Further practice and application

Key Stage 2

Maths

Equivalent fractions

﻿Hi everyone, my name's Mrs. Furlong, and I'm going to be delivering your lesson today for the NCETM lesson 19 on Fractions for Upper Key Stage Two. Our lesson today is going to focus on consolidating your work that you've been doing on simplifying fractions, as well as bringing together the different calculations that you have met so far in fractions. So yesterday, Mrs. Mull took lesson 18, and she was looking at multiplying fractions, and also simplifying fractions. And she left you with some work. I hope you got on okay with it. We're going to have a look at that just to begin this lesson. So let's have a look at the first question. She left you with 3/8 multiplied by two, and yesterday you learned that when you're multiplying a fraction by a whole number, that the numerator of the fraction is multiplied by the whole number, and the denominator remains the same. So, in this example, the three from the 3/8, will be multiplied by the two to give us 6/8, and we can see that in the picture where it shows two sets of 3/8, okay? And when we multiply them, you can see that they join together to become 6/8. But I wonder, is this the end of the calculation? Have we completed it, or is there something more we need to do? That's right, we haven't simplified it. So lets have a look, six and eight, do they have a highest common factor? They do, you're right, it's two. So we need to make sure that we divide both the numerator and denominator by two to express it in its simplest form. So here we are, I'm just going to write on here that we're going to divide them by two, and then we can express it in its simplest forms, and our answer is 3/4. Next question you're asked to solve is three multiplied by 2/9, which we know if we multiply our numerator and our whole number, or in this case, our whole number and our numerator, the three and the two, we get six. So we get 6/9 because our denominator remains the same. I decided to use a number line to illustrate this one. So you can see on our number line that we have jumped from zero, with jumped on 2/9 and then another 2/9 and then another 2/9, so we've got three lots of 2/9 and we land on 6/9. But again, is this the final answer for our question, or if there something else we need to do? You're right, it hasn't been simplified. The highest common factor of six and nine is three. So we need to make sure that we scale down six and nine by three. So it becomes 2/3. Okay, so the next question that Mrs. Mull left you with is 2 1/4 multiplied by two. I think this one's a little bit more challenging. So I've used some illustrations to help us represent this calculation. So you can see here, we've got 2 1/4 and another set of 2 1/4. So we've got 2 1/4 two times. And then we would need to combine them together. And when we put them together you can see that you have 4 2/4. But again, have we got this in its simplest form? You're right, we haven't. I think a lot of you will probably recognise that 2/4 is the same as a fraction you're very familiar with and that's 1/2, isn't it? So we've got four wholes and 1/2. Okay, so in this example that Mrs. Mull gave you, you needed to do 4/6 multiplied by five. We know that we need to multiply the numerator by the whole number, and our denominator of six remains the same. So 4/6 five times or 4/6 multiplied by five would give us 26. You might have decided to use method one from yesterday where you decided to simplify 26 first. What's the highest common factor of 20 and six? That's right, it's two. So our 20 and six would have simplified to 10/3. But the problem now that we have is that 10/3 is an improper fraction. So we need to think about that as a mixed number. And there you can see that we have three and we've got 1/3. Can you see the 1/3 in that diagram? You might have to think of those triangles in different ways to be able to see it in that way. Or you might have decided to use method two. So we would start in exactly the same way, getting to 26, but you might have decided to change it into three wholes and 2/6 first. And then after that, you might've thought those 2/6 can simplify because two and six have a highest common factor of two. And if you simplify them again, you get the answer of 3 1/3. You could do it either of the ways or you might've had a different way still, but the answer still remains 3 1/3. Okay, so today's session is a review session of this segment, and it's helping us to make decisions about simplifying fractions when carrying out calculations. We're going to start our session with the question, I have two planks of wood. One is 6/8 of a metre long, the other is 12/8 of a metre long. How long is my wood all together? I wonder what kind of calculation you think we need to carry out to solve this? That's right, it's an addition. Well, I'm looking at 6/8 and I'm also looking at 12/8, and I'm wondering, could I simplify them before I start my addition? Let's take a look. Let's think about that. Here's our 6/8. What's our highest common factor of six and eight? Because if we're going to simplify it, I need to know that first. That's right, it's two. So I'm going to need to divide my numerator and my denominator by, by two. All right, once I've done that it's simplified to 3/4. You could probably tell me that faster than I could use the pens on my screen. What about with 12/8? What are the common factors of 12 and eight? That's right, it's four. They're both multiples of four, so the common factor is four. So I can use four to help me to simplify 12 and eight. So I'm going to divide both the numerator and denominator by four. I'll just sort my four out at the top of the page, that's better! Right, and you're right, that gives us 3/2. How easy is it to do 3/4 add 3/2? My numerators are the same. Normally when I've added fractions my denominators have been the same, not my numerators. So I don't think this is going to help us today. I think it's made it harder, not easier. Let's think then. I think we'll get rid of that one and go back to the original calculation, which is 6/8 and 12/8. 6/8 and 12/8, that's quite straight forward. Altogether we have 18/8, which I've shown you on a number line here. We jumped from 6/8 to 12, sorry, we've jumped from 6/8, we've jumped on 12/8 to land on 18/8. Is this going to be our final answer? What could we do? Could we use the method from yesterday where we simplify first and then we change it to a mixed number? Or would you prefer to use the method where we change it to a mixed number first and then simplify, the choice is yours. I'll show you what I decided to do, but you may have chosen to do it the other way round. So I decided when I was doing this one that I would think of it as the mixed number fraction first. So I converted my number line to have mixed number fractions on instead. And that meant that when I jumped on from my 6/8 and I jumped on 12/8, I landed on 2 2/8, as you can see on the number line. Okay, but remember, we're thinking about simplifying fractions and that means we need to check our final answer and decide is it in its simplest terms? And hopefully most of you did spot that no, 2/8 is not in its simplest terms because two and eight have that highest common factor of two, don't they? Yep, so 2/8 can be simplified to 1/4. So our final answer is 2 1/4. Okay, now we're on to example two. We've got 3 7/10 and we need to add together 2 9/10. When you were in year four, you may have been taught that if you were adding mixed number fractions, that you can first at the whole numbers and then add the fractional parts. My friend did this. They added the three from the 3 7/10, and they added the two from the 2 9/10 to make five. Next they added the seven from the 7/10, or sorry, they added the 7/10 and they added the 9/10 together to get 16/10. And then my friend expressed their answer in this way, 5 16/10. What do you think of that? You might want to pause the video here and discuss 5 16/10 with someone else in your house, or just have a little thought about it to yourself. Okay, let's have a look at 5 16/10 in a little bit more detail. I did this with my friend and I drew some representation to talk through them. I drew the five wholes and I drew a diagram of my 16/10. My friend noticed something really important about those 16/10. You might have noticed it too. That's right, the 10 tenths make up one whole. So we haven't actually got five wholes and 16/10, we've actually got six wholes and 6/10. In maths there are some rules for how we express things. And when we're expressing a mixed number fraction, and mixed number fractions should only be presented with a proper fraction. It's not correct to present it with an improper fraction. So we're going to get rid of that answer and change it for 6 6/10. I wonder though, is that the end of our answer to 3 7/10 plus 2 9/10, or can we do something more? You might have spotted that the 6/10 has a highest common factor of two. If you spotted that you were ahead of me. But have a look at the diagram and you can see that our 6/10 is equivalent to 3/5. So our final answer simplified is 6 3/5. Okay, so now it's time for you to have a look at something and have a go at something on your own for a moment. I'd like you to have a look at these four equations. And I'd like you to decide whether you think they are correct or incorrect. You might decide that you want to pause the video to do this, and then come back to it in a moment. Okay, so hopefully you've had time to have a look at those and maybe you've made some jottings too. What did you notice? I wonder if you got the same as me? So I found that the two on the left-hand side were correct. The 6 1/5 is correct, and the six is correct. I wonder what's wrong with the 3 4/3? Did you spot it? That's right. When we had a look earlier, we realised that mixed number fractions are made up of whole numbers and proper fractions only, and 3 4/3 has an improper fraction in it. The 4/3 is actually one whole and 1/3, isn't it? So in actual fact, we've got four wholes and 1/3. Why do you think I've got an orange tick next to 4 2/10? Did you spot it? That's right, 2/10 can be simplified, so it is correct, 4 2/10 is not an incorrect answer, but I'd really like it to be expressed in its simplest terms. So in that case, 4 2/10, we can see the highest common factor of two and 10 is two, that's right. So our final answer is really 4 1/5. Okay, so now we're going to have a look at example three, which is a subtraction calculation. Earlier when we looked at an addition calculation, we considered whether it would be a good idea to simplify before we carried out an addition and we found it wasn't a good idea because our denominators were already the same, and when we add or subtract, we want same denominators. In this example, our denominators are already the same, but actually 17/3, its highest common factor's one so we can't simplify it anyway. And 5/3, that's highest common factor is also one so we can't simplify that. So let's think about these numerators of 17 and five. And when we subtract them we get 12/3. Okay, is there anything you notice about 12/3? Okay, maybe you might have noticed that they both have a highest common factor of three so we can simplify them. And here I started to think about that simplifying and we get 4/1. What does 4/1 actually mean? Hmm, we've got four parts and for the whole it's made up of one part. If a whole was made up of one part and we've got four of them then you're right, we've got four wholes, so we just express it as four. So in example four, we're going to have a look at 7 1/6, subtract 1 2/6. Hmm, this one's making me think a little bit more. I wonder what you think of this question. I'd really like you to get a piece of paper and a pen and see if you can have a go at solving it. What methods might you use? And then we'll come back together and have a look at what I did and see whether we did the same. Okay, so let's have a look, I started with the area model. I quite like the area model. I like to be able to visualise things. Some of you might have tried the same, but there's a bit of a problem here. I need to subtract 1 2/6 from 7 1/6. I can easily subtract the one but I'm really struggling to subtract the 2/6 because I'd have to break into one of my bars and that's not so easy. So although it is possible, I decided to come away from the area model and see what else I could do. So next I came up with a number line. I wonder if any of you have used a number line. You might have done. So my number line, I started in 7 1/6, and jumped back one whole to land on 6 1/6 And then I jumped a back further 2/6 and that's right, we got the answer of 5 5/6. Do we need to simplify that? You're right, we don't. Don't need to simplify it because 5/6 is already in its simplest terms. And then had a think in another way, I thought 7 1/6 and 1 2/6 as mixed number fractions I think make it a little bit hard. So I wondered whether it might be an idea to express them as improper fractions instead. Perhaps that might make it easier. So I converted 7 1/6 by thinking about six. So I need seven wholes expressed in six. Well, I know that 6/6 make one whole, so seven lots of six is 42, so 42/6 must make seven wholes. So I've got 42/6 and the 1/6 that gives me my 43/6. And then for the 1 2/6, again, 6/6 make my whole, my one whole, plus those 2/6 in that mixed number fraction gets me all together, 8/6 at the bottom as an improper fraction. Hmm, now I can find this a little bit easier because I can do 43 and subtract eight for my improper fractions. So 43/6, subtract 8/6, gives me 35/6. My only problem now is it still expresses an improper fraction and I really do need to get it back to that mixed number. So I now need to think in six again. I've got 35/6. I know the 6/6 make one whole, so five six are 30, so 36 must make five wholes. And then I've got five more six. So that's another way of getting to the answer. Maybe you've had some other methods too. I'm sure there's lots and lots that you can do with that calculation. Okay, so example five is a little bit of a revisit from yesterday. You were doing some multiplying of fractions yesterday, weren't you, with Mrs. Mull. And yesterday you used the sentence, "The numerator of the fraction is multiplied by the whole number and the denominator remains the same." So I've popped that on here just to remind us for today. So in this question, 4 1/9 multiplied by three might be a little bit tricky, but I think yesterday you were shown that you can partition that mixed number. So our 4 1/9 can be partitioned into four at one side and 1/9 at the other. So now we can do four multiplied by three and 1/9 multiplied by three. Four multiplied by three is quite straightforward , isn't it? We've got 12. what's 1/9 multiplied by three? That's right, we multiply the numerator by the whole number so we get 3/9. What's the final part of this calculation? We need to recombine those numbers. So the 12 and the 3/9, so we need to represent in our answer. So they recombine to give us 12 3/9. Is this the end of our calculations? It's correct, but is there anything else we can do? I heard some of you shouting at the screen, yes, 3/9 can be simplified, you're right. Okay, so what's that highest common factor of three and nine? It's three, you're right. So we can simplify 12 3/9 to become 12 1/3, well done. So this is example six. This is our final example for our lesson before I leave you with some practise. Okay, so in this example, we're doing 10 multiplied by 5/7. Some of you might want to think of that as 10 groups of 5/7 or 10 lots of 5/7. There's lots of ways to think of that, aren't there? The numerator, remember, of the fraction is multiplied by the whole number and the denominator remains the same. So we are going to be dealing with sevenths. That's not going to change. Our denominator of seven will stay the same. So in this example, we just need to think of our whole number, 10 multiplied by our numerator, five, to help us with the calculation. And 10 multiplied by those 5/7 gives us 50/7. Hmm, 50/7, it's a little bit tricky to visualise, isn't it? I always think improper fractions are hard to imagine in your head. So it's always best to put them back to that mixed number fraction. Hmm, how many wholes are we going to get out of 50/7? Yes, you're right, 7/7 make the whole. So how many wholes, how many lots of 7/7 are there in 50/7? Yes, you're right, 49/7 makes seven wholes and then I've got one more seventh. Do I need to simplify? No, of course I don't because one seventh can't be simplified, can it? They're both prime numbers, so that would be a final answer. Okay, so this is now the end of our session. I'm just going to leave you with some work that I'd really like you to practise at home. I've created five calculations. I'd like you to have a try at solving them and remember to express each answer as a mixed number and in its simplest form. After this, what I'd like you to do is I'd like you to put the calculations in order from the ones you thought were easiest to the ones you found the hardest. And then I want you to justify your order with an explanation. This could be a written explanation, or you could explain it to somebody else in your house, or maybe if you wanted to, and perhaps you're able to talk to some friends, perhaps it might be something you discuss with some classmates who might be accessing this as well. Anyway, I hope you've enjoyed today's session. Good luck with the practise. And hopefully I'll see you again soon, take care.

Further practice and application

Further practice and application

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UnitMaths / Equivalent fractions

Maths