Complements #5 : Fractions

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:

rat

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:

fb1

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.

fs2

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:

fs3

Nothing lines up nicely. The answer cannot be in terms of halves, nor can it be in terms of thirds.

fs4

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.

However:

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”

Job done.

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

When it would be far more intuitive and understandable to a child if you started with these:
frf
frrr

and showed solutions in this way:

frr

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?

simplify-before-multiply-example1

Well, that’s linked to commutative properties of multiplication again.

fcc

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.

Regardez:

1/3 x 3 = 1

1/3 x 3/1 = 1

7 x 1/7 = 1

7/1 x 1/7 = 1

etc etc

So… here we go.

recip

Thank you, and good night *tosses microphone into crowd, leaves*.

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7 thoughts on “Complements #5 : Fractions

  1. I like it. Finally somebody bites the bullet on fractions. I have done quite a few posts on fractions, a topic which is mangled so often in schools. ‘Regarding the adding of fractions I would suggest that the question to ask is “What number can you find that you can have halves of and thirds of?”, and any valid answer will do. Then do it, and find what fraction of the chosen number you end up with. This works for multiplying as well (not exactly the same !!!!). There seems to be a reluctance to apply fractions to quantities, I have no idea why this is.

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  3. What a great post. I’m a trainee and have just completed FDP with a low ability year 8 group and I have to confess that in the absence of any guidance as to how to do it from anybody in the department (don’t get me started) I quite frankly made a hash of it leaving them with the “grid method” for adding and subtracting fractions of different denominators which I resorted to after getting nowhere in 3 hours of trying other approaches and “flip then multiply” for dividing.
    I will be at a different school for the rest of my training as part of the course and wonder of wonders, I have the chance to do it again with a not dissimilar group. I was planning on taking two or three lessons to make sure that they were really confident with equivalent fractions and converting between improper fractions and mixed numbers (is slices of pizza OK for visualising this? I cant see why not) before entering the world of adding and subtracting using some of the guidance in this post.
    If anybody has any tips they could offer with regards what they would do if the fraction strips don’t resonate with the students or if they wobble at the step from visualising it using fraction strips to being able to calculate a common denominator (1/5 + 1/7) when the denominator is larger than what you have on the fraction strips…..

      • If I’m understanding you correctly, that means that I’ll take them as far as being able to do it with fractions strips and leave it there at this stage. Is that what you mean?
        I have a bottom set and middle set y7 so that might be appropriate for this year but I thought I would check.
        (This new school is very different to the last school so there will be masses of advice and guidance – I just thought I would check).

  4. Pingback: Fractions, Fractions, Fractions | Solve My Maths

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