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Equivalent fractions: chains

Although the fractions are different, the amount they represent is the same. We call these equivalent fractions.
MICHAEL ANDERSON: So let’s take a look at some equivalent fractions.
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.
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.
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.
MICHAEL ANDERSON: I see, so slightly smaller amounts each time, but you’ve eaten the same amount of chocolate.
PAULA KELLY: Yeah, smaller pieces, but same amount.
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.
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.
MICHAEL ANDERSON: So in this diagram, we’ve got 15ths. How have we got to that?
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.
MICHAEL ANDERSON: OK, and because there’s 5 pieces that have each been split into 3, three lots of 5 are 15.
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.
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.
MICHAEL ANDERSON: Right, so I can put 6 up here, because we’ve eaten the equivalent of 6/15.
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.
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.
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.
MICHAEL ANDERSON: I see, yeah, just the same again.
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.
Should give me, if each slice is half the size– should give me 8. Shall we see?
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.
PAULA KELLY: Fantastic.
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.
So by that pattern, I think that this one should give us 10 over 25, because 8 plus 2 would give us 10.
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.
MICHAEL ANDERSON: Hopefully my prediction will be correct ..
PAULA KELLY: We’ll hope for 10. OK, I’ll let you do the counting.
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.
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.
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.
PAULA KELLY: So each one of those cut in half. We’ll have twice as many here. Should be 12.
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.
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.
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?
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.
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.
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.
PAULA KELLY: Absolutely.
Numbers can be written in different ways, but represent the same value. In primary school students learn number bonds to ten. In other words, pairs of numbers which sum to ten: 1 plus 9, 2 plus 8, 3 plus 7 etc. The combination of two numbers, when added together, equal 10. This is just one illustration of how the number ten can be represented in many different ways.
Starting with the video above, over the next three steps we see how the same amount can be represented by different fractions. Although the fractions are different, the amount they represent is the same. We call these equivalent fractions.
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 \(\frac{2}{10}\).
In this video we consider how to create a string of equivalent fractions.
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Maths Subject Knowledge: Fractions, Decimals, and Percentages

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