Adding and Subtracting Fractions (Part 3)

Key Stage 3

Maths

Manipulating and calculating with fractions

﻿Hello, I'm Mr Langton, and today we're going to look in more detail at adding and subtracting 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. The whole rectangle below represents one. Which fraction does each colour represent? And how many different ways can you write each fraction? Pause the video and have a go, when you're ready, unpause it and we can go through it together. You can start in three, two, one. So these are some of the ones that I came up with. You might have found a few more as well. Here we're going to compare these two different methods for adding together 1/4 and 1/6. On the left hand side we've got Xavier, who's done a 1/4 and 1/6. He said that a 1/4 is equivalent to 12/48 and a 1/6 is equivalent to 8/48. So adding them together gives 20/48. Yasmin has said that a 1/4 is equivalent to 3/12 and that 1/6 is equivalent to 2/12. And so adding those together, you've got 5/12. Now two questions here, which method do you prefer and is there a particular method that you would prefer if you didn't have a grid to help you? In both cases, what Yasmin and Xavier did was find a common denominator so they could add their numerators together. That bit's really important. Like Xavier said that a common denominator for four and six would be 48 and he's correct. The whole grid is out of 48, isn't it? Yasmin looks at her fractions and made them a little bit simpler and she's said, well, a 1/4 is equivalent to 3/12 and 1/6 is equivalent to 2/12, so she's going to work in 12ths. Both of them are right, both of them give the same answer, but is there one that you prefer? If I didn't have a grid, I'd be looking very closely at my multiples of four and six. So multiples of four would be four, eight, 12, 16, 20, 24 and so on. Now you can see the 12 is in there already. Yasmin got 12, didn't she? Multiples of six would be six, 12, as Yasmin got, 18, 24. So we could have used 24, neither of them had 24 and keep going 32, 40 and 48 and so on. So in Yasmin's case, Yasmin found the smallest number that was a multiple of four and a multiple of six. And in doing that, that meant that when she got to her answer, 5/12, she didn't need to simplify the fraction. Xavier did nothing wrong, he got 20/48 and that does simplify down to 5/12. They are equivalent, that's fine. But by getting that common denominator in the first place not lowest common denominator, that made the final answer much easier to get to cause there was no simplifying to do afterwards. So let's have a look at how we could do these. The first question is 1/3 and 1/4. So we're going to look for the lowest common denominator for these two fractions. So the lowest common multiple of three and four. So my three times table is three, six, nine, 12, 15, 18. I'm just going to stop there, I might need to keep going. My four times table is four, eight, 12, oh as it happens, there we go. Our lowest common denominator is going to be 12. I'm going to rewrite both of these fractions out of 12, so a 1/3 written in 12ths a 1/3 we've gone three, six, nine, 12, we multiply that by four, haven't we? So we need to multiply that by four, 1/3 is the same as 4/12. A 1/4 four, eight, 12, then multiply by three to get to 12ths. So we multiply that by three as well. 4/12 and 3/12 is 7/12. Now let's look at the one underneath. Once again, we're working in 1/3 and 1/4. So we know already from what we've just done that our lowest common denominator will be 12. And this time, we're doing a subtraction question, aren't we? So 2/3 into 12ths. Suppose you're talking to four times as many bits. So we're going to have 8/12 is the same as 2/3. And to turn 1/4 into 1/12, let's split them into three times as many pieces, that gives us 3/12. And 8/12 subtract 3/12 is 5/12. Let's have a look on the right hand side. So the lowest common multiple of five and four. So 10, 20. I'm not going to write that one out. What we could do could wait, lowest common multiple is 20. So to turn 1/5 into 20, we're going to have to split them into four equal pieces for each one. That's going to give me 8/20. To turn turn 1/4 into 20ths I'll just put them into five extra pieces, each time and 8/20 plus 5/20 is 13/20. Now on each case so far, the lowest common multiple has actually been the two denominators multiplied together. Three multiplied by four, five multiplied it by four. That's not always the case, is it? We saw it on the last slide, doing Xavier's and Yasmin's examples Yasmin found a common denominator that didn't involve multiplying them together. In the case of this last one, we're doing the same again. The lowest common multiple, the lowest number, that's in a 16 times table and the 12 times table. Well, the 16 times table is going to be 16, 32, 48, 64 and so on. I don't know how much more I'm going to need. Hopefully I've got enough already. 12 times table is 12, 24, 36, not yet 48, 48. There we go. That's in both of them. So I need to change both of these fractions into 48ths. So to get from 16 to 48, seem we have some multiply by three, didn't we? Gives us a 1/3 and 1/3 multiple. So splitting each part into three equal pieces is going to give me 33/48. Now for the 12 times table 48 is one, two, three, four, that's the fourth multiple of 12. So I'm multiplying by four, multiply that by four I'm going to get 20 and 33 take away 20 is 13. Okay, now it's your turn. Pause the video and have a go at the questions. When you done, unpause it and we can go through it together. Good luck. How did you get on? I put some of the answers on the screen for you now, I'm going to go through the rest of them with you. So start off with question 1d. So 30 over something is equivalent to 10/8, which is equivalent to one and some quarters. Now starting off, if I've a 10 and it's going to 30, to keep an equivalent, I've multiplied by three. So I want you to do the same there, multiply by three, eight three is a 24. Now, if I'm turning my 8 into 1/4 then I'm halfing everything, aren't I? I'm going to half my denominator and half my numerator, that will be 5/4, but 5/4 is one whole one and 1/4. So we got to write it in that format there. And question e, to get from seven to 84, I've multiplied by 12. So I want you to do the same here. I need to multiply by 12, but 11, 12 are 132. And finally to turn from 1/11s into 1/33s I multiply it by three. So I'll do the same there, 21. Okay, let's go to question two. 2/3 to take away 1/6 So I'm turning them both into sixths. So 2/3 taking on 2/3 is going to be same as 4/6, So I'm doing 4/6 take away 1/6. And part d, 3/4 plus 5/12, so I'm working in 12th. So I've got to multiply by three, to each of those so that gives me 9/12. Now I've done that, let's answer are all four of them. So 2/4 add 1/4 is 3/4. 2/8 and 1/8 is 3/8. 4/6 take away 1/6 is 3/6. And if you're feeling really clever, you can simplify that to a 1/2. 9/12 and 5/12 is 14/12, which is also a 7/6, or if you're feeling particularly clever, one and 1/6. Any of those three answers is correct. Now, let's go to question three, says that you can use the grids to help. I'm not sure if I necessarily would, but let's do some examples with them anyway. That grid is six by four that, so there are 24 squares altogether. So if I'm going to be shading 5/12 then I'm going to need to shade 10/24 so that's times two. One, two, three, four, five, six, seven, eight, nine, 10. So I've shaded in 5/12 there, let's pick a different colour. Let's go from blue to do 3/8, going to get to 24, It's going to go three times, so three threes and nine. So one, two, three, four, five, six, seven, eight, nine. So all together, I've shaded in six, 12, 18, I've shaded in 19/24. And so question three of question three, working in 1/8 and 1/6 this time, well, if we're going to to work in eighths, 7/8 is going to be most of it, isn't it? It's going to be 21 of them. Wow, that's a lot. So one, two, three, four, five, six, 12, 18, 19, 20, 21. I'm going to take away 5/6, so that's going to be in 24, that's going to be 20 of them, isn't it? We're taking out 20 of them, previously we did 21. So actually we're going to take off all of these here, apart from that one. All of these are going away. Just going to leave this one here and shade it in red to make it nice and obvious 1/24. Okay, and now for the last two, so we've got no help at all here, we're just given the questions. Lowest common denominator for 12 and three, well, I think that's going to be 12 for each one, we've already got the 1/3 in 12th and we know that the 12th is in the three times table. So 11/12 takeaway, I'm going to multiply that by four, take away 8/12, which is 3/12. And finally 2/7 takeaway 3/4. So lowest common multiple of seven four, seven and four, it's going to be 28. So we make both of these fractions out of 28, we're going to multiply that by four. So multiply that by four and then get 8/28 Now three multiplied by seven, that's going to give me 21. So I now need to do 8/28 subtract 21/28, that's going to be negative 13, isn't? It's a negative 13/28. A little bit sneaky to make that one a negative, but nothing wrong with that. It can happen. How did you get on? Okay, now for the last activity. Using the digits four, five, six and eight, just once each, put them into these four fractions. So I'm going to see if you can make the greatest possible answer and the smallest possible answer for both the additions and subtractions. Pause the video and have a go. When you're ready, you can unpause it and we'll go through the answers together. Pause in three, two, one. How did you get on? These are my answers. Hopefully got the same ones. That's the last bit for this lesson. We'll see you soon. Goodbye.

Adding and Subtracting Fractions (Part 3)

Adding and Subtracting Fractions (Part 3)

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UnitMaths / Manipulating and calculating with fractions

Maths