BEAMS KPM

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: To Accept or To Reject Answer
Accept +
Sign Reject – 
(+) × (+) Accept – 
(–) × (–) Reject +
(+) × (–) –
(–) × (+) EXAMPLES –

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

Curriculum Development Division 38
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF F

Solve the following. 2. –4 × (–8) = 3. 6 × (5) =
1. 3 × (–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 39
Ministry of Education Malaysia

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 40
Ministry of Education Malaysia

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

(–3) × (–2) = 6, then 6 ÷ (–3) = –2

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

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

Curriculum Development Division 41
Ministry of Education Malaysia

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 42
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF G

Solve the following. 2. 8 ÷ (–4) 3. (–21) ÷ (–7)
1. (–24) ÷ (–8)

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 43
Ministry of Education Malaysia

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 44
Ministry of Education Malaysia

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: To Accept or To Reject Answer
Sign Accept + +
Reject – +
(+) ÷ (+) Accept – –
(–) ÷ (–) Reject + –
(+) ÷ (–)
(–) ÷ (+)

Curriculum Development Division 45
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

EXAMPLES

(6) ÷ (3) To Accept or To Reject Answer
(–6) ÷ (–3) Accept + 2
(+6) ÷ (–3) Reject – 2
(–6) ÷ (3) Accept – –2
Reject + –2

Division [Fraction Form]: To Accept or To Reject Answer
Sign +
+
(  ) Accept + –
() –

() Reject –
()

() Accept –
()

()
(  ) Reject +

Curriculum Development Division 46
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

EXAMPLES

(8) To Accept or To Reject Answer
(  2) Accept + 4
Reject – 4
( 8) Accept – –4
( 2) Reject + –4

( 8)
( 2)

( 8)
( 2)

Curriculum Development Division 47
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF H

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

4.  25 5.  6 6. – (–35) ÷ 7
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 48
Ministry of Education Malaysia

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 49
Ministry of Education Malaysia

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

Curriculum Development Division 50
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF I

Solve the following. 2. (–3 – 5) × 2 3. 4 – (16 ÷ 2) × 2
1. 12 + (8 ÷ 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 51
Ministry of Education Malaysia

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 52

–5 –4 –3 –2 –1 0 1 2 3 4 5 6

Curriculum Development Division
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF B: 2) –12 3) 5
5) –6 6) –6
1) 4 8) 12 9) 7
4) –10
7) 0

TEST YOURSELF C: 2) –102 3) –92
5) –548 6) 9
1) –42 8) –282 9) –514
4) –908
7) –843

TEST YOURSELF D: 2) 12 3) –19
5) 8 6) 0
1) –12 8) 0 9) –1
4) –10 11) 161 12) –202
7) 8 14) 238 15) –606
10) –125 17) 19 18) –125
13) –364
16) 790

TEST YOURSELF E: 2) –32 3) 84
5) 140 6) –84
1) 32 8) –96 9) 72
4) 25
7) 84

Curriculum Development Division 53
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers

TEST YOURSELF F: 2) 32 3) 30
5) 35 6) 120
1) –15 8) 90 9) –108
4) –48 11) –42 12) –80
7) –216
10) 60

TEST YOURSELF G: 2) –2 3) 3
5) 3 6) –3
1) 3 8) –1 9) 2
4) 1
7) 2

TEST YOURSELF H: 2. –6 3. 3
5. –2 6. 5
1. –3 8. 1 9. 5
4. 5 11. 2 12. 2
7. 8
10. –16

TEST YOURSELF I: 2. –16 3. –12
5. –5 6. –34
1. 16 8. 2 9. 0
4. 10
7. –4

Curriculum Development Division 54
Ministry of Education Malaysia

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 1objectives which can be taught separately. Teachers may use any parts of the

module as and when it is required.

Curriculum Development Division 1
Ministry of Education Malaysia

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 2
Ministry of Education Malaysia

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 3
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

LESSON NOTES

Fraction is written in the form of: numerator
denominator
a
b

Proper Fraction Examples: Mixed Numbers

2, 4
33

Improper Fraction

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

Examples: Examples: Examples:

2, 9 15 , 108 2 1 , 8 5
3 20 4 12 7 6

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 4
Ministry of Education Malaysia

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

i) 1  4  5 Add only the numerators and keep the
88 8 denominator same.

1 4 5
88 8

ii) 1  3  4 Add only the numerators and keep the
88 8 denominator the same.
1
2 Write the fraction in its simplest form.

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

Curriculum Development Division 5
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

1.2 Subtraction of Fractions with The Same Denominators

i) 5  1  4 Subtract only the numerators and keep
88 8 the denominator the same.
1
2 Write the fraction in its simplest form.

51  41
82
88

ii) 1  5   4 Subtract only the numerators and keep
77 7 the denominator the same.

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

Curriculum Development Division 6
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

1.3 Addition and Subtraction Involving Whole Numbers and Fractions
i) Calculate 1  1 .
8

1 +1 9
8 8 8

8 +1  11
8 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.

4  1  28  1 4  2  20  2 4  1 y  12  1 y
7 77 55 5 3 33
 29  12  y
7  18 3
 41 5
7
 33
5

Curriculum Development Division 7
Ministry of Education Malaysia

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.

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

 2n  5  2  3k
n k

Curriculum Development Division 8
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

1.4 Addition or Subtraction Involving Mixed Numbers and Fractions
i) Calculate 11  4 .
88

11 +4  13  1 5
8 8 88

9 +4
8 8

 First, convert the mixed number to improper fraction.
 Then, add or subtract only the numerators and keep the

denominator the same.


2 1  5  15  5 3 2  4  29  4 13  x  11  x
77 7 7 99 9 9 88 8 8

= 20 = 2 6 = 25 = 2 7 = 11 x
77 99 8

Curriculum Development Division 9
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

2.0 Addition and Subtraction of Fractions with Different Denominators

i) Calculate 1  1 . The denominators are not the same.
82 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
28 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 10
Ministry of Education Malaysia

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.

(i) 1  5 Change the first fraction to an equivalent
36 fraction with denominator 6.

 2 5 (Multiply both the numerator and the
66 denominator of the first fraction by 2):

7 2
6 1 2
36
= 11
6 2

Add only the numerators and keep the
denominator the same.

Convert the fraction to a mixed number.

(ii) 7  3 Change the second fraction to an equivalent
12 4 fraction with denominator 12.

79 (Multiply both the numerator and the
12 12 denominator of the second fraction by 3):

2 3
12 3 9
4 12
 1
6 3

Subtract only the numerators and keep the
denominator the same.

Write the fraction in its simplest form.

Curriculum Development Division 11
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

(iii) 1  9 Change the first fraction to an equivalent
v 5v fraction with denominator 5v.

5  9 (Multiply both the numerator and the
5v 5v denominator of the first fraction by 5):

 14 5
5v 1 5
v 5v

5

Add only the numerators and keep the
denominator the same.

Curriculum Development Division 12
Ministry of Education Malaysia

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
64 13
(i) Find the Least Common Multiple (LCM) 64
of the denominators.
(i) Multiply the numerator and the
denominator of the first fraction with
the denominator of the second fraction
and vice versa.

2) 4 , 6 = 1 4  3 6
2) 2 , 3 6 4 4 6
3) 1 , 3
= 4  18
- ,1 24 24

LCM = 2  2  3 = 12

The LCM of 4 and 6 is 12.

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

=2  9
12 12

= 11
12

Curriculum Development Division 13
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

EXAMPLES

1. 2  1
3 5

= 2 5 + 1 3 Multiply the first fraction with the second denominator
3 5 5 3 and
multiply the second fraction with the first denominator.
 10 3 Multiply the first fraction by the
15 15 denominator of the second fraction and
multiply the second fraction by the
= 13 denominator of the first fraction.
15
Add only the numerators and keep the
denominator the same.

2. 5  3
68

8 – 6
=5 3

6 8 8 6 Multiply the first fraction by the
denominator of the second fraction and
= 40  18
48 48 multiply the second fraction by the
denominator of the first fraction.
= 22
48 Subtract only the numerators and keep
the denominator the same.
= 11
24 Write the fraction in its simplest form.

Curriculum Development Division 14
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

3. 2 g  1
37

= 2g 7  1 3
3 7 7 3
Multiply the first fraction by the
= 14g  3 denominator of the second fraction and
21 21
multiply the second fraction by the
= 14g  3 denominator of the first fraction.
21
Write as a single fraction.
4. 2g  h
35 Multiply the first fraction by the
denominator of the second fraction and
 2g 5  h 3
3 5 5 3 multiply the second fraction by the
denominator of the first fraction.
 10g  3h
15 15 Write as a single fraction.

 10g  3h
15

5. 6  4
cd

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

Curriculum Development Division 15
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

2.3 Addition or Subtraction of Mixed Numbers with Different Denominators

1. 2 1  2 3 Convert the mixed numbers to improper fractions.
24 Convert the mixed numbers to improper fractions.

= 5  11 Change the first fraction to an equivalent fraction
24 with denominator 4.

= 5 2  11 (Multiply both the numerator and the denominator
2 2 4 of the first fraction by 2)

= 10  11 Add only the numerators and keep the
44 denominator the same.

= 21 Change the fraction back to a mixed number.
4
Convert the mixed numbers to improper fractions.
 51 Convert the mixed numbers to improper fractions.
4
The denominators are not multiples of one another:
2. 3 5  1 3  Multiply the first fraction by the denominator
64
of the second fraction.
= 23  7  Multiply the second fraction by the
64
denominator of the first fraction.
= 23 4  7 6
6 4 4 6 Add only the numerators and keep the
denominator the same.
= 92  42
24 24 Write the fraction in its simplest form.

= 50 Change the fraction back to a mixed number.
24

= 25
12

= 21
12

Curriculum Development Division 16
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

2.4 Addition or Subtraction of Algebraic Expression with Different Denominators

1. m  m ThTehdeedneonmoimnaintoartsorasrearneontomt umltuipltliepsleosfoofnoenaenaonthoethr:er
m2 2 Multiply the first fraction with the second denominator

=m 2 m (m2)  MMuultlitpilpylythethseecfoinrsdtfrfaracctitoinonwibthyththeefirdsetndoemnoimnaintoartor
2 (m2) of the second fraction.
m  2 2 
 Multiply the second fraction by the
denominator of the first fraction.

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

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

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

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

= y2  ( y 1)(y  1) Write the fractions as a single fraction.
y ( y  1)

= y2  ( y2 1) Expand:
y( y 1) (y – 1) (y + 1) = y2 + y – y – 12
= y2 – 1

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

1

=

y( y 1)

Curriculum Development Division 17
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

3. 3  5n The denominators are not multiples of one another:
8n 4n2
 MThueltdipelnyomthienafitrosrts farraectniootnmbuylttihpelesdeonfoomnienaantoorther
= 3  4n2  5  n 8n oMfutlhtiepslyecthoenfdirfsrtafcraticotino.n with the second denominator
8n  4n2 4n2 8n
 MMuullttiippllyythe tsheecondsfercaocntidon wfitrhacthtieofnirst dbeynomtihneator
denominator of the first fraction.

= 12n2  8n (5  n)
8n(4n2 ) 8n(4n2 )

= 12 n 2  8n (5  n) Write as a single fraction.
8n(4n2 )

Expand:

= 12n2  40n  8n2 – 8n (5 + n) = –40n – 8n2
8n(4n2 )

= 4n2  40n Subtract the like terms.
8n(4n2 )

4n (n  10) Factorise and simplify the fraction by canceling
= 4n(8n2 ) out the common factors.

= n  10
8n2

Curriculum Development Division 18
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

Calculate each of the following. TEST YOURSELF A
1. 2  1  2. 11  5 
77 12 12

3. 2  1  4. 2  5 
7 14 3 12

5. 2  4  6. 1  5 
75 27

7. 2 2  3  8. 4 2  2 7 
13 59

9. 2  1  10. 11  5 
ss ww

Curriculum Development Division 19
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

11. 2  1  12. 2  5 
a 2a f 3f

13. 2  4  14. 1  5 
ab pq

15. 5 m  2 n  2 m  3 n  16. p  1  (2  p) 
7 57 5 2

17. 2x  3y  3x  y  18. 12  4x  5 
25 2x x

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

Curriculum Development Division 20
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

21. 6x  3y  4x  8y  22. 2  4  n 
24 3n 9n2

23. r  5  2r2  24. p  3  p  2 
5 15r p2 2p

25. 2n  3  4n  3  26. 3m  n  n  3 
5n2 10n mn n

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

29. 3  5  n  30. p  1 p 
8n 4n2 3m m

Curriculum Development Division 21
Ministry of Education Malaysia

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 22
Ministry of Education Malaysia

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 23
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

LESSON NOTES

1.0 Multiplication of Fractions

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

23

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

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

2 groups of 3 units

3 +3 = 6

This means that 3 units are being repeated twice, or mathematically can be written as:

23 33
6

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

2 groups of 2 units

2 +2 = 4

This means that 2 units are being repeated twice, or mathematically can be written as:

22  2  2
4

Curriculum Development Division 24
Ministry of Education Malaysia

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?

Let’s consider 2  1 .
2

Since 1
represents 1 whole unit, therefore unit can be represented by the

2

following shaded area:

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

1

2 groups of unit

2

11 = 2 1
+ 2

22

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

2

2 1  1  1
22 2
2
2
1

Curriculum Development Division 25
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

Let’s consider again 1  2.What does it mean? It means ‘ 1 out of 2 units’ and the
22

visualization will be like this:

1 1 2 1
2
out of 2 units

2

Notice that the multiplications 2 1 and 1  2 will give the same answer, that is, 1.
22

How about 1  2 ?
3

Since represents 1 whole unit, therefore 1 unit can be represented by the
3

following shaded area:

1

The shaded area is unit.

3

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

11 = 2
3
+

33

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

3

12 1  1
3 33

2
3

Curriculum Development Division 26
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

Let’s consider 1  2 . What does it mean? It means ‘ 1 out of 2 units’ and the visualization
33

will be like this:

1 12 2
33
out of 2 units

3

Notice that the multiplications 2 1 and 1  2 will give the same answer, that is, 2 .
33 3

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

11
32

This means ‘ 1 1 units’ and the visualization will be like this:
out of
32

1 unit 11 11  1
2 32 6
out of units

32

Consider now this multiplication:

21
32

This means ‘ 2 1 units’ and the visualization will be like this:
out of
32

1 unit
2

21 21  2
32 6
out of units

32

Curriculum Development Division 27
Ministry of Education Malaysia

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 26 3 39

So, what do you think the answer for 1 1 ? Do you get 1 as the answer?
43 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 b) 2   3   6
Examples: 75 35

a) 2  3  6
5 7 35

c)  6  2   12 d)  6   2  12
75 35 7 5 35

Multiply the two numerators together and the two denominators together.

Curriculum Development Division 28
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

1.2 Multiplication of Fractions with Common Factors

12  5 or 12  5 
76 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
12  5 = 60 7 61
7 6 42
(i) Then, multiply the two
(ii) Then, simplify. numerators together and the two
6010 10  1 3 denominators together, and
42 7 7 convert to a mixed number, if
needed.
7
2 12  5  10  1 3
76 77

1

1.3 Multiplication of a Whole Number and a Fraction

Remember 2   5 1 
6
2= 2
1

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

= 1 2   31  Simplify by canceling out the common
1 6  factors.

3 Multiply the two numerators together and
the two denominators together.
=  31 Remember: (+)  (–) = (–)
3
Change the fraction back to a mixed number.
= 10 1
3

Curriculum Development Division 29
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

EXAMPLES

1. Find  5  15
12 10

Solution: 1 5  15 5 Simplify by canceling out the common factors.

 10 2 Multiply the two numerators together and the
12 two denominators together.
Remember: (+)  (–) = (–)
4

= 5
8

2. Find 21  2 Simplify by canceling out the common
65 factors.

Solution : 21  2 1 21
36 5 Note that can be further simplified.

= 7 21  2 1 3
65
Simplify further by canceling out the
3 common factors.

 17 Multiply the two numerators together and
5 the two denominators together.
Remember: (+)  (–) = (–)
= 12
5 Change the fraction back to a mixed
number.

Curriculum Development Division 30
Ministry of Education Malaysia

Basic Essential Additional Mathematics Skills (BEAMS) Module
UNIT 2: Fractions

1.4 Multiplication of Algebraic Fractions

1. Simplify 2  5x Simplify the fraction by canceling out the x’s.
x4
Multiply the two numerators together and
Solution : 12  5x 1 the two denominators together.
x4
Change the fraction back to a mixed
12 number.

=5
2

= 21
2

2. Simplify n  9  4m 
2 n 

Solution: n  9  4m 
2 n 

= n1  9   2 Simplify the fraction by canceling the
common factor and the n.
n  4m 
2  n 1 1 2  1  Multiply the two numerators together
and the two denominators together.
= 9  n ( 2m)
21 Write the fraction in its simplest form.

= 9  2nm
2

Curriculum Development Division 31
Ministry of Education Malaysia