© boardworks ltd 2004 1 of 56 n6 calculating with fractions ks3 mathematics

$: © Boardworks Ltd 2004 1 of 56 N6 Calculating with fractions KS3 Mathematics$
© Boardworks Ltd 20041 of 56

N6 Calculating with fractions

KS3 Mathematics


N6.1 Adding and subtracting fractions

Contents


N6.4 Dividing by fractions

N6.2 Finding a fraction of an amount

N6.3 Multiplying fractions


Fraction counter


When fractions have the same denominator it is quite easy to add them together and to subtract them.

For example,

3

5+

1

5=

3 + 1

5=

4

5

We can show this calculation in a diagram:

+ =

Adding and subtracting simple fractions


7

8–

3

8=

7 – 3

8=

4

8

Fractions should always be cancelled down to their lowest terms.

1

2 =1

2

We can show this calculation in a diagram:

– =



1

9+

7

9+

4

9=

1 + 7 + 4

9=

12

9

Top-heavy or improper fractions should be written as mixed numbers.

= 1 3

9

1

3 = 1 1

3

Again, we can show this calculation in a diagram:

+ + =



What is + ?12

14

12

+

+14

=

=34



What is + ?12

34

12

+

+34

=

= 1 14



What is – ?12

38

12

–

–38

=

=18



Fractional magic square


Fractions with common denominators

Fractions are said to have a common denominator if they have the same denominator.

For example,

1112

,412

and512

all have a common denominator of 12.

We can add them together:

1112

+412

+512

=11 + 4 + 5

12=

2012

= 1 812

= 1 23


Fractions with different denominators

Fractions with different denominators are more difficult to add and subtract.

For example,

What is +12

13

?

We can show this sum using diagrams:

+

36

+26

=

=3 + 2

6=

56


Using diagrams

–

1518

–418

=

=15 – 4

18=

1118

What is 56

–29

?


+

1220

+1520

=

=12 + 15

20=

2720

= 1 720

What is 35

+34

?

Using diagrams


–

2520

–1420

=

=25 – 14

20=

1120

What is –710

?1 14

Using diagrams


Using a common denominator

1) Write any mixed numbers as improper fractions.

1 34

= 74

2) Find the lowest common multiple of 4, 9 and 12.

The multiples of 12 are: 12, 24, 36 . . .

36 is the lowest common denominator.

What is +19

?134

+512


3) Write each fraction over the lowest common denominator.

74

=36

×9

×9

63 19

=36

×4

×4

4 512

=36

×3

×3

15

4) Add the fractions together.

3663

+364

+3615

=36

63 + 4 + 15=

3682

= 2 3610

= 2 185

What is +19

?134

+512

Using a common denominator


Adding and subtracting fractions


Using a calculator

It is also possible to add and subtract fractions using the

key on a calculator.abc

For example, to enter 84

we can key in abc4 8

The calculator displays this as:

Pressing the = key converts this to:


To calculate: 23

+45

using a calculator, we key in:

abc2 3 + abc4 5 =

The calculator will display the answer as:

We write this as 1 157

Using a calculator


Fraction Puzzle


Drag and drop fraction sum


Fraction cards



Contents






Finding a fraction of an amount

We can see this in a diagram:

23

of £18 = £18 ÷ 3 × 2 = £12

23

of £18?What is


Let’s look at this in a diagram again:

710

of £20 = £20 ÷ 10 × 7 = £14

710

of £20?What is



56

of £24 =16

of £24 × 5

= £24 ÷ 6 × 5

= £4 × 5

= £20

56

of £24?What is



What is of 9 kg?4

7

To find of an amount we can multiply by 4 and divide by 7.4

7

We could also divide by 7 and then multiply by 4.

4 × 9 kg = 36 kg

36 kg ÷ 7 = =36

7kg 5

1

7kg



When we work out a fraction of an amount we

multiply by the numerator

and

divide by the denominator

For example,

23

of 18 litres = 18 litres ÷ 3 × 2

= 6 litres × 2

= 12 litres



To find of an amount we need to add 1 times the amount to two fifths of the amount.

251

1 × 3.5 m =3.5 m and2

5of 3.5 m = 1.4 m

so, of 3.5 m =2

51 3.5 m + 1.4 m =4.9 m

What is of 3.5m?2

51



MathsBlox



Contents






Counting on and back using fractions


Multiplying fractions by integers

We can illustrate this calculation on a number line:

0

14

14

14

12

14

34

14

1

14

141

14

121

14

341

14

2

14

What is 8 × ?


Again, we can illustrate this calculation on a number line:

0

34

34

34

121

34

142

34

3

34

343

34

124

34

145

34

6

34

346

34

127

34

148

34

9

34

What is 12 × ?



Let’s use a number line again:

0

13

13

13

23

13

1

13

131

13

231

13

2

13

132

13

232

13

3

13

What is 9 × ?



So,

8 × = 214

12 × = 934

9 × = 313

What do you notice?

What do you notice?

and



Following the rules of arithmetic, we know that,

8 ×14

=14

× 8

In maths the word ‘of’ means ‘times’.

=14

of 8 = 8 ÷ 4

These are equivalent calculations.



Means the same as:

35

20 ×

35

× 2035

of 2015

3 × of 20

20 × 3 ÷ 5 20 ÷ 5 × 3 3 ÷ 5 × 20

Equivalent calculations


When we multiply a fraction by an integer we:


and


For example,

49

54 × = 54 ÷ 9 × 4

= 6 × 4

= 24



57

12 × ?What is

57

12 × = 12 × 5 ÷ 7

= 60 ÷ 7

=607

= 8 47



Using cancellation to simplify calculations

712

What is 16 × ?

We can write 16 × as:712

161

×712

4

3=

283

=139


825

What is × 40?

We can write × 40 as:825

825

×401

8

5=

645

=4512

Using cancellation to simplify calculations


Multiplying a fraction by a fraction

To multiply two fractions together, multiply the numerators together and multiply the denominators together:

38

What is × ?25

38

45

× =1240

3

10

=310


56

What is × ?12255

Start by writing the calculation with any mixed numbers as improper fractions.

To make the calculation easier, cancel any numerators with any denominators.

1225

356

× =

7

5

2

1

145

= 2 45

Multiplying a fraction by a fraction


Multiplying fractions mentally



Contents






Multiplying and dividing fractions


Dividing an integer by a fraction

13

What is 4 ÷ ?

13

4 ÷ means, “How many thirds are there in 4?”

Here are 4 rectangles:

Let’s divide them into thirds.

4 ÷ = 1213


25

What is 4 ÷ ?

25

4 ÷ means, “How many two fifths are there in 4?”

Here are 4 rectangles:

Let’s divide them into fifths, and count the number of two fifths.

4 ÷ = 1025



34

6 ÷ means, ‘How many three quarters are there in six?’

6 ÷ = 6 × 414

= 24

So,

6 ÷ = 24 ÷ 334

= 8

We can check this by multiplying.

8 × = 8 ÷ 4 × 334

= 6

34

What is 6 ÷ ?



Dividing a fraction by a fraction

18

What is ÷ ?12

means, ‘How many eighths are there in one half?’18

÷12

Here is of a rectangle:12

Now, let’s divide the shape into eighths.

= 418

÷12


45

What is ÷ ?23

To divide by a fraction we multiply by the denominator and divide by the numerator.

45

23

÷ can be written as54

23

×

Swap the numerator and the denominator and multiply.

54

23

× =1012

=56



67

What is ÷ ?353

Start by writing as an improper fraction. 353

185

353 =

185

÷67

=185

×76

3

1

=215

=154



Multiplying and dividing are inverse operations.

When we multiply by a fraction we:


and


When we divide by a fraction we:

divide by the numerator

and

multiply by the denominator

Multiplying and dividing by fractions


Dividing fractions

© boardworks ltd 2004 1 of 56 n6 calculating with fractions ks3 mathematics

Documents

fractions n6

simple fractions

improper fractions

multiplying fractions

common denominator slide

common denominators

fractions contents n6

fraction counter slide