What symbol would go in the blanks: 4/3 [    ] 3/4

What symbol would go in the blanks: 4/11 [    ] 5/11

What symbol would go in the blanks: 3/17 [    ] 3/18

What symbol would go in the blanks: 6/11 [    ] 3/7

What symbol would go in the blanks: 7/12 [    ] 9/16

Comparing fractions (Part 1)

Key Stage 3

Maths

Conceptualising and comparing fractions

How many parts need to be shaded to make them equivalent

﻿Hello, I'm Mr. Coward and welcome to the first of a two-part lesson on comparing fractions. For today's lesson, all you'll need is a pen and paper or something to write on and with. If you could please take a moment to clear away any distractions, including turning off any notifications. And if you can, please try and find a quiet space to work where you won't be disturbed. Okay, when you're ready, let's begin. Okay, so time for the try this task. Now, what you need to do is you need to place these fractions here onto the number line, okay? In A, B, C, D and E. And you need to suggest possible values for X, Y and Z. Okay. So one is here by the way, and that's quite an important point in this task. So I'd just like to pause the video and have a go. Pause in three, two, one. Okay, welcome back. Now, here is where I placed everything. And I'll talk through as to why. So these two here, these are both bigger than one. So it's between these two that were deciding. The rest are smaller than one so the rest must be on this side. Now, how much bigger than one is it? Well, this is one quarter bigger than one, and this is two fifths bigger than one. So you've got to decide what is bigger, sorry, one quarter or two fifths? And we'll explore exactly how you can work that out why, but hopefully you had a sense that two fifths is bigger. And you can see that if I draw it out, here we've got something like this. And so that's into quarters roughly. And the same thing into fifths. That will be something like this. Okay, and you can hopefully see that two fifths is going to be bigger than one quarter. So that's why seven fifths goes there and five quarters goes there. Now, nine tenths, that's quite close to one. It's only one 10th away from one. And one 10th is not very big. This is a quarter away from one and a quarter is bigger than one 10th. So nine tenths must be closer to one than three quarters. And here we have a sixth and a sixth is not very big. It's half of a third so it's not particularly big. And you can see that if the number line was split into six equal pieces, like so, that would be around a sixth. Now these markings may have helped you decide positions for X and Y. I thought X was around a third. Roughly a third. Now, if you can see here and here they look like they look about the same distance to me. So I think that this is about two thirds. And then that's three thirds. And then that is, I would say, that's roughly the same distance from there to there as to there to there. So that's one and a third or four thirds. Now that's what I thought. And you may have thought something slightly different. You may have thought they're not quite the same distance and you may have thought of another fraction. So, you know, this is also maybe around three fifths, something like that. So it doesn't have to be exact, but this is just to give you a sense of size. Okay. I'll just like you to pause the video and have a read of this. Okay. Which student do you agree with? Why? Well, I agree with Zacky. Zacky is correct. Carla is not correct. The reason why Carla is not correct is she's not comparing the same whole. And when we're comparing fractions, we must compare the same whole. So she is not comparing the same whole. And how do I know that? How do I know she's not comparing the same whole? Well, because her bars are different lengths. Whereas Zacky's bars, they are the same length so he is comparing the same whole. So from Zacky's diagram, you can clearly see that three out of five equal parts is smaller than three out of four equal parts. Now this brings us to an important point. Okay, imagine you had something. Okay, just say it's a chocolate bar for easiness and you split that into four pieces. Okay, now imagine that same chocolate bar and you split that into five pieces. In which of the splits is the pieces going to be bigger? The one that's split into less pieces. So each piece is bigger. So if you could either have two pieces from the bar that was split into five or two pieces from the bar that was split into four, and you really liked chocolates, say, then you'd rather have the bigger pieces, the two bigger pieces. So two quarters is bigger than two fifths 'cause quarters are bigger than fifths. Okay, so with that idea in mind, try and work out what fractions of each shape is shaded. What do you notice? Pause the video and have a go. Pause in three, two, one. Okay, so some things that I noticed here. Well, this is a half, this is a third. Six equal pieces, five equal pieces, four equal pieces, three equal pieces, two equal pieces. And one of which is shared. As our denominator gets bigger, our fraction, our amount gets smaller. So the bigger the denominator, the smaller the fraction. So if you were to compare, I don't know, maybe three seventeenths, and you were to compare that to three elevenths, which one is bigger? Or which one has bigger pieces? This one has bigger pieces. And do they have the same amount of pieces? Yes. So the three pieces that are bigger is the greater amount, the larger number. So I'd like you to kind of use that idea and have a go at this. So pause the video and have a go. Pause in three, two, one. Okay, so here one third or one quarter? Well, which one has bigger pieces? 'Cause there's the same amount of pieces. And in this one each piece are bigger. The same amount of pieces, each piece is bigger. The same amount of pieces, each piece here is bigger. So that's the greater amount. What about this one? Well, we've got the same amount of pieces so we can compare the size of the pieces. And here, the size of the pieces are bigger. What about here? We've got the same number of pieces and here each piece is bigger. Okay, here, we've got the same number of pieces and each piece is bigger here. So really well done if you've got them correct. Okay, what about this? Which is bigger now? Well, they've got different numerators and different denominators. What about this? And this is not the same as before 'cause this time I'm asking you what fraction of the shape is not shaded. Okay, so what fraction is not shaded? So pause the video and have a go. Pause in three, two, one. Okay, so these, this is my answers. One half, two thirds, three quarters, four fifths, five sixths. What do you notice here? Well, each one is one piece away from the whole. Okay. Each one is one piece from the whole. Now, if this is one piece away from one and each piece is smaller than over here, well, it's going to be closer to one. So imagine it almost as like taking away. One take away a small piece is going to be a bigger number than one take away a bigger piece. So going back to that last question, which is bigger? Can you explain? Well, this one, they're both one piece away from the whole. Oops, sorry. So they're both one piece away from the whole but for six sevenths, sevenths are greater than sixths. Sorry, sevenths are not greater. Sevenths are smaller. Oops. So sevenths are smaller than sixths. So if we're one piece away and with the sevenths, each piece is smaller, then this is closer to the whole. So one piece away from the whole but for six sevenths, sevenths are smaller than sixths, so six sevenths is closer to one. Because it's closer to one, it will be bigger therefore. Okay, because they're both less than one, well the six sevenths is the closer one to one. So that's going to be the larger one. Okay, so what about this? Can you find ways using diagrams or manipulatives to show which fraction is greater? Pause the video and have a go. Pause in three, two, one. Okay, welcome back. Now we could have tried a few different ways. Well, we can use the one piece away from the whole here. So that's one piece away from the hole and that's one piece away from the whole, but here are smaller pieces, so that's closer to the whole. Kind of like we did, and you could have drawn a bar model like we did on kind of that I showed you before. Now this one. Well, you can think of it as four pieces away, so that writing four there, that doesn't really help there. That's four pieces from the whole and that's four pieces from the whole. Well, which pieces are bigger? These pieces are bigger so that's further away from the whole. Or you could have thought about this a different way. Three sevenths, that is less than half 'cause three is less than half of seven. So that number is less than a half . But here, five ninths, well, five is more than half of nine. So this number here is more than a half. So we don't always have to compare to one. We can sometimes compare to a half and that's a really useful strategy. Okay, what about this one? Well, this one, this one is bigger than one. This one is one away from one but it's less than one. So that's less than one and that's bigger than one. So this is obviously the bigger number and I forgot to circle that over there. Now what about this one? Hmm. Trickier this one. So what do you think? Well, here, we've got two pieces away, and here we've got three pieces away. Does that make a difference? Does that help? Hmm. How'd you compare fifths and sevenths? Well, I'm going to compare these to a half. What is a half of five? What is a half of seven? So here, this is half a piece, greater than a half. And this one is half a piece, bigger than a half. Now, sorry, half a piece, is half a piece plus a half. Half a piece plus a half. That's probably a better way to write it than what I did before. And which piece is bigger? Well, fifths are bigger. So half of a fifth is going to be bigger than half of a seventh. So half of a piece here is bigger than half of a piece here. So because it's a half plus half a piece, lots of halves going on, I know, this one will be bigger than this one. And that's quite tricky this one. It's really kind of tough. And we're going to look at other approaches in the future that you may find easier than comparing it to a half. However, that's one way that we can do it. Okay, so now it's time for the independent task. So what I'd like you to do is I'd like you to make a path between this one and this one here. And you can only go down across, you cannot go diagonal. Okay, so if you're here you can go that, that way, that way, that way. Now you can't just make any old path. You're only allowed to move to a square with lower value. And some of these squares maybe impossible to go to 'cause you get trapped there. So the things I want you to think about are how many paths can you do? And is there any impossible squares and which ones are they? So pause the video to complete your task. Resume once you're finished. Okay, and here are my answers. The red, these squares here. This one, this one, this one and this one, these are my impossible squares. And here are some possible paths. So there's quite a few different ways that you could have done it. And you can, the arrows are a bit small, so you could have gone this way, this way like that. Or instead of going down then, you could have gone like that around. You can go down here and then down that way, or go there and down there. And then there and down. Or there, there, there and there. Or there, there, there, there and there. Okay, you can see how there's quite a few different paths there. So hopefully you've managed to find a couple of them and really well done if you did. Okay, so now it's just time for the explore task. So I want you to have find a convincing argument of which fraction is closest to one. And here, which fraction is closest to a half. So pause the video to complete your task. Resume once you've finished. Okay, so here are my answers. Hopefully you found the same and hopefully you've got a good reason. Okay, so let me just go through a few. This one is two pieces away from nine whereas this one is three pieces away from nine, so this one's closer to nine, 'cause it's less pieces away or less of the same size pieces away, should I say. So for this one, you've got five quarters. So that's one quarter away and this one is a third away. Well, what's bigger, a quarter or a third? A quarter is smaller. A third is bigger. So you'd rather be a quarter away. A quarter is closest. What about this one? Well, one 10th. What's that the same as? That's the same as two out of 20. So if you imagine that on a diagram, two twentieths is definitely smaller than three twentieths. And three twentieths is definitely closer to one. This one. Well, that's two pieces from a whole and that's two pieces from the whole. These pieces are smaller. This one, well, that's 0.5 of a piece, half a piece away from a half, and that is one and a half pieces away. So that one's closer to a half. Okay, this one's a bit of a tricky one, this one. Well, this one is kind of one eighth away from a half and this one is half of a third. What's bigger, half of a third or one eighth? Half of a third. 'Cause if you imagine we had our pieces, half of a third is like a sixth. So half of a third is a sixth. So one sixth is bigger than one eighth. Okay, this one I'd say probably good idea is to think about it on the number line. Now I'll draw two number lines here. So here we've got our half. And that would be our eight ninths, okay? So that's that distance there away. There's one seventh. Well, that's going to be that distance there. Can you see how that is smaller than that? Because one seventh is bigger than one ninth. So after one seventh, that distance between there and that distance between there, which one is going to be less? That one because that's further along than that is further along. Okay, and finally, this one is half a piece from a half and this one is a full piece from a half. And ninths are quite in tenths, there's not too much difference in size. So half of a ninth, half of a ninth is actually one 18th. So that one is closer. Sorry, that one is closer, one 18th away. And that one is one 10th away. Okay, so there's some quite tricky ideas and you might not have worked them out in the same way that I did, you might have used your own way. But it's all about just getting a sense of size and comparing to one, comparing to a half, thinking about the distances, using number lines, using your bar models, using those representations will help you. Okay, so that is all for today's lesson. So thank you very much for all your hard work. And I look forward to seeing you next time. Thank you.

Comparing fractions (Part 1)

Comparing fractions (Part 1)

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UnitMaths / Conceptualising and comparing fractions

Maths