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Students first learn how to divide with fractions in year 8. In the past I’ve struggled with teaching how to visualise dividing with fractions in a way that students gain a conceptual understanding, especially when the written method is a relatively simple procedure.

Dividing with integers is much easier to understand on a conceptual level, for instance 12 ÷ 4 can be visualised as 12 split into four equal parts.

However, questions like 2/3 ÷ 1/2 or, how many halves go into two thirds, is much more difficult for students to visualise. Yet, when students are studying GCSE maths, we expect them to be confident and competent with both kinds of division.

To help students visualise what is happening when we divide one fraction by another I do two things. First I keep the fractions simple and second I use proportions of a circle because it is much easier for students to see one circle as a whole.

To keep the fractions simple I use those which are easy to visualise, such as halves, thirds and quarters or those where the denominators have more factors, such as eighths and twelfths. I would avoid using fifths as 5 is a prime, so has only two factors.

I set the scene by using a circle to represent the whole. I then split into a number of sectors, relevant to the question in hand. For the example below, 3/4 ÷ 2, I’ve split a whole circle into eighths and display the proportion of the circle that corresponds to the numerator. Students can see 3/8s make up one half of the three quarter circle.

For the third question I find it is helpful to phrase 3/4 ÷ 1/8 as ‘how many eighths go into three quarters?’ It’s much easier to count the number of eighths within 3/4 than it is to try to calculate it using arithmetic.

3/4÷ 2 = 3/8

3/4 ÷ 1/4 = 3

3/4 ÷ 1/8 = 6

For the next series of questions I ask the students to sketch a circle split into 6 equal sectors on their mini-whiteboard and to wipe off one third. Now we have two thirds of a circle split into four equal sectors.

I ask the class to attempt 2/3 ÷ 4 on their mini whiteboard. Most students present 1/6 as their answer. Some present 1/4 with the argument the shape has been split into four equal sectors. We discuss that while their argument has some validity we need to look at 2/3 of the whole circle and the whole circle is split into sixths. We try the next question.

2/3 ÷ 1/2 is attempted successfully by the vast majority of students. Those who do have some difficulty are helped by their friend.

2/3 ÷ 4 = 1/6

2/3 ÷ 1/3 = 2

2/3 ÷ 1/2 = 4/3

Most of the class found 2/3 ÷ 1/2 = 4/3 difficult to visualise. You can see within the 2/3 there is a complete half and additional 1/3 of the second half. A common misconception here was to calculate 1/2 + 1/3 which is 5/6.

The idea behind this approach was to help students visualise dividing with fractions. The written method follows on from this as a natural progression of their understanding. To link the two approaches together we work through the previous questions using the written method so students can see how the two methods arrive at the same answer, thus consolidating each other.

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## Heather says:

In terms of the questioning, writing quotative division story problems with whole numbers and then with fractions will help students have a context within which to think about the division and in the case of 2/3 divided by 1/2 the remainder and what it means in a real context.

## Jonathan says:

Hi Heather

Thanks for the comment. I like the idea of story problems and putting the skill in context for real world application.

There is so much to explore with division of fractions and mixed numbers!

Jonathan