In what follows I aim to set down a broad overview of some of my current thinking about fractions. The ideas expressed here are just that: ideas and wonderings that are purely personal and not intended as judgements or pronouncements about other people’s views or practice. I don’t pretend to speak with any authority and the thoughts I offer are not based on any empirical evidence beyond my own direct experience and reflections thereon. They are an account of my ongoing attempts to make some personal sense of what fractions are and how they behave as mathematical entities. In particular I have sought to develop a relational, rather than procedural, understanding of working with fractions. The conclusions I’ve reached – provisional and open to refinement and revision as they are – will be my starting point for the current discussion:
fractions are simply numbers and behave like numbers, that is, they are governed by the same laws of arithmetic that enable us to work with all the other numbers we encounter both inside and outside of mathematics lessons.
My thinking and findings have caused me to question the usefulness of much of the common discourse in enabling learners and teachers alike to come to see fractions in this way. Some aspects of this discourse have, it seems to me, shaped attitudes and behaviour – my own included – in such a way as to become counterproductive, making fractions seem mysterious and forbidding. I will argue here that any differences between fractions and integers – in particular, on how we understand and operate upon them – are essentially superficial.
My current understanding of what fractions are is as follows:
A fraction is a number that results from dividing one integer by another.
This working definition was all I needed to enable me to extend my understanding and ability to calculate using integers and apply them to fractions .
The personal definition offered above is by no means a new one. Moreover, it is widely known that the set of rational numbers, encompassing fractions alongside integers, was discovered (if that’s the right word) when it was necessary to divide one integer by another where the divisor was not a factor of the dividend. However, I think people – including many teachers – can often lose sight of this fact. I know that I had. I can now see that I don’t need to get too caught up in the language of numerators and denominators, parts and wholes or in pictorial representations that often rely heavily for their interpretation on geometric concepts. I’m not saying that such representations aren’t meaningful, but I think we can reach a perfectly good conceptual and practical understanding of fractions without them.
I’ve come to believe that an effective way to understand fractions as numbers is to recognise and treat them as numbers from the outset. Supplementing this understanding later by adding further representations to our conceptual schema is, of course, of value. However, from my perspective, many of the commonly used pictorial representations can interfere with the business of constructing a conceptual framework that allows fractions to be understood alongside existing knowledge about other numbers. (I should say at this point that I have found certain pictorial representations involving number lines to be very helpful in scaffolding the learning and teaching of fractions. These are featured in my next post “More Thoughts (and Some Pictures) on Working With Fractions (and Other Numbers)”.)
By my earlier definition, we create fractions when we divide any one integer by any other (except, of course, dividing by zero). For simplicity I shall restrict this discussion to positive integers. Hence the following are, by definition, true:
1 ÷ 2 = 1/2
2 ÷ 3 = 2/3.
The symbols and notation we use are the result of culture and convention. These, for example, are equally true:
2 ÷ 4 = 2/4
8 ÷ 2 = 8/2
Let’s resist the temptation to see the last two examples as somehow incomplete or otherwise imperfect and just leave them as they are for now: they are certainly mathematically valid statements.
With these ideas in mind we shall now perform some operations on integers. To make it interesting – and, more importantly, informative – let’s express the integers 2 and 4 in fractional (or rational) form, that is, as one integer divided by another.
For example, 4 + 2 can be written as
20/5 + 10/5
(20 ÷ 5) + (10 ÷ 5)
Recalling that division by 5 is the same as multiplication by one-fifth and invoking the distributive law, this is equivalent to
(20 + 10) ÷ 5
= 30 ÷ 5
So, adding two rational numbers gives a fraction which, in this case, is equivalent to the integer 6.
Now let’s add the fractions 4/5 and 2/5 or, alternatively
(4 ÷ 5) + (2 ÷ 5)
= (4 + 2) ÷ 5
= 6 ÷ 5
The only difference here is that the sum of the two fractions can only be written in rational form, i.e. as a fraction but not as an integer.
A similar approach to subtraction of 2 from 4 and of 2/5 from 4/5 produces answers of 10/5 (or 2) and 2/5 : again two rational numbers, one of which can be written as an integer and one which cannot.
What if we had chosen to write 4 + 2 as
(8 ÷ 2) + (10 ÷ 5)?
One consequence would be that we no longer have a common divisor and so can’t use the distributive law to find an answer. However, we could apply some proportional reasoning to reframe both fractional representations, for example, as follows:
(40 ÷ 10) + (20 ÷ 10)
= (40 + 20) ÷ 10
= 60 ÷ 10
= 60/10 , or 6.
By analogy, we can now add the fractions 1/2 and 2/5 as
(1 ÷ 2) + (2 ÷ 5)
= (5 ÷ 10) + (4 ÷ 10)
= (5 + 4) ÷ 10
= 9 ÷ 10
The corresponding subtractions this time would give answers of 20/10 (or 2) and 1/10 . Again, two fractions, one of which can be written as an integer and one which cannot.
The point I’m trying to make here is that addition and subtraction with fractions is structurally the same as with integers that happen to have been written as quotients of other integer pairs. Importantly, when working with fractions we are, in essence, working with integers. There is no particular need to appeal to any specialised language or other representational forms, as long as we understand how to operate with integers.
One source of confusion is the idea that a fraction, say 2/3, needs to be defined in relation to a quantity: we might ask “ 2/3 of what?”. But we’re less likely to wonder, on encountering the number 6, “6 of what?” By my earlier definition, 2/3 is the number that results when 2 is divided by 3. There is no ambiguity about its value. 2/3 means 2/3 of one, in the same way that 6 means 6 ones. In other words, numbers are routinely understood as multipliers of units, the default unit being the number 1.
Recall that multiplication is commutative. Hence 2/3 of 6 means the same thing as 6 of 2/3 or, more customarily, 6 × 2/3. Both, of course give an answer of 4. This demonstrates the two main ways to conceptualise multiplication: repeatedly adding six of the number 2/3 together, or scaling 2/3 up by a factor of 6. (Or scaling 6 down by a factor of 2/3 , for that matter.)
Let’s multiply 2 and 4 together using the rational representations 6/3 and 8/2.
2 × 4
= 6/3 × 8/2
= (6 ÷ 3) × (8 ÷ 2)
= 6 × 8 ÷ 3 ÷ 2
= 6 × 8 ÷ (3 × 2)
= 6 × 8 ÷ 6
Here we’re exploiting both the associative law to re-order the chain of operations (again, since division is multiplication by reciprocal) and the fact that successive division by two integers is equivalent to a single division by their product. In my experience, most pupils arriving in secondary schools already know that to divide by four – or multiply by one quarter – you “halve then halve again”.
Let’s now multiply 2/3 and 4/5 together.
= 2/3 × 4/5
= (2 ÷ 3) × (4 ÷ 5)
= 2 × 4 ÷ 3 ÷ 5
= 2 × 4 ÷ (3 × 5)
= 8 ÷ 15
Notice that this relational approach explains the algorithmic rule about multiplying numerators and denominators together when multiplying fractions. In addition, the processes for multiplying both fractions and integers written in rational form are structurally identical. The only difference, as it was with addition and subtraction, is that the resulting value is not an integer in the former case. So, we can multiply together fractions by appealing to our understanding of multiplication and division of integers and without having to memorise blind algorithmic procedures or decode geometric area models. Many such models I have come across seem complicated and somewhat removed from my existing understanding of number.
Before looking at division of rational numbers, I want to highlight something I discovered that I wasn’t explicitly aware of before. Consider this calculation:
24 ÷ (6 ÷ 2)
= 24 ÷ 3
If we want to do this left-to-right it becomes
24 ÷ (6 ÷ 2)
= 24 ÷ 6 × 2
= 4 × 2
Bearing this observation in mind, we can divide 4 by 2 as follows
4 ÷ 2
= 8/2 ÷ 6/3
= (8 ÷ 2) ÷ (6 ÷ 3)
= 8 ÷ 2 ÷ 6 × 3
= 8 × 3 ÷ 2 ÷ 6
= 8 × 3 ÷ (2 × 6)
= 24/12 = 2
Let’s now divide 2/3 by 4/5 using the same approach.
2/3 ÷ 4/5
= (2 ÷ 3) ÷ (4 ÷ 5)
= 2 ÷ 3 ÷ 4 × 5
= 2 × 5 ÷ 3 ÷ 4
= 2 × 5 ÷ (3 × 4)
= 10 ÷ 12
Again we have uncovered a rationale for a widely used procedural algorithm, namely that of “flipping and multiplying” when dividing fractions. You will also have noticed that this answer is not in simplest form. Here is an interesting way to simplify it, which I discovered through taking this more relational approach to interpreting and operating on fractions.
= 10 ÷ 12
= 10 ÷ (2 x 6)
= 10 ÷ 2 ÷ 6
= 5 ÷ 6
To conclude, then, having looked at the four arithmetic operations upon rational numbers, these are just some of the ways in which taking a relational approach has helped me develop my understanding and techniques for working with fractions. As a teacher in a secondary school, I still have much work to do in order to devise a coherent approach to using my personal sense-making in ways that will help my pupils progress from understanding and being able to work fluently with integers to being able to do the same with fractions.
Once again, thank you for joining me in thinking about learning, teaching and doing mathematics. I hope it proves worth your while.