Modelling fractions by division (Part 2)

Key Stage 3

Maths

Manipulating and calculating with fractions

﻿Hello, I'm Mr. Langton. And today we're going to do some more work on modelling dividing fractions. All you're going to need is something to write with and something to write on. Try and make sure you're in a quiet space with no distractions. And when you're ready, we'll begin. We'll start with the Try This activity. Students use this model, to calculate 4 divided by 1/3. How many 1/3's makeup 4? 12, so 4 divided by 1/3 is 12. You can see that on the diagram. Each of these little blocks here represents 1/3. And how many 1/3's make up four? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So 12 1/3's make up 4. So 4 divided by 1/3 is 12. Can you use the model to work out the answers to these three questions here? Pause the video and have a go, when you're ready unpause it and we'll go through it together. And you're pausing in 3, 2, 1. Okay, let's look at the first one. I'll choose a different colour pen. So 3 divided by 1/3. How many times does 1/3 go into 3? So it's going to go once, twice, 3 times, 4 times, 5 times, 5 times, 7 times, 8 times, 9 times. 3 divided by 1/3 is 9. 5/3 divided 1/3, Let's do a different colour. Let's have a green this time. So first off, I just want to find where 5/3 is. 5/3 is going to be here. I'm just going to draw a line down there to make it easier to see. I'll label that 5/3. So that it's going to be 1, 2, 3, 4, 5. And finally 2 divided by 2/3. So here's my 2, here. How many times does 2/3 go into it? Well, 2/3 is going to be two of these blocks, here. So once, twice, three times. Three students draw a diagram to support their answers. Let's look at this first one. 6 divided by 1/2 equals 12. So the top row represents our 1/2's. You can see each whole split into halves. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So 6 divided by 1/2 is 12. And also 6 lots of 2 makes 12. Can you see that connection between the two? Second one: 6 divided by 1/3 is 18. So our 1/3's is our second row down. How many 1/3's go into 6? So there's 3 for each whole 1, isn't there? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18. So 6 divided by 1/3 is 18. And also, 6 lots of 3 makes 18. Each whole 1 is made up of 3 pieces. So if you've got 6 whole 1's with 3 pieces in each 1, you've got 18 altogether. Finally, 6 divided by 1/5 is 30. So this bottom row here is our 1/5's. And I'm not going to draw all over it this time, but we can see obviously 5 1/5's make a whole. If we've got 6 of them, we know that 6 5's a 30. So 6 divided by 1/5 is also 30. It's really useful to spot these patterns when we are dividing by unit fractions. As you've touched on before, if we divide by 1/2, it's the same as multiplying by 2. And dividing by a 1/3 is the same as multiplying by 3. So keep your eye out for those. Kind of a few more questions and you can have a go at these next ones. Now, if you'd like to have a good without me, pause the video now and see what you can do. If not keep it going, we'll go through it together. If you'd like to go without me starting 3, 2, 1. So we're going to start off with the one on the left hand side. 6 divided by 2/3. I'm going to use the diagram to help me. We've already said this row here represents our 1/3's. So how many lots of 2/3 are going to make up 6? That set there, so 2 of them make up 2/3. So 1, 2, 3, 4, 5, 6, 7, 8, 9. So 6 divided by 2/3 is 9. Now, do you remember what happens? 6 divided by 1/3... We said dividing by 1/3 is kind of the same as multiplying by 3, isn't it? And it gives us 18, and there are 18 1/3's all the way along that makes 6. So can you see that link between dividing by 1/3, and dividing by 2/3? What's happened to the answer? What's the difference between the answers? Out of interest, before we do this one, what is 6 divided by 1/4? How many 1/4's are that make up 6. 4 1/4's make up a whole 1. We've got 6 whole 1's, that's going to be 24, isn't it? I wonder if we can see a connection between dividing by 1/4, and dividing by 3/4? Do you want to make a prediction? Well, let's see. Let's pick a different colour. How many, lots of 3/4 are going to go into 6? So there's 3/4 there. Once, twice, 3 times, 4 times, 5 times, 6 times, 7 times, 8 times. That's it, exactly. 8 times. You see the link between dividing by 1/4, and dividing by 3/4, in relation to our answers? Finally, let's have a look at the last one: 6 divided by 6/5. And if we were splitting 6 into 1/5's, there are 5 1/5's in a whole 1, so if we had 6 whole 1's, that's going to be 30. Do you think we could make a prediction on what this is? Look back at the first one, the difference between divided by one 1/3 and 2/3? That's halved. Look at the second one, the difference between divided by 1/4 and 3/4? The answer became split into thirds, it was divided by 3. So I'm going to predict the difference between divided by 1/5 and 6/5, or divided by 6, I'm going to predict the answer is going to be 5. Now we'll see, I might be wrong, but let's check. How many times does 6/5 go into 6? Now 1/5's are along the bottom. We need 6 of them, 1, 2, 3, 4, 5, 6. So this red block here that I've just drawn there, that represents 6/5. So that's once. 1, 2, 3, 4, 5, 6: That's twice. 1, 2, 3, 4, 5, 6: That's 3 times. 1, 2, 3, 4, 5, 6: And that's 4 times, it's looking good. 1, 2, 3, 4, 5, 6: There we go, 5 times! So we can start to make a connection in how we can do this. If we want to divide by 6/5, what we could do is find 1/5, and multiply by 5 and then split into 6 pieces. So when we divided by 3/4, we divided by 1/4 first. Then times it by 4, then split it to 3 pieces. And when we're finding 2/3, we divided by a 1/3 first, so we multiply by 3, and we divided that by 2. Okay, now it's your turn to have a go. Pause the video and have a go the worksheet. When you're finished unpause it, and we'll go through the answers together. Good luck. So how did you get on? I thought it might be useful to go over a couple of answers with you, before we mark the rest of the work. We'll start off with Question D, here. 3/5 divided by 3/10. And as we said before, there are lots of ways that we could write this, but you could say that 3/5 is 3/10 of something. So I can draw up our model. Split into 10 equal pieces. One more there. And here we've got 3/10. Now that represents 3/5. Each of those individual ones is 1/5. Now, altogether, I'm going to have 10 of those 1/5's, aren't I? So 3/5 divided by 3/10, is 10 1/5's and 10 1/5's is two whole 1's. Now that method is not a very easy one to apply to Question G. We need to think of some other ways. One way that we discussed, is that when we divide by 4/5, we'll divide by 4 and then we'll multiply by 5. Now that's quite tricky in this case as well, because we've got to do 3/5 divided by 4. I would rather do that multiply by 5 first. If we start off with 3/5, and we multiply that by 5, we're going to get 15 over 5, which is 3. If we then take that 3 and divide it by the 4, 3 divided by 4 is 3/4. You see, that's how we can get that answer. So put up all the answers for you, now. Just pause it for a moment, check what you've got, and mark all your work. We'll finish with the Explore activity. Binh is painting her bedroom. The tin says at 3/7 of a litre of paint can cover one metre squared. Complete the following four questions. Pause the video to have to go without me, and when you're ready unpause it, and we'll go through it together. We start in 3, 2, 1. Okay. I'm going to start off by looking at what calculation I'm going to need to do for each one. If I've got 3 litres of paint, I want you to share those 3 litres into the 3/7 to work out how many metres squared it's going to cover. I've got 1 litre of paint. I need to do 1 divided by 3/7, 3/4 of a litre of paint. I'm doing 3/4 divided by 3/7. And n litres of paint is going to be n divided by 3/7. And then don't forget, there are lots of ways that we can try to do this. Let's look at the first one, 3 divided by 3/7. Now we've said that you can work out the answers when we're dividing, by dividing by the numerator and then multiplying by the denominator. That's one way that we can do it. Does that work for this one? We're going to do 3 divided by 3 is 1, and 1 times 7 is 7. So 3 litres of paint are going to cover 7 square metres. Second one: We're going to do 1 divided by 3 and times that by 7. Little bit tricky, but I think I can do this. 1 divided by 3, rather than writing it as a decimal, write it as a fraction. It's a 1/3, which I then need to multiply by 7. That's going to get me 7/3, metres squared. 3/4 divided by 3/7. So that's talking about 3/4, and I divide it by 3. I'm going to get 1/4 each, times my 1/4 by 7, and that's going to get me 7/4 metres squared. So for this last one, I can't write a number as an answer, but I can write the calculation that we'll do. However many litres I've got, n, I'm going to divide it by 3, and I'm going to need to multiply it by 7 metres squared. Right, that's it from me, but feel free to spend a little bit more time seeing if you can come up with any more similar questions. I'll see you later, goodbye.

Modelling fractions by division (Part 2)

Modelling fractions by division (Part 2)

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UnitMaths / Manipulating and calculating with fractions

Maths