How are you? Whereabouts in the world are you at the moment? It looks like you're sort of orbiting earth because you've got a very cool background on. But where are you? Even China. You in the uk? Yeah, I'm in China now. Very nice. And Jack, how's life with you? Good, good. I'm very, very glad. Well, I'm sorry about last week. I hope you got my message. I did try to send. I had Nora virus, and this class was right in the very middle of it. I was basically unable to get up from the bathroom floor. I was just lying there, my head against the cold. The cold flooring, it was deeply, deeply unpleasant, but I'm feeling much better now. So I think let's. Let's Stella we once try and paste things so that you're both kind of at the at the same level. There isn't anything that we're talking about which is is passing you by, so to speak. So we're going to just back up slightly and do a kind of little tour of some of the fraction arithmetic that we've been doing. I'm sure that you will know how to do a lot of this. We're just going na check that it's all all understood. So starting off with your fraction's. Not so good. Okay, great. Well, let some in that case, Jack, you can you can take the driving seat here maybe and we're just going to explain a few things and I think within no time you'll be able to jump in Stella and a few questions yourself. So starting off with addition and subtraction. So the important the really important idea about addition and subtraction is that. The denominator do you know what the denominator means, Stella? I. Think I know this words means let me it's it's just it's just the let like Yeah Yeah Yeah like here here's number and the the button here's this Yeah Yeah Yeah good absolutely right. So importantly, the denominator of both fractions has to be the same. When we're adding or subtracting, it's not the case when we're times zing or or dividing. But when we're adding and subtracting, bottom number has to be the same for both. So for example, you know two over five plus one over five is three over five. But you know two over seven plus three over eleven we we can't. Add until we find a way of creating the same denominator. So sometimes you're going to be given these questions, which is super easy. You can see how that works. It's just the bottom number doesn't really change, but the top number does. When you're given something like this, where the bottom numbers are different, we've got to be clever about and we've got to find a way of creating a bottom number which is the same before we add the fractions together again. I imagine you've come across this idea before, Stella, but let me do stop me if if I'm saying anything that doesn't make sense or what needs more of an explanation. So to find or to create the same denominator. We want to think about what the lowest common multiple of both numbers. Now any idea what the lowest common multiple means? Stellar. No, Jack, can you give an explanation of what the lowest common multiple is? So. The loads coming on. But it's a relatively simple idea. It's a complicated expression. The lowest common multiple is the first number or the smallest. To feature in two or more. Other numbers, times tables. So what I mean by that, again, just to simplify that idea, so for example. If you were asked what's the lowest common multiple of 69. You could think about. The six and the nine times table. And why don't we write a few out? So for the six times table, you know we've got six, twelve, 18, 24, 30, 36, blah, blah, blah, blah, blah. And for the nine times table, we've got nine, 18, 27, 36, blah, blah, blah, blah, blah. Now already having just written out a few numbers, you can see that they have what we call common multiples. Actually 36 and 18 are both numbers which feature in the six times table and the nine times table. But importantly, the lowest common multiple is 18 lowest just meaning the smallest. So. And then pick the smallest number to feature in both homtables. Which in this case is 18. Does that make sense? There's an. Idea. Great. So why don't we just do a little bit of practice of that alone? If I say jwhat is the lowest common multiple of 46. Oh, I think I know. Yestella jump in. Is it two? So two is the highest common factor, which is slightly different. The lowest common multiple. Again, you want to think about your four times table. It's like four, eight, twelve, 16, 20. You don't have to do that many for these questions. I'm not gonna to make them too hard and physiis it. It's twelve. Exactly. Well done. Twelve. And Jack, what is the lowest common multiple? Of eight and twelve. 24, 24. Damn right. Absolutely right. And so what that means is if we take this idea into an addition or subtraction fraction question, I could say, Stella, you know what is three over four plus one over six? And we look at that and we go, huh? Okay, we've got a problem at the moment. If it was three over four plus plus two over four, we don't have to do anything. We just add add the numerators together, you know, and we get five over four perfect here. We've got to do a little bit of work first. And what we do is what you just did stelllar in that we go, okay. So 46. What is the lowest common multiple of 46? What is the smallest number that is in both the four times table and the six times twelve? Exactly. So we're going to write it out as this. And having done that, we just think about what did we times buy for? What do we times four by, I should say, to get to twelve. What did we times four by twelve? Well, we times ed four by three to get to twelve, and we times six by two to get to twelve. So whatever you do to the bottom number, so four times three was twelve, we have to do to the top as well. So on the top we're going to do three times three, which is nine. And here we've actually got two times one which is two, and now we can add these two together and we get eleven over twelve. So this is that's a really important step, Stella. So the thing that you're already getting, which is fantastic, is write out the sum again and write the lowest common multiple as the bottom of the fraction, as the denominator. So something over twelve plus something over twelve, then you need to go back here and you need to go, okay. So a four has become a twelve alders. A four become a twelve. A four becomes a twelve by times, zing hit by three. So that's what we're doing to both top and bottom. And the reason why we can do that is because three over four is the same thing as nine over twelve. They're the same number. They're just expressed in different, slightly different ways. But you haven't actually changed what the question is asking you. It's just another way of writing the same number, and you just do the same thing to the second fraction and then you're done, Jack. See if you can do the next one. What would five over eight. Minus one over twelve b. Good. Excellent, well done. And you can see how that works. Stella, again, what jdid, as he rewrote the suwe already figured it out. 24 is the lowest common multiple of eight and twelve. So he put over 24, over 24. Then he went back and he said, okay, how did how did eight become 24? We times eight by three to get to 24. How did twelve become 24? Wetimes twelve by two to become 24. So whatever you do to the bottom, you do to the top. So five times three, 15, one times two, two, 15 over 24 minus two over 24 gives us 13 over 24. Making sense. Yeah, great. So let's do a few more of those again. What is stelllar? What is the lowest common multiple of 68? Oh, six, seven, eight. Wait, let me think, okay? I know I. 呃。Good. Absolutely. 24. Bang on. Good. So what about if I now ask you what is five over six. Minus one over eight. See if you can do this. Es ist? Good. Yes. Excellent. Very nice. Absolutely right. Fantastic work. Stellar. Really, really really really impressive. Brilliant. And Jack, what about. Five over nine. Plus five over twelve. 8。That's 24 in the nine times table. Jack is 24, definitely the lowest common multiple of nine and twelve. Good. Good. Nice. Well done. Absolutely right. So hopefully, that's making sense. We're going to be we're going through all of this quite quickly today. But eststella, you're a quick learner, and I think from what I can see, that's already making sense, which is which is fantastic. So we move on to actually what's a lot easier, what's a lot easier. So when multiplying fractions or dividing. You don't need to worry. About the denominator being the same. Multiplication of fractions is in facts much easier. Than additional subtraction. And that's you know, You do it in exactly the way. It feels most intuitive. It's the top number. Gets times ed or multiplied. By the top number, the bottom by the bottom, and it's as easy as that. So when whenever you see a multiplication question, the fractions, you can thank your lucky stars. It's a lot less work. Generally there are some some exceptions to this, but generally a lot less work than additional subtraction. So if you get given something like three over seven times, one over four. It really is as easy as just going three times one is three, seven times four is 28. And that's my answer done. Don't need to find a common denominator, don't need to change the fractions at all. You can just go top, top, bottom, bottom. So it's really, really simple, really simple. So simple in fact, that I'm gonna I'm gonna to go on to division directly and then we'll we'll practice both. So division of fractions. Is similar to multiplication, but there's one added step. Which is. What we cool, keep. Change flip. So the idea behind this is if you get asked three over eight. Divided by one over three. Let's say we follow the rule. This first step of keep, change, flip. So keep means we keep it as it is. So I'm just going to write three over eight. I change the division to a multiplication. I flip the second fraction. So rather than one over three, it's going to be three over one. And then I do the multiplication I get. Three times three is nine, eight times one is eight. And that's my answer. Nine over eight. So keep change, flip is all you've got to remember and then you don't have to remember anything because it's just a multiplication question and multiplication questions are the easiest by far. So let's do do a few of those. Stella, let's start off with you. Can you? Yeah how do you do three over seven times? Five over eleven? What would that be? Do I need to do like change and flate like dithings not for multiplication? No, that's only for division. Absolutely right. Fantastic. Very, very good. And Jack, how would you do five over six divided by. Seven. Over. Three. Right. Nice. So you can see there that what's Stella? What he's done is he's again, because this is a division question for division, you've got to do this. He's done keep change flip and that's given him 15 on the top six times. Seven is 42 on the bottom. Now really importantly, sometimes you can simplify your answer. What would 15 over 42 simplify, Jack, and why would it simplify. Any idea? Actually if we divide everything by three, 15 divided by three is going to be five and 42 divided by three. It's not the most obvious one, but it's gonna to be 14. So five over 14 should technically be your final answer. But if that's not making sense, Stella, don't worry too much about it because we're gonna to come back to that idea in in in a bit. Stella, why don't you do a division question yourself now, can you do three over five divided by. Eight over eleven. Let me write here first and then. Good. Absolutely right. Fantastic works that were really, really, really good. Keep change, flip. Absolutely right. And Jack, see if you can do this multiplication question again. Just be careful about simplification. At the end there will be the opportunity to simplify. If you can spot it, let's do three over five times, eight over nine. Can this simplify? Can you find a way to simplify? Good. Absolutely right. So three is the common factor. Good. 45 divided by three. It's not five. It's. It's 45 divided by three. 15. It is 15. Well done, Stella. Absolutely right. So eight over 15, our final answer, very, very good. So and I imagine it sounds like, again, selyou're already understanding this somewhat. And simplification is basically the opposite idea to what we're doing here in the addition and subtraction questions. With these ones, we're taking a simplified fraction and we're creating an unsimplified fraction out of it. Nine over twelve is the same fraction as three over four. It's the same number. They're completely equivalent. In the same way that two over twelve is the same fraction as one over six. They're the same. There are two different ways of saying the same thing. These are the simplified versions. Because you could do the opposite journey. You could go, okay, well, if I divide top and bottom by three, I get three over four. Likewise, if I divide top and bottom by two, I get one over six. So as long as you do the same thing to both parts of a fraction, if you times top and bottom by five, if you divide top and bottom by two, you're not changing the value of the fraction. The fraction stays the same. And we can demonstrate this really easily if we think of an a nice simple example like you know, we could say one over two, which is a simplified fraction, I'm gonna to create an unsimplified fraction out of that by times in top and bottom by three. So three over six. Now we know that these are the same. It's the same number, one over two, three over six, same thing. The one on the right is unsimplified. The one on the left is simplified because the numbers are as small as they could possibly be. And we could represent this. You know, we if we drew a little circle. And we halved it. And we shaded in one half. You can see that that's that. And if we did the same thing over here, if we took one circle and we divided it into sixth. So something. Like this. Kind of more or less not very good, not very well drawn. Try and make that better. Something like that more or less. Yeah. I need to exactly so and it's the same, it's the same thing. It's still a half. It's just another way of saying a half. If you if you think of that as a pizza, the only difference, there's no difference in the amount of pizza you're eating. It's just more the way that the slices are arranged. One is cut into sixth, the other is cut into halves. They are the same thing. It's still the same amount of pizza. Does that kind of make sense? So this is where simplification comes in because if you're given or if you get an answer, and annoyingly, you're never going to be asked to simplify the maths. Generally sometimes they might, but generally when when when you're doing longer, more complicated questions, they just expect you to simplify. If you get an answer, which is unsimplified you've just got you've got to know that that's part of your job to to simplify. So if you get something like 24 over 30. You've got to over look at those both numbers and start going, okay, 24, 30. Is there a number that I could divide both of these numbers by to create two smaller numbers? And you can either do it, you can either do it in one go or you can ship away at it. It's absolutely fine to chip away at it as well. So you could you might look at that and go 24 and 30. I know that 24 and 30, both in the six times table. So I'm gonna to divide both of those numbers by six and I'm gonna to say that's the same thing as four over five, which would be correct. Four over five is the simplified version of 24 over 30. Again, they're not different numbers. They're the same number in facts, exactly the same. But four over five is simplified. The numbers are as small as they possibly can be. Or you can chip away at it and there's nothing wrong with chipping away at it. Chipping away at it is going, okay. I know that 24 and 30 can both divide by two so I'm gonna to diviise both of them by two. That gets me to twelve over 15 and I know that twelve and 15 are both in the three times tables so I'm going to divide both of those by three. And then I get four over five and you get the same answer. One is not better than the other. Sometimes it's much you kind of have to chip away at it, especially when the numbers are really big because you're like, I don't know, I don't know what you know number goes into 144 and 168. So it makes more sense to to go, okay, well, I'm just going to divide them by two and then I'm gonna to chip away bit gets hopefully that's making sense as an as an idea Jack jump in. Can you can you simplify this? 30 over 45. What would that simplified be? Nice. And can you go even further than that chip to weit? But I think you can go even further. Lovely. Good. Well done Jack. Absolutely right. And you can see Jack chipped away there. He said, I G, I know 1345. I know that they're both in the five times table so I'm gonna to divide by five. First I get six over nine then I'm gonna to divide by three and I get two over three. You could if you spotted it, it's a hard one to spot, go, okay, I know 30 and 45 are both in the 15 times table, so I'm going to divide both by 15 and I get two over three in one go. But again, there's nothing wrong with this. And I think it's a really nice thing to do just to just to break it up slowly. Really, really nice stelllar. See if you can do 18 over 24. What would that simplify to? Very, very nice. Can you chip away? Yes, good, Stella. Lovely. Three over four. Absolutely way. Fantastic. And again, nothing wrong. Really nice just to chip away that you divided by two first and then spotted that nine and twelve are divisible by three. So divide by two, divide by three. Got to the answer. You could also get there and one by dividing by six, 18, 24, both in the six times table. And that goes you to three over four directly. So again, we're really moving fast and I really appreciate stelllar how quickly you're absorbing all of this new information. I know a lot is coming at you, but keeping pace absolutely brilliantly. So that that's all that's meant by simplification. Let's let's do ten minutes just recapping all of those ideas and then we're going to move on to a little bit of mixed numbers towards the end of the class. And then I think we only really need one other class and you'll both be sort of Yeah up to up to speed. So yes, Jack, what what is four over five minus two ninths? Yeah. Good. Well done, Jack. Absolutely, ray. We've gone all the way back to the start of the class, finding the lowest common multiple of 59, figuring out what the new fractions are. So Jack times this one by nine and this one by five, doing the same thing to the top four times nine, 36, two times five, ten, and then mining both of those fractions. Really, really nicely done. Stellar. What would three over ten. Let me write the better ten there three over ten. Plus three over 48. So. Good. Well done. Excellent. Brilliant Stella. So good. Absolutely right. Perfect. Could not have been done any more brilliantly. Very, very good. Shawhat is nine over eleven. Times four over five. 30, 655 damn writers. Very, very good and stellar. What is twelve over eleven times? Three over four. And just be careful. This is a slightly harder question. Just be careful to simplify your answer. If you can think for this one, you will be able to. Good, that's six over 44. Hey, you can check p away. So remember for simplification that the numbers should be getting smaller rather than I don't know if that's a three. Is that 312 that you wrote or is it a three or three over two? It's 300 tough way. I'm not sure what is this means. Like is it times twelve? 36 times twelve? Well, no. So so remember this when when you're simplifying, the numbers are always getting smaller. So if you have something like 20 over 25, you're going, okay, I'm spotting those are both in the five times table. So so that's going to be four over five. And remember, you can chip away at it so you can go, you can spot that 36 and 44 are both even numbers arenthey, so they have to be divisible by two. So we could maybe start there and go, okay, well, what happens if we divide top and bottom by two? Where would that get us? I remembered. So this one is. Nearly 36 divided by two is actually 18 but good good Oh 18, Yeah Yeah 18. 你如说。Good. Yeah 18 over 22 and just notice that you probably do the same thing again. Yeah exactly. Well done. 哎笑。Yes. Well done, Stella. Absolutely right. And just notice now that we've we've kind of we've run out of ideas, we can't get smaller than nine over eleven without the numbers becoming decimals. You know if we diviverted by two again, it would be like 4.5 over 5.5 and we don't want that. So that's that's as far as we go. We go to nine over eleven. Really, really good check. How would we do three over five divided by. Seven over eleven. So very nice and just notice there again Stella, we flew through these ideas quickly, but division is this idea of keep change, flip. So that's what Jack just did there three times eleven, 30 35 times seven. Oh sorry, I saw I saw I saw this is Yeah Yeah so that's how he got his answer of 33 over 35. Very good. Absolutely. So see if you can do your own now what would what would one over. Nine divided by five over six. The actually out versary let's do five over seven. Beautiful. Absolutely right. Seven over 45. Fantastic. And Jack over to you. This is gonna to be a question that you can simplify for. What would five over six divided by ten over nine b? And just again, remember to simplify your answer. It will be simplifiable this one. Good. Can you go even further? Yeah three over four. Absolutely. Very, very nice. Brilliant. So this has been a kind of round the world tour of fraction arithmetic. We're just going to introduce one little idea before the end of the class just in the last five minutes. But really, really well done Stello for for keeping pace so well. I know we've been moving far, far quicker than we would do normally, but that's just so that you know you and Jack are on the same same kind of playing field, so to speak. One really nice trick. I mentioned the fact that multiplication questions are the easiest Bylong shot in terms of just not having to remember much. You just go, okay, it's the top times. The top is bottom time bottom. Occasionally, you know in a mental arithmetic test, they might give you something like this. They might say, what is 240 over 567 times seven or 77 over. 2400, right? And you look at that and and you've got and theysay, you've got to answer that in a minute and you go, how I don't know. How am I meant to do 240 times 70. I mean, I could I could write it out, but it's going to take a long time. You know then I'm going to have to simplify. It's all taking a really, really long time. So I would advise you could do that. Yeah, I'm going to advise that you don't do that in a thing like this. This is just to demonstrate a really sneaky little trick when it comes to multiplication. And this is called something called, this is cross simplification. So cross simplification. Is a little trick that you can use when the numbers get tough. The multiplication questions. You can also use it you know when the numbers aren't tough. And it can be a really useful thing to do just to get you to it and answer quicker. I'll just show you how it works and then we're going to wrap it up today. So the way it works. Is to simplify diagonally. So what do I mean by that? I mean, you're looking at these numbers here. 77 and 567. And you're also looking at these numbers here, the diagonals. And what you're allowed to do is if you spot that, for example, 240 over 2000, let's start with that diagonal, we would be able to divide both by 240. If you think about it, we divided both by 240. We would end up with one over ten. So what I can do then is I can write one and ten in those positions with 77 and 576. It's a hard one to spot there, but 77 over 576, 67. So I can't speak today. They're both actually divisible by seven and that would give you eleven over 81, I think. So suddenly when I rewrite these, we've got something that is a lot easier to do because I just go, okay, one times eleven is 1181 times ten is 810. And that's my answer. There's a little bit of a weird idea. The numbers that I picked just then were too difficult, I think. But let's do let's do an example, which is a lot easier to do. So let's do something like five over, twelve times six over now, as you've already been doing, we could do this the regular way and we could just go five times six is 30 and twelve times 20. It's going to be Yeah nice. Thank you. Seller 240 perfect. We could simplify that. We could go what does 30 over 240 simplify to? Both divisible by ten, aren't they? Probably divide? No, wait, it's not gonna be. Oh, okay, I know. Good like Oh they they can divithree, so it's gonna be ten and 80. Yeah lovely. Maybe two. Yep or just ten, right? Ten and 80 are both in the ten times table. Yeah but you can divide by two. There's nothing wrong with that. Five over 40 goods. Can you go even further? 51, I mean five and 40, both in the five times table. Oh Yeah, Oh, Oh Yeah. One over eight, exactly. So you can do it that way and you get to an answer of one over eight. There's a little bit of simplification work to do. You get the answer. Notice that we could get exactly the same answer by using this weird cross multiplication trick across simplification, I should say, just by spotting the okay, if we look at this diagonal for six and twelve now, they're both in the six times table. So I could go, six divided by six is one. Twelve divided by six is two. Same thing on this diagonal. Five and 20. Five divided by five is one. 20 divided by five is four. I end up with one times one, one over two times four, eight. I got the same answer. So it's not essential this thing here, but it can really help you get out of a hairy situation when the numbers get enormous and you're like, Oh my God, I don't know what to do with this. The cross simplication trick can sometimes wipe. So it's it's not I'm not saying you have to do it, but it's worth just having up your sleeve. Alright guys, we've gone slightly over time and sorry for that. A big round of applause for the both of you. That was absolutely excellent. We did way more than we should have done to be honest, but you managed to keep pace brilliantly. Stelllar, very, very well done to you. Especially well done for you, Jack, for flexing your skills and showing what you've already leart. We're going to Carry on with fractions next time, looking at mixed numbers. And then actually, I think we're we're going to be moving on to percentage and and decimals and that sort of thing after. So have a wonderful rest of your Mondays, guys. It was lovely to see both of you and see you next time. Take care of yourselves. Bye.