n m n 5 3 7 2 5 2 7 5 16. (2 ) 2 1 p p 17. 5 3 2 2x 3y x y 18. x x x 5 2 12 4 19. x x x x 1 1 20. 2 4 2 x x x x
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 21 Curriculum Development Division Ministry of Education Malaysia 21. 4 4 8 2 6x 3y x y 22. 2 9 4 3 2 n n n 23. r r r 15 5 2 5 2 24. p p p p 2 3 2 2 25. n n n n 10 4 3 5 2 3 2 26. n n mn 3m n 3 27. mn m n m m 5 5 28. mn n m m m 3 3 29. 2 4 5 8 3 n n n 30. m p m p 1 3
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 22 Curriculum Development Division Ministry of Education Malaysia 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.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 23 Curriculum Development Division Ministry of Education Malaysia 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 (+) (+) = + (+) (+) = + (+) (–) = – (+) (–) = – (–) (+) = – (–) (+) = – (–) (–) = + (–) (–) = +
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 24 Curriculum Development Division Ministry of Education Malaysia 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 23 can be represented visually as follows: This means that 3 units are being repeated twice, or mathematically can be written as: 6 2 3 3 3 Now, let’s calculate 2 x 2. This multiplication can be represented visually as: This means that 2 units are being repeated twice, or mathematically can be written as: 4 2 2 2 2 LESSON NOTES 3 + 3 = 6 2 + 2 = 4 2 groups of 3 units 2 groups of 2 units
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 25 Curriculum Development Division Ministry of Education Malaysia Now, let’s calculate 2 x 1. This multiplication can be represented visually as: This means that 1 unit is being repeated twice, or mathematically can be written as: 2111 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? Let’s consider . 2 1 2 Since represents 1 whole unit, therefore 2 1 unit can be represented by the following shaded area: Then, we can represent visually the multiplication of 2 1 2 as follows: This means that 2 1 unit is being repeated twice, or mathematically can be written as: 1 2 2 2 1 2 1 2 1 2 1 + 1 = 2 2 1 + 2 1 = 1 2 2 2 groups of 1 unit 2 groups of 2 1 unit
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 26 Curriculum Development Division Ministry of Education Malaysia Let’s consider again 2. 2 1 What does it mean? It means ‘ 2 1 out of 2 units’ and the visualization will be like this: Notice that the multiplications 2 1 2 and 2 2 1 will give the same answer, that is, 1. How about 2 ? 3 1 Since represents 1 whole unit, therefore 3 1 unit can be represented by the following shaded area: Then, we can represent visually the multiplication 2 3 1 as follows: This means that 3 1 unit is being repeated twice, or mathematically can be written as: 3 2 3 1 3 1 2 3 1 3 1 + 3 1 = 3 2 The shaded area is 3 1 unit. 2 1 out of 2 units 2 1 2 1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 27 Curriculum Development Division Ministry of Education Malaysia Let’s consider 2 3 1 . What does it mean? It means ‘ 3 1 out of 2 units’ and the visualization will be like this: Notice that the multiplications 3 1 2 and 2 3 1 will give the same answer, that is, 3 2 . Consider now the multiplication of a fraction by a fraction, like this: 2 1 3 1 This means ‘ 3 1 out of 2 1 units’ and the visualization will be like this: Consider now this multiplication: 2 1 3 2 This means ‘ 3 2 out of 2 1 units’ and the visualization will be like this: 2 1 unit 3 1 out of 2 units 3 2 2 3 1 3 1 out of 2 1 units 6 1 2 1 3 1 2 1 unit 3 2 out of 2 1 units 6 2 2 1 3 2
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 28 Curriculum Development Division Ministry of Education Malaysia What do you notice so far? The answer to the above multiplication of a fraction by a fraction can be obtained by just multiplying both the numerator together and the denominator together: 6 1 2 1 3 1 9 2 3 1 3 2 So, what do you think the answer for 3 1 4 1 ? Do you get 12 1 as the answer? The steps to multiply a fraction by a fraction can therefore be summarized as follows: 1.1 Multiplication of Simple Fractions Examples: a) 35 6 7 3 5 2 b) 35 6 5 3 7 2 c) 35 12 5 2 7 6 d) 35 12 5 2 7 6 Steps to Multiply Fractions: 1) Multiply the numerators together and multiply the denominators together. 2) Simplify the fraction (if needed). Remember!!! (+) (+) = + (+) (–) = – (–) (+) = – (–) (–) = + Multiply the two numerators together and the two denominators together.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 29 Curriculum Development Division Ministry of Education Malaysia 1.2 Multiplication of Fractions with Common Factors 6 5 7 12 or 6 5 7 12 1.3 Multiplication of a Whole Number and a Fraction 6 1 2 5 = 6 31 1 2 = 6 31 1 2 = 3 31 = 3 1 10 Second Method: (i) Simplify the fraction by canceling out the common factors. 6 5 7 12 (i) Then, multiply the two numerators together and the two denominators together, and convert to a mixed number, if needed. 6 5 7 12 7 3 1 7 10 2 1 Convert the mixed number to improper fraction. Simplify by canceling out the common factors. Remember 2 = 1 2 First Method: (ii) Multiply the two numerators together and the two denominators together: 6 5 7 12 = 42 60 (ii) Then, simplify. 7 3 1 7 10 42 60 10 7 3 Multiply the two numerators together and the two denominators together. Remember: (+) (–) = (–) Change the fraction back to a mixed number. 1 1 2
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 30 Curriculum Development Division Ministry of Education Malaysia 1. Find 10 15 12 5 Solution: 10 15 12 5 = 8 5 2. Find 5 2 6 21 Solution : 5 2 6 21 = 5 2 6 21 5 7 = 5 2 1 Simplify by canceling out the common factors. Note that 3 21 can be further simplified. Simplify further by canceling out the common factors. 3 1 Simplify by canceling out the common factors. EXAMPLES Multiply the two numerators together and the two denominators together. Remember: (+) (–) = (–) Multiply the two numerators together and the two denominators together. Remember: (+) (–) = (–) 3 1 1 7 Change the fraction back to a mixed number. 2 1 4 5
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 31 Curriculum Development Division Ministry of Education Malaysia 1.4 Multiplication of Algebraic Fractions 1. Simplify 4 2 5x x Solution : 4 2 5x x = 2 5 = 2 1 2 2. Simplify m n n 4 9 2 Solution: m n n 4 9 2 = 1 4 2 9 2 n m n n = 1 ( 2 ) 2 9 n m = 2nm 2 9 1 2 1 1 Simplify the fraction by canceling out the x’s. Multiply the two numerators together and the two denominators together. Simplify the fraction by canceling the common factor and the n. Multiply the two numerators together and the two denominators together. Write the fraction in its simplest form. Change the fraction back to a mixed number. 2 1 1 1