1. N6 Calculating with fractions
KS3 Mathematics
The aim of this unit is to teach pupils to:
Calculate fractions of quantities; add, subtract, multiply and divide fractions.
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
N6 Calculating with fractions

2. N6.1 Adding and subtracting fractions
Contents
N6 Calculating with fractions A1
N6.1 Adding and subtracting fractions A1
N6.2 Finding a fraction of an amount A1
N6.3 Multiplying fractions A1
N6.4 Dividing by fractions

3. Fraction counter
This counter can be used to count on and back in fractional steps. It can also be used to aid pupils to add and subtract simple fractions mentally.
Start by setting the step size to 1/5 and clicking the clockwise arrow once to show 1/5 on the board.
What is 1/5 + 1/5 ?
Click the clockwise arrow once to show 2/5.
What is 2/5 + 2/5?
Press the clockwise arrow twice to move to 4/5.
What is 4/5 + 2/5?
Press the clockwise arrow to move to 11/5.
What is 11/5 minus 3/5?
We can think of 11/5 as 6/5.
6/5 - 3/5 is 3/5.
Demonstrate this using the counter.
Next, set the step size to 1/4.
Use the counter to ask questions such as what is ‘1/2 + 1/4?’
Finally, spend some time counting on and back in 1/10s, 1/8s and 3/4s. Link this to the addition and subtraction of simple fractions.

4. Adding and subtracting simple fractions
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:
Talk about adding fractions with the same denominator.
What is three fifths plus one fifth?
What are we adding? (fifths)
Three fifths plus one fifth is four fifths.
Show the example on the slide and emphasize that when the denominator is the same we can add together the numerators. Show that by writing (3 + 1) over 5 using a single bar we can avoid adding the denominators together by mistake.
Show the calculation as a diagram. + =

5. Adding and subtracting simple fractions7 8 – 3
7 – 3 8 4 8 1 1 2 = = = 2
Fractions should always be cancelled down to their lowest terms.
We can show this calculation in a diagram:
Talk through the example on the board and remind pupils that fractions should always be cancelled down to their lowest terms.
The 4 and the 8 in 4/8 are both divisible by 4. Cancelling gives us 1/2. – =

6. Adding and subtracting simple fractions1 9 + 7 4 = 9 = 12 9 = 1 3 9 = 1 1 3 3
Top-heavy or improper fractions should be written as mixed numbers.
Again, we can show this calculation in a diagram:
Improper fractions (top-heavy fractions) should be written as mixed numbers.
In the fraction 12/9, 12 is bigger than 9. This is an improper (or top-heavy) fraction. 9/9 make one whole plus 3/9 left over.
Again, remind pupils that fractions must be cancelled if possible.
3 and 9 are both divisible by 3. Cancelling gives us 1/3. + + =

7. Adding and subtracting simple fractions1 2 1 4
What is ? + =
Tell pupils that they should be able to work out the answer to 1/2 + 1/4 mentally.
We can show this using diagrams.
Click to reveal.
We can think of 1/2 as 2/4..
You can see that 2/ /4 is 3/4.. 1 2 1 4 3 4 + =

8. Adding and subtracting simple fractions1 2 3 4
What is ? + =
This time the answer is more than 1.
If we add 2 of the quarters to 1/2 that makes 1 whole, plus the other quarter makes 11/4.
We can show this with the following diagrams.
Click to reveal. 1 2 3 4 1 4 + =

9. Adding and subtracting simple fractions1 2 3 8
What is – ? – =
Ask pupils how we could work this out mentally.
If we think of 1/2 as 4/8, then the calculation is much easier.
Explain that the diagram shows 1/2 is equivalent to 4/8. 1 2 3 8 1 8 – =

10. Fractional magic square
Explain to pupils that in a magic square each row, column and diagonal adds up to the same amount. This amount is called the magic total.
Start by working out the magic total and then ask pupils for any squares they can work out.

11. Fractions with common denominators
Fractions are said to have a common denominator if they have the same denominator.
For example: 11 12 4 12 5 12 ,
and
all have a common denominator of 12.
We can add them together:
Remind pupils that since the denominators are the same we can write the sum over a single bar with 12 underneath. (Using a single bar will avoid the problem of adding denominators in error).
Give some examples of subtraction verbally.
What is 11/12 – 5/12? (6/12 which is equivalent to 1/2)
Remind pupils that improper fractions should be written as mixed numbers and fractions, where necessary, should be written in their lowest terms (cancelled down). 11 12 4 12 5 12 12 20 12 1 8 12 1 2 3 + + = = = =

12. Fractions with different denominators
Fractions with different denominators are more difficult to add and subtract.
What is + 1 2 3 ?
We can show this sum using diagrams:
Point out, with reference to the diagrams that we cannot add 1/2 + 1/3 directly. The answer is certainly not 2/5, as some pupils may think!
Ask pupils to suggest ways of adding these fractions.
We know that 1/2 is equivalent to 3/6 and 1/3 is equivalent to 2/6.
Click to divide the shapes into sixths.
6 is the lowest common multiple of 2 and 3.
Once we have written both fractions with a common denominator, we can add them together. + = 3 6 2 6
3 + 2 6 5 6 + = =

13. Using diagrams 5 6 2 9 What is – ? – = 15 18 4 18 15 – 4 18 11 18 – =
Here are some more examples using diagrams.
Reveal the diagrams of 5/6 and 2/9.
What is the lowest common multiple of 6 and 9?
As a mental method for finding the lowest common multiple instruct pupils to take the larger of the two denominators, in this case 9, and go through multiples of 9 (in order) until they find a multiple of 9 which is also a multiple of 6.
9 is not a multiple of 6, but the next multiple of 9, 18, is.
Click to divide the diagrams into 1/18s. 15 18 4 18
15 – 4 18 11 18 – = =

14. 1 Using diagrams What is 3 5 + 4 ? + = 12 20 15 20 12 + 15 20 27 20 7
Reveal the diagrams of 3/5 and 3/4.
What is the lowest common multiple of 5 and 4?
Again, take the larger of the two denominators, in this case 5, and go through multiples of 5 (in order) until we find a multiple of 5 which is also a multiple of 4.
10 is not a multiple of 4, nor is 15, but the next multiple of 5, 20, is.
Let’s divide these rectangles into 1/20s.
Click to reveal. 12 20 15 20 20 27 20 1 7 20 + = = =

15. 1 Using diagrams 4 7 10 What is – ? – = 25 20 14 20 25 – 14 20 11 20 –
Talk through the example as before. 25 20 14 20
25 – 14 20 11 20 – = =

16. Using a common denominator
What is + 1 9 ? 3 4 5 12
1) Write any mixed numbers as improper fractions. 1 3 4 7 4 =
2) Find the lowest common multiple of 4, 9 and 12.
Let’s add these fractions without using a diagram.
We must follow these steps.
Revise method for changing mixed numbers to improper fractions. (This step could be left out if we add whole number parts together separately from fractional parts).
2) We need to write the fractions over a common denominator. The lowest common denominator is equal to the lowest common multiple of the denominators.
Tell pupils that one way to do this is to go through the multiples of the highest denominator, in this case 12, until we find a multiple of 12 which is divisible by the other denominators, in this case 4 and 9. Going through multiples of 12.
12 is divisible by 4 but not by 9, 24 is divisible by 4 but not by 9, 36 is divisible by both 4 and 9. Therefore, 36 is the lowest common multiple.
The multiples of 12 are:
12,
24,
36 is the lowest common denominator.

17. Using a common denominator
What is + 1 9 ? 3 4 5 12
3) Write each fraction over the lowest common denominator. ×9 ×4 ×3 7 4 63 1 9 4 5 12 15 = = = 36 36 36 ×9 ×4 ×3
3) Review the method for finding equivalent fractions.
4) Remind pupils that the answer should be written as a mixed number and cancelled down if possible.
4) Add the fractions together. 36 63 36 4 36 15 36 36 82 2 36 10 2 18 5 + + = = = =

18. Adding and subtracting fractions
Use this activity to talk through as many examples as necessary.

19. Using a calculator
It is also possible to add and subtract fractions using the
key on a calculator. a b c
For example, to enter 8 4
we can key in a b c
The calculator displays this as:
Allow pupils a few moments to locate the fraction key on their calculators. Demonstrate how this key on the calculator can be used to simplify fractions (both to cancel fractions and to write improper fractions as mixed numbers).
Pressing the =
key converts this to:

20. 1 Using a calculator To calculate: 2 3 + 4 5
using a calculator, we key in: a b c 2 3 + 4 5 =
The calculator will display the answer as:
Point out that although it is easy to use a calculator to add and subtract fractions they must be able to do it, demonstrating the full method, without using a calculator. Pupils can then use the calculator to check their answers.
We write this as 1 15 7

21. Fraction Puzzle
Ask pupils how they can solve the puzzle on the board so that each row, column and diagonal has the same sum. Suggest that pupils convert all the fractions shown so that they have a common denominator.
Each puzzle is based on the whole number magic square:
4 9 2
3 5 7
8 1 6
Clicking on the check button will reveal the sum of each completed row column and diagonal.

22. Drag and drop fraction sum
Possible solutions are:
1/2 + 3/6
1/2 + 4/8
2/4 + 3/6
3/6 + 4/8
1/3 + 4/6
3/9 + 4/6
1/4 + 6/8
2/8 + 3/4

23. Fraction cards
Start by asking pupils for the lowest common denominator of the set of fractions on the board. Go on to establish that every fraction on the board can therefore be expressed as an equivalent fraction over that denominator.
Review the fact that to add or subtract two fractions they must be written over a common denominator.
Ask volunteers to come to the board and show different possible solutions to the sum or difference required.
If required you may drag one of the fraction cards down and ask pupils for the other card to make the required sum or difference.
Ensure, during the course of the activity, that pupils are able to quickly and confidently convert fractions to equivalent fractions over a common denominator (either mentally or using jottings).

24. N6.2 Finding a fraction of an amount
Contents
N6 Calculating with fractions A1
N6.1 Adding and subtracting fractions A1
N6.2 Finding a fraction of an amount A1
N6.3 Multiplying fractions A1
N6.4 Dividing by fractions

25. Finding a fraction of an amount2 3
of £18?
What is
We can see this in a diagram:
1/3 of £18 is £6. Click to reveal.
Remember 2/3 means 2 lots of 1/3.
So, 2/3 of £18 is 2 x £6.
Click to reveal £12.
We divide by 3 to find 1/3 and then we multiply by 2 to find 2/3. 2 3
of £18
= £18 ÷ 3 × 2
= £12

26. Finding a fraction of an amount7 10
of £20?
What is
Let’s look at this in a diagram again:
When the diagram appears say:
Here is £20.
1/10 of 20 is £2. Click to reveal.
2/10 of £20 is £4. Click to reveal and continue stating the proportion shaded until you reach 7/10 of £20 is £14. 7 10
of £20
= £20 ÷ 10 × 7
= £14

27. Finding a fraction of an amount5 6
of £24?
What is 5 6
of £24 = 1 6
of £24
× 5
= £24 ÷ 6 × 5
This time we’ll do it without the help of a diagram.
Remember 5/6 means ‘5 lots of 1/6’ so we need to work out 1/6 of 24, by dividing 24 by 6, and then multiply that answer by 5.
Reveal each stage in the calculation.
= £4 × 5
= £20

28. Finding a fraction of an amount4 7
What is of 9 kg?
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
Remind pupils that we can multiply by the numerator and divide by the denominator in either order.
In this example, point out that 9 kg is not divisible by 7.
Let’s try multiplying by the numerator first and then dividing by the denominator.
Emphasize that 36 ÷ 7 is equivalent to 36/7. Writing this as an improper fraction allows us to write the answer as a mixed number. 36 ÷ 7 is 5 remainder 1 or 51/7.
We could also write 51/7 as 5.14 (to 2 decimal places). = 36 7 kg 5 1 7 kg
36 kg ÷ 7 =

29. Finding a fraction of an amount
When we work out a fraction of an amount we
multiply by the numerator
and
divide by the denominator
For example: 2 3
of 18 litres
Tell pupils that it doesn’t make any difference whether you multiply first or divide first. It depends on the problem.
Demonstrate using the numbers in the example, that multiplying 18 by 2 and then dividing by 3 (this gives us 36 ÷ 3 = 12) is the same as dividing 18 by 3 and then multiplying by 2 (this gives us 6 × 2 = 12).
Establish that in this example if we divide first, the numbers will be easier to work with.
= 18 litres ÷ 3 × 2
= 6 litres × 2
= 12 litres

30. Finding a fraction of an amount1 2 5
What is of 3.5m?
To find of an amount we need to add 1 times the amount to two fifths of the amount. 2 5 1 2 5
of 3.5 m =
1 × 3.5 m =
3.5 m
and
1.4 m
Discuss the meaning of 12/5 of an amount.
Go through the example on the board.
An alternative would be to convert 12/5 to an improper fraction, 7/5. We could then divide by 5 to get 0.7 m and then multiply by 7 to get the answer 4.9 m.
Ask pupils if they can give you an equivalent decimal calculation (1.2 × 3.5 m) or an equivalent percentage calculation. (120% of 3.5 m)
How would we calculate 22/5 of 3.5 m?
Establish would need to add 2 × 3.5 m to 2/5 of 3.5 m, that’s 7 m plus 1.4 m, 8.4 m. (Alternatively, we could find 12/5 of 3.5 m).
Compare this to finding 2.2 × 3.5 m or 220% of 3.5 m.
so,
of 3.5 m = 2 5 1
3.5 m m =
4.9 m

31. MathsBlox
Divide the class into two teams – red and blue – and nominate a spokesperson for each team.
The spokesperson from the first team chooses a number on the grid. A question appears when the number is selected and everyone on that team must try to work out the answer. They may use jottings if they wish. Select a member of the team at random to give you their answer (if they pass the hexagon is coloured in the opposing team’s colour).
Click on the ‘show answer’ button. If it is correct, colour the hexagon in the team colour if it is incorrect the hexagon is coloured in the opposing team colour. Spend a few moments discussing mental strategies to answer the question.
Play then passes to the other team.
The winning team is the first team to connect a line of hexagons either from top to bottom or from left to right.

32. N6.3 Multiplying fractions
Contents
N6 Calculating with fractions A1
N6.1 Adding and subtracting fractions A1
N6.2 Finding a fraction of an amount A1
N6.3 Multiplying fractions A1
N6.4 Dividing by fractions

33. Counting on and back using fractions
Spend some time counting on and back using fractions. Start with unit fractions such as 1/4 and 1/8 and progress to other proper fractions such as 3/4.
Ask pupils how we could use the counter to work out, say, 10 × 1/3.
Demonstrate, using the counter, that 10 × 1/3 is 31/3.
Establish that 1/3 of 10 is also 31/3 and that 10 × 1/3 is equivalent to 1/3 of 10.
Tell pupils that in maths ‘of’ means the same as ‘times’. 1/3 of 10 is equivalent to 1/3 × 10 which is equivalent to 10 × 1/3. These calculations are also equivalent to 10 ÷ 3.
Repeat for other examples.

34. Multiplying fractions by integers1 4
What is 8 × ?
We can illustrate this calculation on a number line: 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4
Remember - an integer is a whole positive or negative number.
As the numbers appear on the number line, ask pupils what the next number will be. This will revise counting on in steps of 1/4.
8 × 1/4 is 2. 1 4 1 2 3 4 1 1 4 1 2 3 4 1 2

35. Multiplying fractions by integers3 4
What is 12 × ?
Again, we can illustrate this calculation on a number line: 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4
Again, as the numbers appear on the number line, ask pupils what the next number will be. This will revise counting on in steps of 3/4.
12 × 3/4 is 9. 3 4 1 2 1 4 2 3 3 4 1 2 4 1 4 5 6 3 4 6 1 2 7 1 4 8 9

36. Multiplying fractions by integers1 3
What is 9 × ?
Let’s use a number line to illustrate this sum: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3
As before, as the numbers appear on the number line, ask pupils what the next number will be. This will revise counting on in steps of 1/3.
9 × 1/3 is 3. 1 3 2 3 1 1 3 2 3 1 2 1 3 2 2 3 3

37. Multiplying fractions by integers
So:
8 × = 2 1 4
12 × = 9 3 4
and
9 × = 3. 1 3
Ask pupils to discuss the meaning of these examples.
8 × 1/4 means 8 lots of 1/4 or 8/4 which is equal to 2.
Also, 1/4 of 8 is 2.
Remember, 1/4 of 8 means 8 divided into 4 equal parts, in other words 8 ÷ 4.
12 × 3/4 is 36/4 which is equal to 36 divided by 4, which is 9.
Also 3/4 of 12 is 9.
1/4 of 12 is 3, so 3/4 of 12 is 9.
Remember - the bottom number in the fraction, the denominator, tells you what they are: quarters. The top number in the fraction, the numerator, tells you how many of them there are: three quarters.
In the last example, 9 × 1/3 is 9/3 which is equal to 3. Also 1/3 of 9 is 3.
What do you notice?

38. Multiplying fractions by integers
Following the rules of arithmetic, we know that:
In maths the word ‘of’ means ‘times’.
8 × 1 4 1 4
× 8 1 4
of 8
8 ÷ 4 = = =
Explain that when we use the multiplication symbol with fractions it means the same thing as ‘of’.
Give a few examples verbally.
1/5 × £25 means 1/5 of £25.
2/3 × 60m means 2/3 of 60m.
Finding a quarter of a given amount is the same as dividing it by 4.
Emphasize that all of the expressions on the board are equivalent.
These are equivalent calculations.

39. Equivalent calculations3 5
20 ×
Means the same as: 3 5
× 20 3 5
of 20 1 5
3 ×
of 20
Discuss the equivalence of these expressions.
In maths ‘of’ means ×.
Tell pupils that the intermediate step in each calculation may give different numbers, but the final answer is always the same.
We can use a different order of operations to check our calculations.
For example:
20 × 3 ÷ 5 = 60 ÷ 5 = 12
20 ÷ 5 × 3 = 4 × 3 = 12
3 ÷ 5 × 20 = 0.6 × 20 = 12
20 × 3 ÷ 5
20 ÷ 5 × 3
3 ÷ 5 × 20

40. Multiplying fractions by integers
When we multiply a fraction by an integer we:
multiply by the numerator
and
divide by the denominator
For example: 4 9
54 ×
Remind pupils that an integer is a whole number that can be positive or negative or 0.
Point out that it does not matter what order we use to multiply and divide.
When the denominator divides exactly into the number we are multiplying by, it is easiest to divide first and then multiply.
In this example, we would get the same answer if we multiplied 54 by 4 and then divided by 9. However, if we divide first the numbers are smaller and easier to work out mentally.
If the denominator does not divide into the number we are multiplying by, we can multiply first and then divide, to write the answer as a mixed number.
Compare this to finding a fraction of an amount.
= 54 ÷ 9 × 4
= 6 × 4
= 24

41. Multiplying fractions by integers5 7
12 × ?
What is 5 7
12 ×
= 12 × 5 ÷ 7
= 60 ÷ 7
In this case, we cannot easily divide 12 by 7, therefore we need to multiply by the numerator, 5, first.
We could think of 12 × 5/7 as 12 × 5 sevenths, 60 sevenths, and go straight to this step (indicate the third step in the calculation). = 60 7 = 8 4 7

42. Using cancellation to simplify calculations7 12
What is 16 × ?
We can write 16 × as: 7 12 4 16 1 × 7 12 28 =
If the number we are multiplying by and the denominator of the fraction share a common factor, then we can cancel the common factor before we multiply.
Spend a few moments making sure that pupils are convinced that 16 can be written as a fraction as 16/1.
When we multiply two fractions together we can cancel any of the numerators with any of the denominators.
16 and 12 are both divisible by 4, we can therefore cancel.
We then multiply the numerators together to get 28. Click to reveal.
Multiplying the denominators together gives us 3. Click to reveal.
28/3 is an improper fraction so we have to write it as a mixed number.
Point out that if we didn’t cancel in the first step we would still get the same answer, but the numbers would be more difficult to work out mentally and we would still have to cancel at the end. 16 × 7 is 112. We would then have to cancel 112/12 to 28/3. 3 3 = 1 3 9

43. Using cancellation to simplify calculations8 25
What is × 40?
We can write × 40 as: 8 25 8 8 25 × 40 1 64
Talk through this method once more.
Again, point out that if we didn’t cancel in the first step, we would still get the same answer, but the numbers would be more difficult to work out mentally and we would still have to cancel at the end. (8 × 40 is 320. We would then have to cancel 320/25 to 64/5) = 5 5 = 4 5 12

44. Multiplying a fraction by a fraction3 8
What is × ? 2 5
To multiply two fractions together, multiply the numerators together and multiply the denominators together: 3 3 8 4 5 × = 12
Point out that we could also cancel before multiplying. 40 10 = 3 10

45. Multiplying a fraction by a fraction5 6
What is × ? 12 25
Start by writing the calculation with any mixed numbers as improper fractions.
To make the calculation easier, cancel any numerators with any denominators. 7 2 12 25 35 6 × = 14 5 1 5 = 2 4 5

46. Multiplying fractions mentally
Practise multiplying fractions mentally. Ask pupils for the solution to the calculation, and then check it by revealing the answer.

47. N6.4 Dividing by fractions
Contents
N6 Calculating with fractions A1
N6.1 Adding and subtracting fractions A1
N6.2 Finding a fraction of an amount A1
N6.3 Multiplying fractions A1
N6.4 Dividing by fractions

48. Multiplying and dividing fractions
Ask pupils to answer each problem working mentally or using jottings before revealing the solution behind the curtains.

49. Dividing an integer by a fraction1 3
What is 4 ÷ ? 1 3
4 ÷ means, “How many thirds are there in 4?”.
Here are 4 rectangles:
We could also think of this as, ‘What number times 1/3 will give us an answer of 4?’ or,
‘If we divided four apples into thirds, how many pieces would we have?’
Reveal the rectangles and ask pupils to count the number of thirds in four or deduce the number by working out 4 × 3, 3 thirds in each whole one.
Dividing by a third is equivalent to multiplying by three.
Point out that when we divide a whole number by a fraction less than one, the result will be a larger number.
Let’s divide them into thirds.
4 ÷ = 12 1 3

50. Dividing an integer by a fraction2 5
What is 4 ÷ ? 2 5
4 ÷ means, “How many two fifths are there in 4?”.
Here are 4 rectangles:
We could also think of this as, ‘what number times 2/5 will give us an answer of 4?’.
Reveal the rectangles and ask pupils to deduce the number of fifths in 4 by working out 4 × 5, five fifths in each whole one.
So how many two fifths are there in four whole ones?
Click to count the number of two fifths, 10.
We could have worked this out by dividing the number of fifths, 20, by two to get 10.
What is 2/5 × 10? or 2/5 of 10?
This is four, so we know we are correct.
Let’s divide them into fifths,
and count the number of two fifths.
4 ÷ = 10 2 5

51. Dividing an integer by a fraction3 4
What is 6 ÷ ? 3 4
6 ÷ means, “How many three quarters are there in six?”.
6 ÷ = 6 × 4 1 4
= 24
So:
6 ÷ = 24 ÷ 3 3 4
Go through each step in the calculation.
Establish that when we divide a whole number by a fraction, we multiply by the denominator and divide by the numerator.
= 8
We can check this by multiplying.
8 × = 8 ÷ 4 × 3 3 4
= 6

52. Dividing a fraction by a fraction1 8
What is ÷ ? 2
means, “How many eighths are there in one half?”. 1 8 ÷ 2
Here is of a rectangle: 1 2
Reveal the shape divided into eighths and ask pupils how many eighths there are in one half.
Reveal the calculation and the answer 4.
State that dividing by 1/8 is equivalent to multiplying by 8.
When we divide by a fraction we multiply by denominator and divide by the numerator.
Now, let’s divide the shape into eighths. 1 8 ÷ 2
= 4

53. Dividing a fraction by a fraction2 3 4 5
What is ÷ ?
To divide by a fraction we multiply by the denominator and divide by the numerator.
Swap the numerator and the denominator and multiply. 4 5 2 3 ÷ 5 4 2 3 ×
can be written as
Explain that when we are dividing by a fraction we can write an equivalent calculation by swapping the numerator and the denominator around (turning the fraction upside-down) and multiplying.
This works because when we multiply by a fraction we multiply by the numerator and divide by the denominator.
Multiplying by a fraction is straight forward because we simply multiply the numerators together and multiply the denominators together. 5 4 2 3 × = 10 12 = 5 6

54. Dividing a fraction by a fraction3 3 5 6 7
What is ÷ ?
Start by writing as an improper fraction. 3 5 3 5 = 18 5 3 18 5 ÷ 6 7 = 18 5 × 7 6
Point out that cancelling before we multiply makes the calculation easier. If we did not cancel at this stage we would have to cancel later on. 1 = 21 5 = 1 5 4

55. Multiplying and dividing by fractions
Multiplying and dividing are inverse operations.
When we multiply by a fraction we:
multiply by the numerator
and
divide by the denominator
When we divide by a fraction we:
divide by the numerator
and
multiply by the denominator

56. Dividing fractions
Practise dividing fractions mentally. Ask pupils for the solution to the calculation, and then check it by revealing the answer.