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Unit 1:
Negative Numbers
UNIT 1
NEGATIVE NUMBERS
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Addition and Subtraction of Integers Using Number Lines 2
1.0 Representing Integers on a Number Line 3
2.0 Addition and Subtraction of Positive Integers 3
3.0 Addition and Subtraction of Negative Integers 8
Part B: Addition and Subtraction of Integers Using the Sign Model 15
Part C: Further Practice on Addition and Subtraction of Integers 19
Part D: Addition and Subtraction of Integers Including the Use of Brackets 25
Part E: Multiplication of Integers 33
Part F: Multiplication of Integers Using the Accept-Reject Model 37
Part G: Division of Integers 40
Part H: Division of Integers Using the Accept-Reject Model 44
Part I: Combined Operations Involving Integers 49
Answers 52
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
1
Curriculum Development Division
Ministry of Education Malaysia
MODULE OVERVIEW
1. Negative Numbers is the very basic topic which must be mastered by every
pupil.
2. The concept of negative numbers is widely used in many Additional
Mathematics topics, for example:
(a) Functions (b) Quadratic Equations
(c) Quadratic Functions (d) Coordinate Geometry
(e) Differentiation (f) Trigonometry
Thus, pupils must master negative numbers in order to cope with topics in
Additional Mathematics.
3. The aim of this module is to reinforce pupils‟ understanding on the concept of
negative numbers.
4. This module is designed to enhance the pupils‟ skills in
using the concept of number line;
using the arithmetic operations involving negative numbers;
solving problems involving addition, subtraction, multiplication and
division of negative numbers; and
applying the order of operations to solve problems.
5. It is hoped that this module will enhance pupils‟ understanding on negative
numbers using the Sign Model and the Accept-Reject Model.
6. This module consists of nine parts and each part consists of learning objectives
which can be taught separately. Teachers may use any parts of the module as
and when it is required.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
2
Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
The concept of negative numbers can be confusing and difficult for pupils to
grasp. Pupils face difficulty when dealing with operations involving positive and
negative integers.
Strategy:
Teacher should ensure that pupils understand the concept of positive and negative
integers using number lines. Pupils are also expected to be able to perform
computations involving addition and subtraction of integers with the use of the
number line.
PART A:
ADDITION AND SUBTRACTION
OF INTEGERS USING
NUMBER LINES
LEARNING OBJECTIVE
Upon completion of Part A, pupils will be able to perform computations
involving combined operations of addition and subtraction of integers using a
number lines.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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PART A:
ADDITION AND SUBTRACTION OF INTEGERS
USING NUMBER LINES
1.0 Representing Integers on a Number Line
Positive whole numbers, negative numbers and zero are all integers.
Integers can be represented on a number line.
Note: i) –3 is the opposite of +3
ii) – (–2) becomes the opposite of negative 2, that is, positive 2.
2.0 Addition and Subtraction of Positive Integers
–3 –2 –1 0 1 2 3 4
LESSON NOTES
Rules for Adding and Subtracting Positive Integers
When adding a positive integer, you move to the right on a
number line.
When subtracting a positive integer, you move to the left
on a number line.
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
Positive integers
may have a plus sign
in front of them,
like +3, or no sign in
front, like 3.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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(i) 2 + 3
Alternative Method:
EXAMPLES
Adding a positive integer:
Start by drawing an arrow from 0 to 2, and then,
draw an arrow of 3 units to the right:
2 + 3 = 5
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Start
with 2
Add a
positive 3
Adding a positive integer:
Start at 2 and move 3 units to the right:
2 + 3 = 5
Make sure you start from
the position of the first
integer.
–5 –4
–3 –2 –1 0 1 2 3 4 5 6
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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(ii) –2 + 5
Alternative Method:
Adding a positive integer:
Start by drawing an arrow from 0 to –2, and then,
draw an arrow of 5 units to the right:
–2 + 5 = 3
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Add a
positive 5
Make sure you start from
the position of the first
integer.
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Adding a positive integer:
Start at –2 and move 5 units to the right:
–2 + 5 = 3
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Unit 1: Negative Numbers
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(iii) 2 – 5 = –3
Alternative Method:
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtracting a positive integer:
Start by drawing an arrow from 0 to 2, and then,
draw an arrow of 5 units to the left:
2 – 5 = –3
Subtract a
positive 5
Subtracting a positive integer:
Start at 2 and move 5 units to the left:
2 – 5 = –3
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Make sure you start from
the position of the first
integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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(iv) –3 – 2 = –5
Alternative Method:
Subtracting a positive integer:
Start by drawing an arrow from 0 to –3, and
then, draw an arrow of 2 units to the left:
–3 – 2 = –5
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtract a
positive 2
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtracting a positive integer:
Start at –3 and move 2 units to the left:
–3 – 2 = –5
Make sure you start from
the position of the first
integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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3.0 Addition and Subtraction of Negative Integers
Consider the following operations:
4 – 1 = 3
4 – 2 = 2
4 – 3 = 1
4 – 4 = 0
4 – 5 = –1
4 – 6 = –2
Note that subtracting an integer gives the same result as adding its opposite. Adding or
subtracting a negative integer goes in the opposite direction to adding or subtracting a positive
integer.
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
4 + (–5) = –1
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
4 + (–6) = –2
4 + (–1) = 3
4 + (–2) = 2
4 + (–3) = 1
4 + (–4) = 0
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Rules for Adding and Subtracting Negative Integers
When adding a negative integer, you move to the left on a
number line.
When subtracting a negative integer, you move to the right
on a number line.
–3 –2 –1 0 1 2 3 4
–3 –2 –1 0 1 2 3 4
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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(i) –2 + (–1) = –3
Alternative Method:
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Adding a negative integer:
Start at –2 and move 1 unit to the left:
–2 + (–1) = –3
EXAMPLES
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Adding a negative integer:
Start by drawing an arrow from 0 to –2, and
then, draw an arrow of 1 unit to the left:
–2 + (–1) = –3
Add a
negative 1
Make sure you start from
the position of the first
integer.
This operation of
–2 + (–1) = –3
is the same as
–2 –1 = –3.
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Unit 1: Negative Numbers
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(ii) 1 + (–3) = –2
Alternative Method:
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Adding a negative integer:
Start at 1 and move 3 units to the left:
1 + (–3) = –2
Add a
negative 3
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Adding a negative integer:
Start by drawing an arrow from 0 to 1, then, draw an arrow of
3 units to the left:
1 + (–3) = –2
Make sure you start from
the position of the first
integer.
This operation of
1 + (–3) = –2
is the same as
1 – 3 = –2
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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(iii) 3 – (–3) = 6
Alternative Method:
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtracting a negative integer:
Start at 3 and move 3 units to the right:
3 – (–3) = 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtracting a negative integer:
Start by drawing an arrow from 0 to 3, and
then, draw an arrow of 3 units to the right:
3 – (–3) = 6
Subtract a
negative 3
This operation of
3 – (–3) = 6
is the same as
3 + 3 = 6
Make sure you start from
the position of the first
integer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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(iv) –5 – (–8) = 3
Alternative Method:
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtracting a negative integer:
Start at –5 and move 8 units to the right:
–5 – (–8) = 3
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
Subtract a
negative 8
This operation of
–5 – (–8) = 3
is the same as
–5 + 8 = 3
3 + 3 = 6
Subtracting a negative integer:
Start by drawing an arrow from 0 to –5, and
then, draw an arrow of 8 units to the right:
–5 – (–8) = 3
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Solve the following.
1. –2 + 4
2. 3 + (–6)
3. 2 – (–4)
4. 3 – 5 + (–2)
5. –5 + 8 + (–5)
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
–5 –4 –3 –2 –1 0 1 2 3 4 5 6
TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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TEACHING AND LEARNING STRATEGIES
This part emphasises the first alternative method which include activities and
mathematical games that can help pupils understand further and master the
operations of positive and negative integers.
Strategy:
Teacher should ensure that pupils are able to perform computations involving
addition and subtraction of integers using the Sign Model.
PART B:
ADDITION AND SUBTRACTION
OF INTEGERS USING
THE SIGN MODEL
LEARNING OBJECTIVE
Upon completion of Part B, pupils will be able to perform computations
involving combined operations of addition and subtraction of integers using
the Sign Model.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Curriculum Development Division
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PART B:
ADDITION AND SUBTRACTION OF INTEGERS
USING THE SIGN MODEL
In order to help pupils have a better understanding of positive and negative integers, we have
designed the Sign Model.
Example 1
What is the value of 3 – 5?
NUMBER SIGN
3 + + +
–5 – – – – –
WORKINGS
i. Pair up the opposite signs.
ii. The number of the unpaired signs is
the answer.
Answer –2
+
+
+
LESSON NOTES
EXAMPLES
The Sign Model
This model uses the „+‟ and „–‟ signs.
A positive number is represented by „+‟ sign.
A negative number is represented by „–‟ sign.
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Unit 1: Negative Numbers
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Example 2
What is the value of 53 ?
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 53 ?
NUMBER SIGN
–3 – – –
+5 + + + + +
WORKINGS
i. Pair up the opposite signs.
ii. The number of unpaired signs is the
answer.
Answer 2
_
+ + +
_
+
_
+
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Solve the following.
1. –4 + 8
2. –8 – 4
3. 12 – 7
4. –5 – 5
5. 5 – 7 – 4
6. –7 + 4 – 3
7. 4 + 3 – 7
8. 6 – 2 + 8 9. –3 + 4 + 6
TEST YOURSELF B
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Unit 1: Negative Numbers
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PART C:
FURTHER PRACTICE ON
ADDITION AND SUBTRACTION
OF INTEGERS
TEACHING AND LEARNING STRATEGIES
This part emphasises addition and subtraction of large positive and negative integers.
Strategy:
Teacher should ensure the pupils are able to perform computation involving addition
and subtraction of large integers.
LEARNING OBJECTIVE
Upon completion of Part C, pupils will be able to perform computations
involving addition and subtraction of large integers.
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Unit 1: Negative Numbers
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PART C:
FURTHER PRACTICE ON ADDITION AND SUBTRACTION OF INTEGERS
In Part A and Part B, the method of counting off the answer on a number line and the Sign
Model were used to perform computations involving addition and subtraction of small integers.
However, these methods are not suitable if we are dealing with large integers. We can use the
following Table Model in order to perform computations involving addition and subtraction
of large integers.
LESSON NOTES
Steps for Adding and Subtracting
Integers
1. Draw a table that has a column for + and a column
for –.
2. Write down all the numbers accordingly in the
column.
3. If the operation involves numbers with the same
signs, simply add the numbers and then put the
respective sign in the answer. (Note that we
normally do not put positive sign in front of a
positive number)
4. If the operation involves numbers with different
signs, always subtract the smaller number from
the larger number and then put the sign of the
larger number in the answer.
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Unit 1: Negative Numbers
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Examples:
i) 34 + 37 =
+ –
34
37
+71
ii) 65 – 20 =
+ –
65 20
+45
iii) –73 + 22 =
+ –
22 73
–51
iv) 228 – 338 =
+ –
228 338
–110
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
We can just write the answer as
45 instead of +45.
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
Add the numbers and then put the
positive sign in the answer.
We can just write the answer as
71 instead of +71.
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Unit 1: Negative Numbers
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v) –428 – 316 =
+ –
428
316
–744
vi) –863 – 127 + 225 =
+ –
225
863
127
225 990
–765
vii) 234 – 675 – 567 =
+ –
234
675
567
234 1242
–1008
Add the numbers and then put the
negative sign in the answer.
Add the two numbers in the „–‟
column and bring down the number
in the „+‟ column.
Subtract the smaller number from
the larger number in the third row
and put the sign of the larger
number in the answer.
Add the two numbers in the „–‟
column and bring down the number
in the „+‟ column.
Subtract the smaller number from
the larger number in the third row
and put the sign of the larger
number in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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viii) –482 + 236 – 718 =
+ –
236
482
718
236 1200
–964
ix) –765 – 984 + 432 =
+ –
432
765
984
432
1749
–1317
x) –1782 + 436 + 652 =
+ –
436
652
1782
1088 1782
–694
Add the two numbers in the „–‟
column and bring down the number
in the „+‟ column.
Subtract the smaller number from
the larger number in the third row
and put the sign of the larger
number in the answer.
Add the two numbers in the „–‟
column and bring down the number
in the „+‟ column.
Subtract the smaller number from
the larger number in the third row
and put the sign of the larger
number in the answer.
Add the two numbers in the „+‟
column and bring down the number
in the „–‟ column.
Subtract the smaller number from
the larger number in the third row
and put the sign of the larger
number in the answer.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Solve the following.
1. 47 – 89
2. –54 – 48
3. 33 – 125
4. –352 – 556
5. 345 – 437 – 456
6. –237 + 564 – 318
7. –431 + 366 – 778
8. –652 – 517 + 887 9. –233 + 408 – 689
TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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TEACHING AND LEARNING STRATEGIES
This part emphasises the second alternative method which include activities to
enhance pupils‟ understanding and mastery of the addition and subtraction of
integers, including the use of brackets.
Strategy:
Teacher should ensure that pupils understand the concept of addition and subtraction
of integers, including the use of brackets, using the Accept-Reject Model.
PART D:
ADDITION AND SUBTRACTION
OF INTEGERS INCLUDING THE
USE OF BRACKETS
LEARNING OBJECTIVE
Upon completion of Part D, pupils will be able to perform computations
involving combined operations of addition and subtraction of integers, including
the use of brackets, using the Accept-Reject Model.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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PART D:
ADDITION AND SUBTRACTION OF INTEGERS
INCLUDING THE USE OF BRACKETS
To Accept or To Reject? Answer
+ ( 5 ) Accept +5 +5
– ( 2 ) Reject +2 –2
+ (–4) Accept –4 –4
– (–8) Reject –8 +8
LESSON NOTES
The Accept - Reject Model
„+‟ sign means to accept.
„–‟ sign means to reject.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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i) 5 + (–1) =
Number To Accept or To Reject? Answer
5
+ (–1)
Accept 5
Accept –1
+5
–1
+ + + + +
–
5 + (–1) = 4
We can also solve this question by using the Table Model as follows:
5 + (–1) = 5 – 1
+ –
5 1
+4
EXAMPLES
This operation of
5 + (–1) = 4
is the same as
5 – 1 = 4
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
We can just write the answer as 4
instead of +4.
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Unit 1: Negative Numbers
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ii) –6 + (–3) =
Number To Accept or To Reject? Answer
–6
+ (–3)
Reject 6
Accept –3
–6
–3
– – – – – –
– – –
–6 + (–3) = –9
We can also solve this question by using the Table Model as follows:
–6 + (–3) = –6 – 3 =
+ –
6
3
–9
This operation of
–6 + (–3) = –9
is the same as
–6 –3 = –9
Add the numbers and then put the
negative sign in the answer.
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Unit 1: Negative Numbers
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iii) –7 – (–4) =
Number To Accept or To Reject? Answer
–7
– (–4)
Reject 7
Reject –4
–7
+4
– – – – – – –
+ + + +
–7 – (–4) = –3
We can also solve this question by using the Table Model as follows:
–7 – (–4) = –7 + 4 =
+ –
4
7
–3
This operation of
–7 – (–4) = –3
is the same as
–7 + 4 = –3
Subtract the smaller number from
the larger number and put the sign
of the larger number in the
answer.
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Unit 1: Negative Numbers
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iv) –5 – (3) =
Number To Accept or To Reject? Answer
–5
– (3)
Reject 5
Reject 3
–5
–3
– – – – –
– – –
– 5 – (3) = –8
We can also solve this question by using the Table Model as follows:
–5 – (3) = –5 – 3 =
+ –
5
3
–8
This operation of
–5 – (3) = –8
is the same as
–5 – 3 = –8
Add the numbers and then put the
negative sign in the answer.
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Unit 1: Negative Numbers
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v) –35 + (–57) = –35 – 57 =
Using the Table Model:
+ –
35
57
–92
vi) –123 – (–62) = –123 + 62 =
Using the Table Model:
+ –
62
123
–61
This operation of
–35 + (–57)
is the same as
–35 – 57
Add the numbers and then put the
negative sign in the answer.
Subtract the smaller number from
the larger number and put the sign
of the larger number in the answer.
This operation of
–123 – (–62)
is the same as
–123 + 62
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Solve the following.
1. –4 + (–8)
2. 8 – (–4)
3. –12 + (–7)
4. –5 + (–5)
5. 5 – (–7) + (–4)
6. 7 + (–4) – (3)
7. 4 + (–3) – (–7)
8. –6 – (2) + (8) 9. –3 + (–4) + (6)
10. –44 + (–81)
11. 118 – (–43)
12. –125 + (–77)
13. –125 + (–239)
14. 125 – (–347) + (–234)
15. 237 + (–465) – (378)
16. 412 + (–334) – (–712)
17. –612 – (245) + (876) 18. –319 + (–412) + (606)
TEST YOURSELF D
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Unit 1: Negative Numbers
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PART E:
MULTIPLICATION OF
INTEGERS
TEACHING AND LEARNING STRATEGIES
This part emphasises the multiplication rules of integers.
Strategy:
Teacher should ensure that pupils understand the multiplication rules to perform
computations involving multiplication of integers.
LEARNING OBJECTIVE
Upon completion of Part E, pupils will be able to perform computations
involving multiplication of integers.
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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PART E:
MULTIPLICATION OF INTEGERS
Consider the following pattern:
3 × 3 = 9
623
313
003 The result is reduced by 3 in
3)1(3 every step.
6)2(3
9)3(3
93)3(
62)3(
31)3(
00)3( The result is increased by 3 in
3)1()3( every step.
6)2()3(
9)3()3(
Multiplication Rules of Integers
1. When multiplying two integers of the same signs, the answer is positive integer.
2. When multiplying two integers of different signs, the answer is negative integer.
3. When any integer is multiplied by zero, the answer is always zero.
positive × positive = positive
(+) × (+) = (+)
positive × negative = negative
(+) × (–) = (–)
negative × positive = negative
(–) × (+) = (–)
negative × negative = positive
(–) × (–) = (+)
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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1. When multiplying two integers of the same signs, the answer is positive integer.
(a) 4 × 3 = 12
(b) –8 × –6 = 48
2. When multiplying two integers of the different signs, the answer is negative integer.
(a) –4 × (3) = –12
(b) 8 × (–6) = –48
3. When any integer is multiplied by zero, the answer is always zero.
(a) (4) × 0 = 0
(b) (–8) × 0 = 0
(c) 0 × (5) = 0
(d) 0 × (–7) = 0
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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Solve the following.
1. –4 × (–8)
2. 8 × (–4)
3. –12 × (–7)
4. –5 × (–5)
5. 5 × (–7) × (–4)
6. 7 × (–4) × (3)
7. 4 × (–3) × (–7)
8. (–6) × (2) × (8) 9. (–3) × (–4) × (6)
TEST YOURSELF E
Basic Essential Additional Mathematics Skills (BEAMS) Module
Unit 1: Negative Numbers
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PART F:
MULTIPLICATION OF INTEGERS
USING
THE ACCEPT-REJECT MODEL
TEACHING AND LEARNING STRATEGIES
This part emphasises the second alternative method which include activities to
enhance the pupils‟ understanding and mastery of the multiplication of integers.
Strategy:
Teacher should ensure that pupils understand the multiplication rules of integers
using the Accept-Reject Model. Pupils can then perform computations involving
multiplication of integers.
LEARNING OBJECTIVE
Upon completion of Part F, pupils will be able to perform computations
involving multiplication of integers using the Accept-Reject Model.
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
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