1. KS3 Mathematics N5 Using Fractions
The aim of this unit is to teach pupils to:
Use fraction notation; recognise and use the equivalence of fractions and decimals.
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
N5 Using Fractions

2. N5 Using fractions Contents A N5.1 Fractions of shapes A
N5.2 Equivalent fractions A
Improper Fractions and Mixed Numbers

3. Quarter or not?
The aim of this activity is to ensure that pupils are able to identify which shapes have one quarter shaded. For shapes drawn in three dimensions pupils will need to imagine the parts of the shape that are not visible in the two dimensional drawing.

4. 1 4 Fractions of shapes Remember, one quarter is written: one thing
divided into 4
four equal parts
Remind pupils of the meaning of fraction notation.

5. Fractions of shapes What fraction of this diagram is shaded?
Briefly discuss how much is shaded.
The diagram is divided into five equal parts called fifths. Two parts are shaded.
We call this fraction two fifths.
How much is unshaded? (3/5)
Three-fifths is the complement to 1 of two fifths.

6. 2 5 Fractions of shapes Two fifths is written as: two parts numerator
out of 5
five parts altogether
denominator
Ask pupils if they can remember the name given to the number at the top of a fraction and the number at the bottom of a fraction.
Reveal these key words on the board.

7. Fractions of shapes activity
Start by selecting a shape.
Activity 1 – Shading in a given fraction
Choose how many parts to divide the shape into.
Ask a volunteer to shade in a given fraction of the shape. Repeat for other examples.
For example, select the square and divide it into 12 equal parts. Ask one volunteer to shade in 1/4, another to shade in 2/3 and another to shade in 5/6.
Activity 2 – Finding the fraction shaded
Choose another shape and shade in a random number of parts. Make sure that the numerator is hidden by clicking on it.
Ask pupils to tell you how many parts have been shaded.
This activity can also be used to introduce equivalent fractions.
Extend this activity by hiding the division lines and asking pupils to estimate the fraction shaded.

8. N5.2 Equivalent fractions
Contents
N5 Using fractions A
N5.1 Fractions of shapes A
N5.2 Equivalent fractions A
N5.3 One number as a fraction of another A
N5.4 Fractions and decimals A
N5.5 Ordering fractions

9. Equivalent fractions Start by showing two bars.
Set one bar to show 1/8s and the other to show 1/4s.
Ask a volunteer to shade in half of the bar showing 1/8s.
How many 1/8s make 1/2? (4)
4/8 is the same as one half.
How many sixths make one half? (3)
3/6 is also the same as one half.
Show this amount on the bar.
Turn on another bar. Set it to show fifths.
Can any number of fifths make a half?
Now set the bar to show sevenths.
What about sevenths?
Establish that to make a fraction the same as a half the bar must be divided into an even number of equal pieces.
Ask pupils to state any other fractions they can think of equal to a half.
Turn on all four bars and ask pupils to show their suggestions on the board.
Repeat the exercise for 2/3. Start by setting two of the bars to thirds and sixths. Ask a volunteer to shade in two thirds of the bar showing thirds and continue as before.
Establish that to show a fraction equal to 2/3 the bar must be divided into a number of parts equal to a multiple of three.
Ask pupils what they notice about the numerator (the top number) for all fractions equal to 2/3.
The numerator is always a multiple of 2 (an even number) and the denominator is always a multiple of 3.
Repeat the exercise for ¾.
Establish that, in this case, the denominator is always a multiple of 4 and the numerator is always a multiple of 3.
Pupils should notice that the numerator and the denominator are multiplied by the same number. Ask a volunteer to justify this using the fraction bars.

10. What does equivalent mean?
Equivalent fractions
What does equivalent mean?
Ask pupils what equivalent means.
Equal to or the same as.
Many words that start with equ- have something to do with things being equal. Can you think of any?
Some examples include equilateral, equation, equidistant.

11. 3 6 18 = = 4 8 24 Equivalent fractions Look at this diagram: ×2 ×3 ×2
Ask pupils what proportion of the diagram is shaded (3/4).
Look what happens if we cut each quarter into two equal parts.
Click to divide.
We now have 1/8s.
Exactly the same amount is shaded, but you can see how we can call this amount 6/8?
What have we done by cutting each quarter into two equal parts?
Explain that we have multiplied the number of shaded sections by two (we had three shaded sections; now we have six) and we have multiplied the number of equal parts by two (we had four; now we have eight).
Click to reveal the arrows showing the numerator and the denominator being multiplied by 2.
We’ve multiplied the numerator by 2 and the denominator by 2.
The numbers have changed but exactly the same proportion of the circle has been shaded. 3/4 and 6/8 are equivalent fractions.
We could divide each of these eights into three equal parts. Look what happens. Click to reveal.
Now, how many equal parts are there altogether? (3 x 8, 24)
How many of those equal parts are shaded? (3 x 6, 18)
So we now have 18 out of 24 parts shaded.
Click to reveal this fraction.
Explain that we have multiplied both the numerator and the denominator by three. The numbers have changed but exactly the same proportion of the circle has been shaded.
What would we multiply the numerator and the denominator of 3/4 by to get 18/24? (6)
You can se that each quarter of our original diagram has been divided into six equal parts.
3/4, 6/8 and 18/24 are equivalent fractions. Can you think of any other fractions that are equal to ¾?
In how many different ways could we write ¾? (Infinitely many!) 3 6 18 = = 4 8 24 ×2 ×3

12. 2 6 24 = = 3 9 36 Equivalent fractions Look at this diagram: ×3 ×4 ×3
Explain this set of equivalent fractions as in the previous slide. 2 6 24 = = 3 9 36 ×3 ×4

13. 18 6 3 = = 30 10 5 Equivalent fractions Look at this diagram: ÷3 ÷2 ÷3
Ask pupils what proportion of the diagram is shaded. (18/30)
We could simplify this diagram by removing these horizontal lines.
Click to remove some of the horizontal divisions.
We now have ten equal parts.
Exactly the same amount is shaded, but you can see how we can call this amount 6/10. By removing those horizontal lines we have made every 3/30 into 1/10.
Explain that we have divided the number of shaded sections by 3 (we had 18 shaded sections; now we have 6) and we have divided the number of equal parts by 3 (we had 30; now we have 10).
Click to reveal the arrows showing the numerator and the denominator being divided by 3.
18/30 and 6/10 are equivalent fractions.
Tell pupils that by dividing the numerator and the denominator by the same number, we have simplified the fraction. It is simpler because the numbers are smaller.
Can we simplify this fraction any further?
Yes, 6 and 10 are both even numbers, so we could divide the numerator and the denominator by 2. Remember, if we divide the numerator and the denominator by the same number the numbers that make up the fraction change but the fraction itself has exactly the same value.
Click to show the numerator and the denominator being divided by 2.
6/10 is equivalent to 3/5.
We can see this in the diagram by grouping each 2 tenths into one fifth.
Click to reveal.
Can we simplify 3/5 any further?”
No, 3 and 5 have no common factors, there is no number which divides into both 3 and 5.”
We have expressed the fraction 18/30 in its lowest terms. This is also called cancelling the fraction down.
How could we have cancelled 18/30 to its simplest form in one step?
Establish that we could have divided the numerator and the denominator by 6.
We call 6 the highest common factor of 18 and 30. 18 6 3 = = 30 10 5 ÷3 ÷2

14. Equivalent fractions
Use this activity to generate patterns of equivalent fractions.
Stress that every fraction on the board is exactly the same fraction written in a different way. Establish that there are infinitely many ways to write the same fraction.
When all of the equivalent fractions have been revealed ask pupils how we could convert between any two given fractions on the board by multiplying and/or dividing the numerator and the denominator by the same number.
Link:
N8 Ratio and Proportion – Using scale factors.

15. Exercise A. Find the missing number
6) 14 ÷ ___ = 7
7) 35 ÷ ___ = 7
8) 48 ÷ ___ = 8
1) 6 x ___ = 24
2) 5 x ___ = 50
3) 3 x ___ = 15
4) 5 x ___ = 25

16. Exercise B. Find the missing number

17. Mixed numbers and improper fractions
When the numerator of a fraction is larger than the denominator it is called an improper fraction.
For example, 15 4
is an improper fraction.
We can write improper fractions as mixed numbers. 15 4
Talk through the diagrammatic representation of 15/4. Every four quarters are grouped into one whole, and there are three quarters left over.
can be shown as 15 4 3 4 =

18. Improper fractions to mixed numbers37 8
Convert to a mixed number. 37 8 = 8 + 5 5 8 1 + = = 4 5 8
Explain that to convert an improper fraction to a mixed number we can divide the numerator by the denominator to find the value of the whole number part. Any remainder is written as a fraction.
Relate fractions to division. 37/8 means 37 ÷ 8.
Talk through the division of 37 by 8. Discuss the meaning of the remainder in this context. We are dividing by 8 and so the 5 represents 5/8.
This number is the remainder. 37 8 = 4 5 8 4 5
37 ÷ 8 =
4 remainder 5
This is the number of times 8 divides into 37.

19. Mixed numbers to improper fractions2 7 3
Convert to a mixed number. 2 7 3 = 2 7 1 + = 7 + 2 = 23 7
We can explain this conversion by asking for the number of 1/7 in 3 whole ones. Explain that there are 21 sevenths in three wholes. Two more sevenths makes 23 sevenths altogether.
Explain that to convert a mixed number to an improper fraction in one step we multiply the whole number part by the denominator of the fractional part and add the numerator of the fractional part (refer to the example). This gives us the numerator of the improper fraction. The denominator of the improper fraction is the same as the fractional part of the mixed number.
… and add this number …
To do this in one step, 3 3 2 2 23
… to get the numerator. = 7 7 7
Multiply these numbers together …

20. Find the missing number
In this activity equivalent fractions, mixed numbers and improper fractions are generated.
Ask pupils to find the value of the missing number, explaining their reasoning.

21. Find the missing number
In this activity equivalent fractions, mixed numbers and improper fractions are generated.
Ask pupils to find the value of the missing number, explaining their reasoning.

22. Summary To find equivalent fractions,
You need to divide or multiply both numerator and denominator by the same number.
Are these pair of fractions equivalent ?
1 and 6
10 and 5
2 and 6