Category: Minimally Different Exercises

Since I first started teaching, I’ve always been somewhat unsure of how to approach the topic of dividing fractions by fractions. While the procedure is relatively easy to perform, giving students the opportunity to see why and how it works always proved much more challenging. This week, I taught this in my favourite way yet, which I will share in this post.

 

Attempt 1

The very first time I taught this, I was keen to avoid sharing any form of ‘keep, flip, change’ mnemonic or ‘just flip the second fraction upside down and multiply’, because I wanted my students to understand what they were doing beyond following a procedure. As such, I did a bit of research (back before I’d joined Twitter!) and decided to present students to the topic by using models such as these. After all, I reasoned, it is always best to start with a pictorial representation.

One of the biggest issues is that I was asking far too much of the students I was teaching: I had incorrectly believed that the simple presentation of a pictorial method would enable the students to see the underlying mathematical structures, but without excellent teacher questioning and exposition, this is unlikely to happen. I (at this point in my career) certainly wasn’t able to make effective use of this model. As such, I think the main outcome of this particular lesson was that students had simply acquired more procedural knowledge – but this procedure had even more opportunities for students to make errors than the standard ‘keep, flip, change’ algorithm.

Moreover, while it worked quite nicely for small fractions, and is excellent for exploring what is happening when fractions with the same denominator are divided, it became quickly convoluted when dealing with mixed numbers and less straightforward problems. Again, this isn’t necessarily an issue with the model itself, but I had failed to plan for an ‘exit strategy’ (that is – a way for students, over time, to be able to answer similar questions without the need for the model). I want to be clear that the reason this sequence of lessons was not as successful as I had hoped was more due to a lack of pedagogical knowledge on my part, rather than an inherent problems with the model, but the consequence of this experience was that I completely changed my approach when I next taught division of fractions.

Attempt 2

The second time I taught division of fractions by fractions, I had made up my mind that I wasn’t going to spend much time on why the procedure that I taught worked. My reasoning was that my first attempt had been unsuccessful, and I was currently exploring the idea that it might be better to teach the how before the why. The rationale was that if students become procedurally fluent with how to divide fractions, and through practice, are able to automatise the procedure, then this success would enable me to return to why it worked afterwards. I’d also been considering the idea that asking students to understand why is often more cognitively demanding than understanding how, so I made the decision to begin with how.

The lesson in question ran relatively successfully: I modelled through some examples, and then students did some practice in a variety of ways. Obviously, multiple students did want to know why this procedure was working, but I promised them that we would return to this later. In reality, though, this never really happened. After that initial lesson, students stopped asking why.

Moreover, when we returned to dividing fractions in subsequent lessons/homeworks/assessments, it became clear that the majority of my class were failing to retain this procedure. I think there are multiple reasons why this was happening, but I think that one important reason is that students spent most of the initial lesson on dividing fractions not thinking. They were answering plenty of questions successfully, but then again, they only needed to think ‘well, I flip this upside down’ – the multiplication of the numerators and the denominators was something that students in that class were already confident with. As such, once they’d completed the first few questions, the rest of the lesson was simply more mindless practice, as although the numbers became more difficult to manipulate, they didn’t have to think about the important key features of dividing fractions by fractions. As before, this attempt had not been successful as it needed to be.

 

Attempt 3

Next time round, I decided to approach the idea from ‘pattern spotting’ and allowing students to hopefully deduce the need to ‘flip the second fraction upside down’. We started with this question:

All students were quite happy with the idea that this would be 2, once it had been shown diagrammatically. We moved onto related questions:

Again, this all seemed to be fairly straightforward. We then discussed this question: ‘if 4 divided by ½ is 8, what other ways can you state the relationship between 4 and 8?’ From this, I tried to draw out the necessary truth that dividing by ½ must be the same as multiplying by 2, and that dividing by 2/3 must be the same as multiplying by 3, and so on. Some students (those of higher prior attainment) grasped this idea really quickly, which was excellent. For other students (particularly those of lower prior attainment), this explanation was evidently not complete. While they were confident with how they could divide whole numbers by unit fractions using simple diagrams, they really struggled with making the link to how this process related to questions such as:

They could replicate the changing of ¼ to 4, and the division sign to a multiplication sign, but I still think something was missing.

 

More Recently

In this last week, I taught division of fractions to my Year 7 class. For some context, my Year 7 class are completely mixed in terms of their prior attainment. In class, we had already looked at addition and subtraction of fractions, multiplication, simplifying and converting between mixed numbers and improper fraction. I am continually astonished at their mathematical capabilities and in awe of their primary school teachers, as the vast majority of this class are remarkably fluent with the manipulation of fractions.

I had a quick look through the Key Stage 2 National Curriculum (always my first point for teaching anything to Year 7 these days!), and noted that between Years 3 and 6 they would have covered:

• Recognising unit fractions as a division of a quantity
• Solving problems involving fractions to divide quantities, including non-unit fractions where the answer is a whole number
• Dividing proper fractions by whole numbers

This gave me a really useful starting point.

I began the lesson by giving them these three related statements:

I then gave them a sequence of multiplication statements, and asked them to write any related division statements. While the first couple of questions involved the use of only positive integers, I also included questions such as:

(This was particularly interesting, as one of my highest attaining students then asked: ‘miss, I’ve written 4 divided by 1/2 = 8, but wouldn’t 4 divided by 1/2 be 2?’). I also included questions such as:

 

This was a nice introduction – the last couple, in particular, caused all sorts of questions and to be asked and conjectures to be made. After this, I brought the class back together, and we looked at questions such as:

After multiplying and simplifying, they noticed that when you have:

Or questions in that form, the answer must be 1. This was a lot of fun: there was a noticeable ‘buzz’ in the room, and at this point, I introduced them to the word reciprocal. I would never have bothered sharing this with a Year 7 class previously, as I mistakenly believed it would have overcomplicated things, and after all, they would go on to meet this terminology later. This was a mistake – it is so important that our students have the opportunity to express things in precise mathematical language as early as possible, and they were so excited about this new word to add to their collection. After this, we did some practice of finding reciprocals.

I brought the class back together again, and explained we were going to look at how this all fitted together. Returning to the earlier question of:

I challenged them to write a related division statement. After a few moments of thinking, pupils shared the following ideas:

At this stage, there were quite a few murmurs around the room: some students were beginning to spot the relationship between division by 4/3 and multiplication by ¾, and so on. Obviously, some students were not yet there, but through some discussion and questioning, we drew out the implication that division by a fraction must necessarily be the same as multiplying by its reciprocal. We went through some examples together, students did some more practice, and every single student left being able to confidently use the word reciprocal, with most being able to articulate the relationship between division and multiplication of a reciprocal.

 

Obviously, I don’t think this lesson was perfect! I’ve still got a lot of learning to do about how best to bring in different representations in order to support all students with their understanding, and I still don’t know how this approach will prove to be in the long-term. Nonetheless, it was one of my favourite lessons I’ve taught this year, and I think it was definitely the best I’ve ever taught this topic. I’m going to continue to reflect on how best to improve this, and I’m definitely going to continue to develop my use of precise mathematical language with all classes and all topics.