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DIRECTED NUMBERS

Subtopic 1Subtopic 1

Perform computations involving multiplication Perform computations involving multiplication

and division of integers to solve problems.and division of integers to solve problems.

At the end of this subtopic, you will be able to:

I ) multiply integers

II ) solve problems involving multiplication of integers

III) divide integers

IV) solve problems involving division of integers

+

How many items are there ?

= ?

Do you still remember ???

14 + 12 = 26

+ X + = ++ X - = - - X + = - - X - = +

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-5 -4 -3 -2 -1 0 1 2 3 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 5

NEGATIVE INTEGERS POSITIVE INTEGERS

NEGATIVE DIRECTION POSITIVE DIRECTION

HORIZONTAL NUMBER LINE

VERTICAL NUMBER LINE

P I O NS T I ET GI EV RE S

N I E NG T A ET GI EV RE S

P DO IS RI ET CI TV IE O N

N DE IG RA ET CI TV IE O N

To multiply integers:1. face the number to be multiplied2. move forward if the sign of the other number is +.3. move backward if the sign of the other number is –.

-2 X 4

sign

Other number

Number to be multiplied

Solve the following:Solve the following:

Face negative direction and move 2 steps

backward 4 times

-2 X 4 = (-2) + (-2) + (-2) + (-2) = -8

-8 -7 -6 -5 -4 -3 -2 -1 0 -8 -7 -6 -5 -4 -3 -2 -1 0


4 X 2 4 X 2

= (2) + (2) + (2) + (2)= (2) + (2) + (2) + (2)

= 8= 8

Face positive direction and move 2 steps forward 4 times

0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8


(a)(a)6 X (-5) 6 X (-5)

(b)(b)-4 X (-9) -4 X (-9)

(c)(c) -4 X (-2) X (-3)-4 X (-2) X (-3)

(d)(d)-7 X 6 X (-2)-7 X 6 X (-2)

HOT TIPS+ X + = +- X + = -+ X - = -- X - = +

= - 30= 36

= -24

= 84

+ + = ++ - = - - + = -- - = +

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÷÷÷÷

If there are like signs, the answer is positive.If there are unlike signs, the answer is negative.

To divide integers

-division is the process of equal sharing or equal grouping

+EXAMPLE:

+ +++

+ + + + +

+ +

+

++

+

+

++

+

++

+ +

12 ÷ 3= 4

Example:

-

- ---

- - - - -

- -

-

--

-

-

--

-

--

- -

-12 ÷ 3 = -4

Division is the inverse of multiplication

12 ÷ 3 = 4 , then, 3 X 4 = 12

-12 ÷ 3 = - 4, then, 3 X ( -4) = -12

Since – 3 X 4 = - 12, then, - 12 ÷ ( - 3 ) = 4.

Since - 3 X ÷ ( - 4 ) = 12, then, 12 ÷ ( - 3 ) = - 4

EXERCISE1. 8 ÷ (-2 )2. -63 ÷ ( -7)3. -40 ÷ 4 ÷ 54. 48 ÷( - 8) ÷(-2)

At the end of this subtopic, you will be At the end of this subtopic, you will be able to:able to:

I ) perform computations involving combined I ) perform computations involving combined operations of addition, subtraction, operations of addition, subtraction,

multiplication multiplication and division of integers and division of integers

II ) solve problems involving combined II ) solve problems involving combined operations of operations of

addition, subtraction, multiplication and addition, subtraction, multiplication and division of division of

integers including the use of brackets integers including the use of brackets

Subtopic 2Subtopic 2 Perform computation involving combined Perform computation involving combined operation of addition, subtraction, operation of addition, subtraction, multiplication and division of integers to solve multiplication and division of integers to solve problemsproblems

Combined Operations on Integers

To perform addition , subtraction, multiplication and division on integers

Examples:

Evaluate : a) -2 + 5 – 7 = 3 – 7 = - 4

b) – 6 X (- 4) ÷ ( - 3 ) = 24 ÷ ( - 3 ) = - 8

Do the operations from left to right

EXERCISE

Calculate:a) 7 + 3 – 15

b) -2 – 11 + 3

c) -3 – ( -2 ) – 8

d) - 8 – 13 – (- 6)

e) - 9 + ( - 6) – 12

f) - 15 + (-7) –(-12)

EXERCISE

a) 4 X (- 3) ÷ 2

b) - 5 X ( - 8) ÷ 10

c) - 12 X (- 2) ÷ (-8)

d) - 40 ÷ 8 X (- 11)

e) - 128 ÷ 4 X ( - 3 )

f) - 9 X (- 8) ÷ (- 12)

At the end of this subtopic, you will be At the end of this subtopic, you will be able to:able to:

I ) compare and order fractionsI ) compare and order fractions

II ) perform addition, subtraction, multiplication II ) perform addition, subtraction, multiplication or or

division on fractions division on fractions


Extend the concept of integers to fractions to Extend the concept of integers to fractions to

solve problemssolve problems

+

?

How much is left, in fraction ?

Positive and Negative Fractions

*A positive fraction has a ‘+’ sign before it or has no sign. Example: 1 , 3 4 7

•A negative fraction has a ‘–’ sign before it. Example: 1 , 5 4 9

+

- -

divide the number line to equal parts according to the LCM of the denominators

-1 2

-1 0 12

12

1-5 6

16

13

23

56

1 3

-1 3

-1 6

-4 6

Complete the fractional number line

-3 6

36

8 6

7 6

46

-2 6

26 1 1-2

3

Arrange the following fractions in ascending order.

5

4

7

5

10

7

,

and

1075

1074

, 1057

1055

7510

757

350

280

350

250350

245

350

245

350

250350

280

10

7

7

55

4< <

To perform addition and subtraction on fractions.

3

2

9

7

3

2

9

7

33

32

9

7

9

6

9

7

9

1

HOT TIPS+ X + = +- X + = -+ X - = -- X - = +

Example 1

Example 2

2

1

8

5

4

3

2

1

8

5

4

3

82

81

28

25

44

43

16

81012

16

6

8

3

HOT TIPS+ X + = +- X + = -+ X - = -- X - = +

To multiply and divide fractions

Example 1

9

11

4

3

9

10

4

3

6

5

Example 2

9

4

3

21

9

4

3

5

4

9

3

5

4

15

4

33

Multiply by the reciprocal

of 4 9

Change to mixed number

Change to improper fraction

HOMEWORK

Exercise 1.3C

No. 1(a), (e)

2(a), (e)

3(a), (f)

4(a), (e)

At the end of this subtopic, you will be At the end of this subtopic, you will be able to: able to:

I ) compare and order decimals.I ) compare and order decimals.

II ) perform addition, subtraction, II ) perform addition, subtraction, multiplication or multiplication or

division on decimals. division on decimals.

Subtopic 4Subtopic 4Extend the concept of integers to decimal Extend the concept of integers to decimal to solve problemsto solve problems

What is the balance of the money?

RM 1.20 each RM 5.60

SPECIMEN

Positive and Negative Decimals

There are positive and negative decimals.

Positive decimals are greater than zero.

Negative decimals are smaller than zero.

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

To compare and order decimals

Decimals can be marked on a number line.

-0.7 < -0.2

To add and subtract decimals.

Example :

a) 6.4 + ( - 4.8 )

= 6.4 – 4.8

= 1.6

b) – 9.25 – ( - 4.36 ) – 5.12

= -9.25 + 4.36 – 5.12

= - 4.89 – 5.12

= - 10.01

Exercise

Calculate:

1. a) - 3.5 + 2.6

b) 4.2 + 1.5

c) -1.1 + 9.9

d) -6.2 + ( -7.9)

e) 2.3 – 1.9

f) -1.8 – 9.2

2. a) -2.51 + 9.62 + 2.1

b) 1.13 + (- 4.78) + (- 5.9)

c) - 0.92 + (- 3.19) + 7.2

d) – 2.16 – 3.47 – 4.8

e) 9.23 – 12.4 – 4.8

To multiply and divide decimals

Example 1

1.3 X ( - 2.5 ) 1.3X 2.5 6 5 2 6 3.2 5

= -3.25

Example 2

2.0

34.6

2

4.63

= -31.7

2) 63.431.7

632141414-

At the end of this subtopic, you will be able At the end of this subtopic, you will be able to:to:

I ) perform addition, subtraction, multiplication or I ) perform addition, subtraction, multiplication or

division involving two directed numbers. division involving two directed numbers.

II ) perform computations involving combination II ) perform computations involving combination of two of two

or more operations on directed numbers or more operations on directed numbers including including

the use of brackets. the use of brackets.

III) pose and solve problems involving directed III) pose and solve problems involving directed numbersnumbers


Perform computation involving directed Perform computation involving directed

numbers (integers, fractions and decimals)numbers (integers, fractions and decimals)

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BODMAS is a mathematical rule to determine the order of calculation

B is for brackets

O is for power of

M is for multiplication

A is for addition

S is for subtraction

D is for division

Evaluate the following:Evaluate the following:

1)1)

2) 8 X (-4.5 )2) 8 X (-4.5 )

3) 3)

65

23

)2(4

15

CalculateCalculate

1) ( -6 + 10 ) X 51) ( -6 + 10 ) X 5

2)2)

3) 3)

8

5)7

6(

12

5

1.2)4

31(5

What have we learnt?

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