Ugh. fractions. Is it me, or are these really a lot more difficult than people assume?? I feel like they’re taught way too early, and that a lot of the complications that arise from them are partly due to the necessary approaches to communicate the in’s and out’s of fractional operations at such an early age.
Aaaanyway let’s begin…
One of the most likely things to completely throw students (and teacher explanations) when it comes to fractions is this: they have multiple meanings in a variety of contexts.
Students are usually first introduced to them as part-to-whole comparisons. Metaphors such as slices of pizza are often used simply because fractions are a fairly abstract concept to get your head around, and when you’re six or seven years old, or any other age for that matter, abstract leads to metaphor.
Later on fractions become divisional sums. No longer is there two slices “out of” three making up a whole pizza. Instead, using exactly the same notation we’re now taking the integer 2 and dividing it by 3 to get 0.6 recurring.
*Hint, show students that the divide symbol is a secret fraction. The dots are numerator and denominator. This isn’t “a thing”, the symbol was originally used for subtraction, but it’s a nice spooky coincidence.
Then finally just to really kill them off, we change our minds again and they become part of the ratio party:
This post will be focused primarily on number operations using fractions.
Too often students are shown a simplified way to add / subtract fractions in schools without ever exploring why it works, where it comes from, or what is actually happening mathematically.
To put this into context, imagine you had never added two fractions together. I tell you that you simply turn the numbers upside down whilst patting your head. You look at me like I’m mad, but do it obligingly. When you get your (weird) answer, I jump for joy and tell you WELL DONE, you’re AMAZING!!! Then give you twenty more questions. Each time I see those lovely sixes becoming nines, and fives becoming undefined symbols I pat you on the head and weep with joy. The method is eventually embedded. The student has no idea what the hell is going on, but they get ‘happy happy joy joy’ response when they do it, so they keep doing it.
That’s essentially what’s going on for a lot of students.
So let’s look at what’s actually being done mathematically.
First, consider why we can’t work with two fractions with different denominators without altering them:
Here are a few representations of 1 as different fractions:
Otherwise known as fraction strips / bars. Easy to visualise, easy to see that two halves make 1, eight eighths make 1 etc. I think it’s important to use the word ‘one’ instead of ‘a whole’. We never say ‘a whole’ anywhere else in mathematics, so why bother here? Just adding to the confusion (pun intended).
What’s also nice about the above picture, (taken from this interactive resource) is that you can visualise the relationships between sizes of fractions with different denominators.
So we see that 1/4 + 1/4 is equal to 2/4, which nicely lines up exactly with 1/2, because, well, it’s equal to 1/2. But notice that the 1/3’s don’t line up quite so nicely. At this point I should mention that I think it’s *really* important to talk about equivalent fractions in some depth. Metaphors are great here. “would you be any more full of chocolate if I cut my bar into six equal pieces and gave you three, or if I cut it into two pieces and gave you one piece?” or “which would you prefer to receive? Half of all of my money, or just three sixths of it?” etc. I fear a lot of students don’t really see that equivalent fractions are ‘worth’ the same amount. </digression>
So if we take 1/2 + 1/3 it’s not quite so straight forward:
Nothing lines up nicely. The answer cannot be in terms of halves, nor can it be in terms of thirds.
Lucky for us though, the handy sixths line up nicely both with halves and thirds. A half is equal to 3/6, and a third is equal to 2/6, so now the sum is achievable without switching to decimals.
But the point is that before we converted to a different denominator (specifically, one whose denominator is a common multiple of both our originals), this sum was going nowhere.
If you object to visual representations (you MONSTER!), then you can show students the logic behind the need to convert denominators like this:
1/2 + 1/3 = 1÷2 + 1÷3
The rules that dictate the order of operations do not allow us to do the addition first, so we cannot ‘add the tops’ or any such like.
3/6 + 2/6 = 3÷6 + 2÷6 = (3+2)÷6
Ooh, see now we’ve trumped those division signs with brackets. In your face, divisor.
I die a little every time I hear a teacher say “because it just does”.
Subtraction is no different to addition, so let’s move onto multiplication.
“Just times the tops and times the bottoms”
Aaah you know me better than that by now. Although even the trusty BBC Bitesize website tells you to ‘just’ do that. (Fix it in the new version @tessmaths !)
How often have you seen teachers jump straight in with this:
1/3 x 1/4
and showed solutions in this way:
These are taken from this webpage, which I like a lot. I rarely see teachers even have a conversation about what 1/3 x 1/4 means. Let’s put the record straight. It means a third of a quarter, or a quarter of a third (multiplication is commutative don’t you know!). Without having that discussion, students are less likely to have any intuition about their answer ‘looking’ right.
This Khan Academy video does a pretty good job of showing all of what I just said:
Before we move onto division, let’s just look at one more case:
Ever wondered why this works?
Well, that’s linked to commutative properties of multiplication again.
Last one… division.
“Just flip the second fraction then multiply”
Just get out.
Here’s the fact you need to know : any number multiplied by its reciprocal is equal to one.
1/3 x 3 = 1
1/3 x 3/1 = 1
7 x 1/7 = 1
7/1 x 1/7 = 1
So… here we go.
Thank you, and good night *tosses microphone into crowd, leaves*.