Getting to grips with fractions

Blog: Ian Davies , Director of Secondary | Posted: 11/03/16
The study of fractions is a key part of mathematics.  It leads onto further proportional reasoning and is in many ways a “gateway” topic allowing access to understanding huge parts of the curriculum.

So let’s start consideration of fractions with a question – what fraction do you see here?

My guess is that the majority of people would say “three-fifths”.  When looking at fractions represented like this we tend to think (or perhaps we have been taught to think?) “left-to-right” and shaded rather than unshaded.  But this could equally be a representation of two-fifths or even other fractions.  It could be five-thirds, with each “equal part” representing a third: the coloured three are “one whole” with the unshaded parts being another two-thirds.  In a similar way, can you see how this could be interpreted as two and a half?

Which brings me to the key question of this blog: “Why is it so hard to understand fractions?”  Well, as the above question shows, fractions can be hard, or at least confusing! For example, even the definition of a fraction can be quite troublesome.  Many people would say “a fraction is part of a whole one” but that doesn’t help with fractions like “five-thirds”!

One solution is to say simply “a fraction is a number” – it’s just a question of how you arrive at that number. Others would (and do!) argue that “a fraction is an operator”, that “three-fifths” is meaningless other than as “three-fifths of” something. Is your head hurting yet?!

Representation, representation, representation

Just to add to the mix of potential confusion, the “area model” used to represent a fraction as above is but one of many possible forms.  Think about all of these ways of showing one half:

How obvious is it that they all show/mean the same thing?  How can we support children to understand this?

At Mathematics Mastery, we believe that looking at and manipulating different representations of the same concept – concrete alongside pictorial and abstract – is key to deepening understanding.  If when you first learn fractions you only look at them in one form it’s much more difficult to appreciate the equivalence of other forms later on.

Although there is a degree of challenge in understanding fractions and fraction notation, even young children have a keen understanding of the concept.  For example, I cottoned on to my elder brother always having the “bigger half” pretty early on!  “Equal parts” and “fair shares” of concrete objects (chocolate, sweets, toys…) are the first step to this understanding.

Language, language, language

We also need to think really carefully about how we describe fractions.  I know that in the past I’ve tended to shift to short-hand too quickly, moving from saying “equal parts” to just saying “parts” when sometimes that equality is not fully appreciated.  I’ve always been more precise using “numerator” and “denominator” rather than “top number” and “bottom number” as this latter naming really doesn’t help that a fraction is itself a single number and some magic combination of two.  Strangely it was only very recently I found that the bar separating the numerator and denominator also has a name – it’s called a vinculum.

It’s really interesting the way that visiting Shanghai teachers write fractions:

• They start with the vinculum (nice and horizontal, not a diagonal solidus!). This gives a signal to the learners that we’re dealing with a fraction here, something to do with splitting wholes into equal parts.
• Next they write the denominator – 5, say. This tells us we’re dealing with fifths; we’re splitting wholes into five equal parts.  The learners now have a sense of what type of fraction they’re dealing with
• Finally, they write the numerator. If this was a 4, we’ve got 4 equal parts out of 5 – oh, that’s nearly a whole one.  If the numerator was a 7, we’ve got 7 equal parts out of 5 – oh, that’s more a whole one.  The learners now have a sense of the size of the fraction they’re dealing with.

If I compare this to how I’ve traditionally written, say, four-fifths:

• I’d have started with the “4”. The learners have no idea what might come next – cm? Another digit?  A string of digits?  A decimal point?
• Then I’d have drawn the vinculum (even though I didn’t know what it was called). What for?  Am I underlining for emphasis, or what?
• Finally, comes the 5…oh, it’s a fraction! Now let’s try and get a picture of what that might mean…

As you can probably tell, I’m a big fan of the Shanghai way of writing fractions and we’ve incorporated this into our Mathematics Mastery fraction materials, which we’ve extensively updated this year after learning, research and feedback.  An example of a Year 7 lesson can be found in our shared resources section.

A little fractions question

Here’s a little question that I like to ask teachers and students alike:

It’s a fairly easy question, and what I’m really after isn’t the answer of course, but how they find it and what alternative methods or reasons they can come up with.

Interestingly, most people look for a common denominator.  Some will convert to decimals or percentages.  Some will say that $$\frac{4}{5}$$ is $$\frac{1}{5}$$ smaller than 1 whereas $$\frac{5}{6}$$ is only $$\frac{1}{6}$$ smaller than 1:

$$\frac{1}{6}$$ is smaller than $$\frac{1}{5}$$, so $$\frac{5}{6}$$ is larger as it’s closer to 1.  I like that way – but what I like more is the flexibility of thinking and appreciation of different approaches that the study of fractions – indeed mathematics – is all about.

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