﻿Hi, everyone, Mr. East here. I'm going to take today's lesson, and we're going to start off by thinking about what Mrs. Kingham asked you to do in the previous lesson. She was talking about and comparing equal parts of the same whole that looked different, and she asked you to think about these three squares. They're all squares, and they all have some parts, but the parts are different in shape, and they're different in colour. So let's think about the fraction of each whole that is shaded. So let's start with the shape with the yellow part that's shaded. So we've got four, sorry, parts, and one of those parts is shaded orange or yellow, and they're all the same size, so that is one quarter of the whole. If we go to the shape with the blue part shaded, this time, my part is a rectangle. There are still four equal parts, so we can still say that a quarter of the shape has been shaded. The final one, the red or the pink part, this time, the part is a square. There are still four equal parts, and because one of them is shaded in, that's still a quarter of that shape that is shaded. So my job is to try and convince you that even though the parts are different in appearance, they must take up the same amounts of space of the whole because they are all a quarter. The first thing I'm going to do is I'm going to prove to you that, here, I've made them, I'm going to prove to you that they are all the same size, that the whole is the same size. I'm going to do that by putting them on top of each other, and we can see, to my best attempts, that they are all the same size, because when I put them on top of each other, we can't see the other ones. Just line them up a little bit better. We can't see that one. So my first job was that the wholes are all the same size. Now that I've proven to you that the wholes are the same size by putting them on top of each other, I'm going to cut out and move the parts to demonstrate that they are all equal in size. I'm going to cut up the orange triangle, the yellow triangle, and I'm going to cut up my blue rectangle to prove it to you. So, always wanted to say this, here's one I made earlier. I have stuck my three wholes onto this piece of cardboard. Now, can you see? I've already cut out a part of this part, and I've taken it off. Can you see? Taken it off, and I'm going to stick it above the other part of the part, and can you see now that I've changed my yellow, my orange triangle, into a square, and it's now equal in size and appearance to the red square. Now at the moment, my blue rectangle, it doesn't look the same, does it? But I'm sure some of you can see that I could also cut this out in the same way. So I have already cut this out. I'm going to take off part of this part, I'm going to keep it the same size, and I'm going to stick it back on top. Can you see now that those three are identical in appearance and size, that that means that when they look like they do on the screen at the moment, even though they look different, they are still equal parts of the whole. And if you're not convinced still, okay, one more thing I'm going to try and do on the screen to prove it to you. So a little bit of magic. I am going to split up my orange triangle. I've cut a bit off, and I'm going to move it on top of there. And can you now see, that looks identical in appearance to the red square, and exactly the same as I did for real, same for my blue part. I'm going to cut it and move it. And can you now see that they are identical in appearance? And that shows us that a quarter of each of those wholes takes up the same amount of space. Okay, so we're going to look at fractions of liquid. Some of you might've done this in a previous lesson before. And if I tilt the screen, in front of me, I've got three different glasses which all contain some squash. I want you to think, could you estimate what fraction of each glass is filled with squash? What do you think about this one? Shout something out, an estimate, the fraction of this glass. And what about this one? What about this one? That's quite difficult, isn't it? Maybe an easier question is thinking which glass has the largest part of the whole that's been filled with squash. Do you think this glass, this glass, or this glass? Which is the largest part of the whole that's been filled with squash? Which one do you think is the smallest part of the whole? It's got to be different one to the one you think is the largest, but which one do you think is the smallest part of the whole? Now, what I'm going to do is I'm going to pour some of them into each other to prove something to you. Did anyone say that they thought the volume, the amount of water, squash, sorry, in each glass was the same? Because, actually, it's identical. Even though they look like there's different amounts, I think lots of you would have thought this one had the largest fraction that was squash 'cause it goes higher up the glass, doesn't it? But because the glasses are different shapes, that's actually not true. Now I'm going to pour all three of them into this one. Oh, see, this one's already there, but I'm going to pour these two into here to prove to you that if I add two of these parts, which I'm saying is the same, to here, it will fill the whole of the glass. About to get messy. Let's see if I can do some good pouring. I'll pour all of that one in. Look at that. Do you think, is there going to be room for all of that part in here? Will have to be. If it was a third full, all of this needs to fit in here. Right up the top. Right the way to the top. So the whole of this glass is full. Teeny bit left, that's just so that it doesn't overflow. Both of these have come together to fill this. Now I'm going to see if I can pour them back out to prove that this fits into each of these. Uh oh. I made a bit of a mess, but can you see the top of this glass, the whole that fills this and this, they are the same size, so these glasses, even though they look different, the whole is the same size. Final one. This is the one that lots of you probably wouldn't have thought was the same size. Let me pour all of the liquid, the whole of the liquid in here, into this glass to prove that it all fits. Ooh, better pouring that time, all the way up to the top. Again, a teeny bit left. Actually, I spilt some before, but all of the wholes are the same size. Now, are the lines still there? I drew a little line on this one. Let me see if I can go back. That's about how much... Oh, that's too much. I'll pour some of that into here. That's back in line. Where's the line I drew in this glass? Oh, there it is, I can see it. Let me pour the rest of this in. Making a big mess on my desk. And there we are, that's how they were at the beginning. They might be in a slightly different order, but they're back at the beginning. The fraction that is filled with squash in each glass is equal in each case. This time, we're going to use glasses on a screen rather than real glasses, because that can get a bit messy. Now all of these glasses can contain 300 millilitres of water. I'm going to pour, virtually, 100 millilitres into each glass. Are you ready? Here you go. I poured 100 millilitres into the first glass, the second glass, and the third glass. Do you believe me? Do you believe me that each of those fractions, each of those parts, are equal in size? Doesn't look it, does it? Just like with my glasses, because the glasses are different shapes, they don't look like they're equal in size. So what I'm going to do to try and convince you this time is I'm going to pour the water out of those three different shaped glasses into three identical glasses. So the first glass goes in there, the second glass goes in the middle one, and the third glass is poured into there. Now, can you see that when they're in identical glasses, it's easier to see that the volume of water is exactly the same in all three? Now I'm going to pour them back into the original glasses. So all of that's gone in, all of that part's gone in, and all of that part has as well. Now, another way, the same way that I tried to prove to you before, I'm going to pour each of those three parts into one glass. If I told you, well, when I told you 100 millilitres was in each glass, if we add those three 100s together, that will be 300, so that should fill one of these glasses. Let's see that. I pour the middle one into the end one, and then the final one, and just like I said, my three lots of 100 millilitres has now filled my first glass. Going to pour that out and pour it into the second one, just like I tried to show you. I prove that they are all the same size. The whole is the same, even though they look different, because if I fill one up, and then I pour that into the next one, each time it fills the whole of the glass. Now I want to show you another example. In front of me, I have two identical glasses. They're the same height, the same shape, but this one is full of rice, and this one has no rice. So the whole of this glass is full, none of this glass, no parts, are full of rice. I want to try and pour half of this into this glass so then there is an equal fraction in each. So if I'm pouring half in, that would mean half of this will be full, and half of this will be full. So have a look on this glass. Where do you think, on this glass, a half would be? I want you to say stop when you think I get to where a half might be. Stop. Think I heard some of you say stop. I think lots of you think that this is going to be half. I'm going to keep my finger here, and I'm going to pour, as carefully as I can, my rice from one cup to the other, until I get to that point. I've lost a few grains of rice, but most of it went in. Let me level this out. Do they look equal to you? Is there half in here and half in here? No, nowhere close at the moment. There is a lot more in this, so we can't compare them. They're not equal parts. I'm going to have to keep on pouring so that these levels become the same. So let me pour in here. Bit neater that time. That might be it. Let me level them out. That equal? No, still a bit more in this glass. I reckon that might be good. Okay, I've levelled them out. Now that shows that there is an equal amount in each. Again, though, this, both of the amounts which have rice in, they look a lot bigger, don't they, than the amounts that don't have rice in? But, hopefully, I've proved to you in another way that two equal parts may be equal in size, but they might look very, very different. Now I want to try one more thing. I'm going to put a lid onto this glass, I'm going to turn it upside down. Lost a few grains of rice, but can you see that, again, if I hold that next to it, they are both the same amount, but again, this looks a lot smaller than this, doesn't it? But actually, they are the same size. I'm going to turn this back over. I'm going to prove to you that I haven't lost loads of rice. I'm now going to pour the rest of this into there, till we've done it all. All the way to the top. Let me level it out, and there, you can see, the whole of this glass has been filled, and it's all come from out of here. Now, again, I thought I'd show you this on the screen as well as just showing you with the glasses of rice. So on the left of the screen here, we can see we have our full glass of rice, just like I had. Then, if we pour half of this into the first glass, we can say that half of this glass is full, and then if we pour that into another identical glass, we can see that it's the same fraction that's full, because they are the same level, so therefore we can say a half of this glass is full, and a half of this glass is full. Even though they don't look the same because this part looks a lot larger than this part, we know from what I've just shown you and from this image that they are actually equal in size. And just like I did, or tried to do, here, that shows us even more that if we turn this glass upside down, so then the rice falls from the bottom of this glass, to then down here, which is now the bottom of the glass, again, these parts look very different, don't they? But we know they are equal in size. So our generalisation is... Can you say this with me? Equal parts of the whole do not have to look the same. That's a really important point. That's the point we're trying to prove in this lesson. A different example for you. What fraction of the square do you think is red? Have a think. Now, I can see that there's three parts. There's a white rectangle, there's a red triangle, and there's another white triangle. But we said before that we can only say that amount is shaded if the parts are equal, and at the moment, these parts don't look equal, do they? But if you cast your mind back to the beginning of the lesson, we said that sometimes, the shapes don't look the same, but they take up the same amount of space of the shape. So I want us to focus on the white rectangle. You can use your visualisation that I know you've been practising . Can you imagine there would be another white rectangle here, and then a final rectangle here. If we change them to be rectangles, then maybe there would be three of them, so maybe then we could say that this middle section would be a third. So let me show you that. I'm saying that, possibly, if we can visualise three of these rectangles filling the whole, that the white rectangle is a third of the whole. I'm then thinking in my mind, if I split this red part, this red triangle, up, could I make another rectangle? And just like I did before, if I rotate that around, now I can see there's another rectangle in the middle, and then a third rectangle on the right. That now shows me that there are three rectangles, so I can say that each of them is one third of the whole, because my parts are equal in size, so we can say that the red fraction of the square is one third. Now this is your practise activity. I want you to use all of the skills that we've spoken about in this lesson to see if you can work out the fraction of the whole shape that is red, the fraction of the whole shape that is green, the fraction of the whole shape that is yellow, and the fraction of the whole shape that is blue. Now, remember, we've learned in this lesson that the parts may look different, but they may be equal in size. So I wonder, just as a guess, just as a prediction, do you think any of the colours are equal in size? Do you think they're the same fraction of the whole? Maybe in your mind, you're already doing some cutting up and some moving of these parts to compare them in size. I want you to think about how you can explain what you think, can explain what you know. If you cast your mind back to previous lessons, we used this sentence scaffold, didn't we? The sentence scaffold says, "The whole has been divided into mm equal parts. One of these parts is one of the whole. So I want you to think about for each of those colours, how many of those parts could you make to fill the whole of the shape? And that will help you to identify the fraction of the whole that is each colour. If you want an extra challenge, can you work out what fraction of the whole shape has not been shaded? So, which fraction... Sorry, what fraction of the shape is white? That's your practise activity and your challenge. Have a go, and at the start of the next lesson, we'll think about this together.

In this lesson, we will understand that the same fraction can look different in the context of shading shapes with different volume or area. We will also investigate how a shaded fraction can look different but be the same proportion in the same shape if we make it in different ways.

In this lesson, we will understand that the same fraction can look different in the context of shading shapes with different volume or area. We will also investigate how a shaded fraction can look different but be the same proportion in the same shape if we make it in different ways. Resources provided by NCETM (CC BY-NC-SA: https://creativecommons.org/licenses/by-nc-sa/4.0)

Maths

Y4 Maths

Key Stage 2

Year 4

Understand that Equal Parts Can Look Different - Volume and Area Contexts

Working with fractions

UNIT

Divide and describe the same whole when divided into differing numbers of equal parts

Understand fraction notation to represent a relationship between part and whole

Begin to use and understand the terms 'Numerator' and 'Denominator'

Name unit fractions and match them with the fraction notation and a representation

Embed previous fraction work using a linear model

Assign unit fraction names and notation to 3D representations

Assign unit fraction names and notation to equal parts of quantities

Recognise and reason about unit fractions in a variety of contexts

Understand that Equal Parts Can Look Different - Area

Compare unit fractions using a fraction wall

Reason about comparing unit fractions

Compare unit fractions in a measure's context

Can we compare unit fractions of different wholes?

Construct a whole from a part and identify the fraction it represents

Build and compare different wholes from the same unit fractions that represent the same length

Build and compare different wholes from the same unit fractions that represent different lengths

Build and compare different wholes from different unit fractions that represent the same quantity

Build and compare different wholes from different unit fractions that represent different quantities

Build and compare wholes when different unit fractions represent different amounts

Lesson overview:Understand that Equal Parts Can Look Different - Volume and Area Contexts

Core Content

Working with fractions: