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1.7

## National STEM Learning Centre

Skip to 0 minutes and 6 seconds MICHAEL ANDERSON: So let’s take a look at some equivalent fractions.

Skip to 0 minutes and 10 seconds PAULA KELLY: So if I had, here– we’ll call it a chocolate bar– split into five equal slices– if I was to eat 2 out of these 5 slices– and I’ll shade it in, just to show you how much of the whole bar I’m going to eat.

Skip to 0 minutes and 26 seconds MICHAEL ANDERSON: That’s pretty restrained, just eating the 2 out of the 5 chocolate pieces. So as a fraction, I can write that as 2 over 5. So you’ve eaten 2 parts out of 5 parts.

Skip to 0 minutes and 38 seconds PAULA KELLY: Ideally, OK. So if I, for example, thought these fifths were too large, I could cut each slice in half, do the same for all of them, So rather than five pieces, I’d now have 10. Same again– I want to eat the equivalent amount. So rather than 2 out of my 5 slices, I would have each one of these halved. I’d actually have 4 out of my 10 slices.

Skip to 1 minute and 6 seconds MICHAEL ANDERSON: I see, so slightly smaller amounts each time, but you’ve eaten the same amount of chocolate.

Skip to 1 minute and 10 seconds PAULA KELLY: Yeah, smaller pieces, but same amount.

Skip to 1 minute and 12 seconds MICHAEL ANDERSON: And here we’ve got 1, 2, 3, 4 out of 10 equal size pieces. So that’s obviously going to be 4 over 10 as a fraction.

Skip to 1 minute and 20 seconds PAULA KELLY: Yeah. So I’ve eaten the same amount, just may appear as more slices. Our slices are half the size, so you have twice as many in total.

Skip to 1 minute and 29 seconds MICHAEL ANDERSON: So in this diagram, we’ve got 15ths. How have we got to that?

Skip to 1 minute and 34 seconds PAULA KELLY: So if we come back to our original diagram, and rather than cut each piece into 2, we’re going to cut each piece into 3 this time.

Skip to 1 minute and 44 seconds MICHAEL ANDERSON: OK, and because there’s 5 pieces that have each been split into 3, three lots of 5 are 15.

Skip to 1 minute and 50 seconds PAULA KELLY: Fantastic, yeah. So we would cut all of these. Our chocolate bar is still the same size. I now have more pieces, but more smaller pieces.

Skip to 1 minute and 59 seconds PAULA KELLY: So I have 15 slices. Now, if I just keep this area the same and shade this in, I’ve actually got 1, 2, 3, 4, 5, 6 pieces.

Skip to 2 minutes and 14 seconds MICHAEL ANDERSON: Right, so I can put 6 up here, because we’ve eaten the equivalent of 6/15.

Skip to 2 minutes and 19 seconds PAULA KELLY: Yeah, perfect. Same amount of chocolate, but slightly small pieces. Each one of these has been split into three so I have three times as many for each one of these.

Skip to 2 minutes and 28 seconds MICHAEL ANDERSON: So what we seem to have done is make three fractions that are actually exactly the same amount. We’ve got 2/5, 4 over 10, and then 6/15. But those three fractions are actually equivalent.

Skip to 2 minutes and 41 seconds PAULA KELLY: Yeah, just the same. They look different, but they are equivalent, and we can see that from my diagram. So we could then come back to our original diagram, and this time cut each slice into four. So I would have 4 lots of 5. I’d have 20 slices altogether. Another way we could do this, rather than having our 10 pieces– if I wanted to make this into 20 slices, I could cut each one of these in half.

Skip to 3 minutes and 9 seconds MICHAEL ANDERSON: I see, yeah, just the same again.

Skip to 3 minutes and 11 seconds PAULA KELLY: Yes, same again. So each one would be half the size. I would have twice as many. So rather than having 4/10, again, I keep my area the same.

Skip to 3 minutes and 27 seconds Should give me, if each slice is half the size– should give me 8. Shall we see?

Skip to 3 minutes and 31 seconds MICHAEL ANDERSON: I’ll count just to see. So we’ve got 1, 2, 3, 4, 5, 6, 7, 8. So it seems to work.

Skip to 3 minutes and 41 seconds MICHAEL ANDERSON: So I think I’ve noticed a bit of a pattern here as well. So if we look at the denominators, we’ve got 5, 10, 15, 20, and then we’re going to look at 25 and 30. They seem to be going up by the same amount– by 5 each time. And looking at those numerators, we started off with 2/5, so 2 is on top. And then we went to 4, 6, 8. So they seem to be going up by 2 each time. So they’re going up consistently just the same way that the denominators are, but they’re going up by 2, and they’re going up by 5 each time.

Skip to 4 minutes and 14 seconds So by that pattern, I think that this one should give us 10 over 25, because 8 plus 2 would give us 10.

Skip to 4 minutes and 23 seconds PAULA KELLY: Yeah. Another way to look at it is how many of our 5’s go into 25? That’s been multiplied by 5. Let’s keep our fractions equivalent. We’ll do the same as our 2, multiply it by 5 to give us 10. So same again. We’ll keep our area exactly the same. We’ll Shade say this pie in.

Skip to 4 minutes and 40 seconds MICHAEL ANDERSON: Hopefully my prediction will be correct ..

Skip to 4 minutes and 43 seconds PAULA KELLY: We’ll hope for 10. OK, I’ll let you do the counting.

Skip to 4 minutes and 47 seconds MICHAEL ANDERSON: All right, so I’m hoping for 10. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So it seems to work.

Skip to 4 minutes and 56 seconds PAULA KELLY: Fantastic, OK. So finally, we have our 30ths. So a couple of ways we could see this. We could see 30 as being double 15.

Skip to 5 minutes and 5 seconds MICHAEL ANDERSON: I see. So for each of our 15 pieces here, we’re going to split each of them in half, because that will give us 30 pieces. Yeah, I can see that one.

Skip to 5 minutes and 15 seconds PAULA KELLY: So each one of those cut in half. We’ll have twice as many here. Should be 12.

Skip to 5 minutes and 19 seconds MICHAEL ANDERSON: I see, yep.

Skip to 5 minutes and 21 seconds PAULA KELLY: Another way to have a look at it is 10 is also a factor of 30. So we could say we’re going to multiply this by 3, or say multiply this by 3.

Skip to 5 minutes and 30 seconds MICHAEL ANDERSON: So what we can do to get from this diagram to this one is split each of these individual pieces into thirds, and then we’ll have 10 lots of 3 to give us 30 pieces here. And those are quite small pieces of chocolate that you’ve eaten.

Skip to 5 minutes and 44 seconds PAULA KELLY: Yes, quite a light snack. So we’ll go for the same area again, OK? And how many are we hoping for this time?

Skip to 5 minutes and 51 seconds MICHAEL ANDERSON: Well, if the pattern holds true, we seem to be going up by 2 each time. Every time the denominator goes up by 5, the numerator’s going up by 2. So I’m going to guess [? for ?] there’s 12, but let’s count them just to check. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. OK, so what we seem to have done here is created a string of equivalent fractions. So the fractions are all the same, but we used, each time, different numbers to represent the same proportion of the whole.

Skip to 6 minutes and 21 seconds PAULA KELLY: And that really helps us, because sometimes a common misconception is just keep doubling and doubling and so on. If we did that method, we’d miss out some very important ones in the middle. So if we think about coming back to our original fraction and thinking of multiplying by 2, by 3, by 6 and so on, we can see a whole range of our equivalent fractions.

Skip to 6 minutes and 41 seconds MICHAEL ANDERSON: And just by looking at the sequences, I suppose we can make more and more and grow the chain as much as we like, just by adding 2 each time and adding 5 each time, to the numerator and the denominator.

Being able to represent the same amount using many different fractions is a very useful skill which is frequently used when manipulating fractions. For example, how $$\frac{1}{5}$$ is the same as $$\frac{2}{10}$$.