As of today, just very, very quickly recap what all of those arithmetic types were. So starting off with addition, Jack, do you want to kick us off with what two fifths plus a quarter would be? But would that be job? 13 over 20. Fantastic. And that's all making sense. We find 20 as the first number that's in both the five and the four times table, the lowest common multiple. There's another word for that. And we just think about what we're times in both of the denominators. So we're times in five by four. That means we've got to times the two by four as well to create eight by the 20 and we're times in four by five. So that means we've got to times the one by five to create five over 20. Then and only then can we add the two fractions together when we get 13 over 20. Good Jack Stella. What would seven over eight. Minus five over twelve. Good. So just be careful there. Remember that your what you said there is that you're time ztop and bottom by three here and we're timzing top and bottom by two here. So what does that make the top numbers? It makes twelve, 21 and ten. Yep. Which gives us. Oh, which is eleven, very, very good. Eleven over 24 is absolutely right. And on multiplication, Jack, and this is going to be an answer that you can simplify. So watch out for that. Six over eleven times three. Now let's do two over three. The whom? Here, that's. Maybe it's eleven and four third 4:11. Absolutely. Well done, Stella for suggesting that interest. Notice here again, you don't have to do it this way, but it's it's nice to have to trick up your your sleif. You want to do it this way. You're also allowed only when multiplying to cross simplify is that little trick that we talked about last time whereby you're allowed to simplify the diagonals of a fraction multiplication. So if you spotted that 36 are both in the three times table, you can divide both by three and you get two there and you get one there. And that's just gonna to get you your answer a little bit more quickly and simply four over eleven, we get exactly the same answer, slightly different method, but it also works. Again, you don't have to do it this way, but I think it's good to know that it's an option cross simplification that's called good stelllar. Let's see if you can do this one here. Three over five. Divided by nine over ten. And again, this is going to be one that you can simplify. So I haven't gotten to an answer, so watch out for that. So just to remind you, the order of this is keep. So keep that one as it is. I saw this one and change any and okay, change. I saw this one. I want to keep change. Yes. Great. No, it cannot divide by two good. So how about 33 maybe? Four. I think they too, maybe. 56 yeequal. Gonna be three. Oh, it's gonna be 23. Good. Two over three is absolutely right. And again, nothing wrong with doing it that way. It's perfect if you do it that way. Again, it's just to demonstrate the alternative. Once you get to this point, once you've made it a multiplication question, again, if you spot that, you've got diagonals that can be simplified. So in other words, 39 are both in the three times table, and five and ten are both in the five times table. If you want to at that point, you can cross simplify so that one is a little bit hard. Yeah, kind well, I mean, 39, nice and easy 13, ten and five, pretty easy two in one. And then you'll just you get to that answer really quickly. Two over three. One times two is two. One times three is three. Again, nothing wrong with the way that you did it, Stella, if you're still there, just to demonstrate that there is a sneaky alternative if you can spot it. Okay. Maybe I'll just stit, can you just give me a sign that you're there? Are you are you okay? You are okay. Great. So we're going to move on to mixed number arithmetic today, which Jack has already done, but I think it will be a good chance to recap it for them. So first of all, what is a mixed number? Stellar? What do you understand by the term mixed number? I see number, I forget it. So mix number is very simply a fraction which has a value. Greater than one. Which is expressed as a whole number. And as well as a raction. On the other hand, an improper fraction, sometimes it gets called a top heavy fraction. But it's not really you know top heavy is we understand it makes more sense as a term, but it's not really the official way to describe them are fractions greater than one which are expressed as single fractions. And again, I'll just demonstrate this with actual numbers because I think it sometimes the worded explanations of maths are more complicated than a reality. But a mix number would be something like two and a quarter. That's a mix number because you've got a whole number and you've got a fraction next to it. If we do the same and I'll try and do the the equivalent improper fractions, two and a quarter as an improper fraction would be written as nine over four. Another one, do you want to give me an example of a mixed number, Stella? Maybe a. Sample. Yeah, three and a six. Great. Fantastic. And Jack, can you give me three and a six written as an improper fraction? What would that be? 36, 31, six, that number that Stella just wrote, what would that be as an improcrofraction? 31. How to do this? So it's. Good, excellent. 19 over six is correct. Let's just put that down there and maybe in blue to show that they are equivalent. So yes, so what's a really important skill is being able to convert from one to the other. Now just to make sense of this, if we imagine two and a quarter as pizzas, right, Stella? So imagine we've got two and a quarter pizzas. I'm going to, we'll draw three circles. I'm going to divide them up into quarters. And I'm just going to shade two whole pizzas and a quarter of the pizza. So two holes is going to be all of that, all of that, all of that, all of that. That's one hopizza. This is two hopizzas. And this is another quarter. Now we can see, can you see, Stella, that if we think of this only in terms of quarters, if I ask you how many quarter slices of pizza are in two and a quarter pizzas, well, we just count the quarters and we go, okay, we've got one, two, three, four, five, six, seven, eight, nine. Exactly. And that's why those two fractions are equivalent. Two and a quarter is the same thing as nine quarters. There are two different ways of saying exactly the same number. Yet go still los, that question. So like I think I know it's like two times four equals eight and then plus one equals nine and then the four here and then the four also like that exactly, exactly. That's all there is to it. Times the bottom number by the whole number. Add the top the the top number and Yeah the bottom number stays the same. You can see how that's works with this one here. Three times six is 18 plus the 1:19 over six. So let's just have a practice of that to begin with. So Stella, see if you can tell me what 32 thirds would be written as an improper fraction, as a single fraction, it would be what's over three. 1111 well done fantastic and Jack, what would twelve and four over five be written as? An improper fraction. Four over five Yeah twelve and four over five. I know. Let's you want to correct five? Good, 64 over five is absolutely right. Yeah. Is that making sense, Jack? So we're just doing the same thing, converting from mixed numbers to improper fractions. Remember, as you did up here perfectly, 31 sixth became 19 over six. This one, we're times zintwelve by five, and we're adding the four to create 64. And the denominator stays as it is. Really, really nice work. Sta, and what about eleven and nine over ten stelllar as a improper? So this is going to be. Yep, just be careful there. So eleven times ten is what? Oh, it was 110 and it was nineteengood. So 119 over ten is correct. Well done. Fantastic. So that's that's really all there is to it. If you ever forget how to do it, just give yourself a really easy example like two and a quarter and you can just write out the pizzas go, okay, so two and a quarter is nine over four. So Yeah, I see what you do. You times the big number by the bottom number, add the top number. What happens? We're going to, again, we're slightly racing through material in these classes, but I don't mind because you're able to keep up stellar. What happens if we go the other way round? What happens if I say what's 20 over three written as a mixed number? Any idea how we do this and let's take again that easy? I know, I know. Okay, I know. Yeah so you need to 20 minus three first equal wait wait, wait wait wait wait, wait, wait maybe like you think what number can times and equal 20 but the things is. We need to times it like two number times times it and then plus one number equals 20. Yeah, absolutely right. So how close to 20 can you get into the three times? Oh, so you can do three times five plus five. Three times five plus five is works but you can get even closer than three times five. We want to get as close as possible. What we're thinking about what times three remember? Oh Yeah. So maybe three times eight. Three times eight is no, three times seven, three times six, six. Three times six is as close as we can get to 20. That's under 20 and it's 18 and f plus. Yeah. Good. Absolutely. Write 62 thirds is your answer. And you can see how we if we use the method that we just learned a second ago to go the other way back to an improper fraction six times three is 18 plus two, 20 over three. So works both ways. That's heder. So just to break down what you did, that seller, which was absolutely brilliant, really, really well, intuitive, we're essentially doing what is 20 divided by three. Ztwenty divided by three is six remainder. Two, six times three is the closest that we can get to 20, which doesn't go over 20. But importantly, we don't want to go over. And then we just we we note down what the remainder is. The remainder is this number here. And this number here is the answer for the division. So you just think what number times three, like very closed at like 20 like three times six because if you three times seven is 21 so it Yeah just can be three times six exactly exactly. And that's gonna get you back. So we'll do a couple more of those, Jack. Let's see if you can do 17 over four. What would that be again as a mixed number, Jack? No, I know. Good four and a quarter is absolutely right. Fantastic work Jack and Stella. What would 32 over five again? Okay, can I do on my board? Like I have my one light board. I will show you. Like if I finish course can work it. You can work on paper. You can work on your own thing. Okay, so maybe it's like five here and then here is six and then it's gonna be over here is two and then here is six well done 62 fits absolutely right. And Jack, what would 43 over seven be? Yep, nearly just be careful. Seven times seven is 49, seven times six is 42. So it's six and we've got the rest correct. Very good. Well done, shaack and Stella. What is 123 over twelve? Again, written as a mixed number twelve times. Okay, I know twelve. So here is twelve and then here is ten and then here three. Very, very it's actually very easy. Yeah, absolutely. And just notice that three over twelve is simplifiable again. They're never going to ask you to do this sort of your job just to spot it if and when it happens. So what would three over twelve simplify to a. Tiaway divide by three. Because this 1:12 is close like four and then this one is one perfect. So final answer, ten and a quarter. Yeah, very, very good. Stand again for a note to both of you. They're never they're generally not not going to ask you to simplify some questions will but often you know it's it's just your job to be really, really attentive when you get to an answer and go, can I simplify this, yes or no? It gets easier and easier the more you practice. All right, fantastic. So let's jump in to some mixed number arithmetic. Starting off with addition, which I think is probably the easiest. So if we have something like three and a quarter. Plus two and a half. My advice, there are some people that like to converts these questions into improper fractions, which is what you have to do for the multiplication and division questions. But my advice would be, for these ones, don't bother. You can really just split it up into two parts. You can just go three plus two. It's five, a quarter plus a half. And you've got to do a little bit of work here. I'm going to say the lowest common multiple of 42 is actually four. Four is in the four times table and the two times table. So that's a quarter plus two quarters, which is going to give me three quarters. And then I just add these two answers together. Got five there. I've got three quarters there. So my final answer is 53 quarters. That's what I'd recommend. If you want to, you can also express them as improper fractions and add them that way. It's more work, I think. And you're always looking for the I would suggest the least amount of work you can possibly do in that's economy of efforts is so, so, so important, especially as the going gets tougher. So I think that for me is the easiest way to do it. But again, just to take you through how you could also do it if you wanted. Some people like to do this. I don't. I don't personally, you could convert to improper fraction. So you could say three and a quarter is the same thing as 13 over four. And two and a half is the same as five over two. I'm now going to add those two fractions together again, common denominator of four. So the first one stays exactly as it is. Second one becomes ten. I add these two together, I get 23 over four, which is equal to four to 53 quarses. So I get the same answer. But I think it's more work to be honest, and especially sometimes adding these two numbers can be really, really difficult. And that denial inate are tricky. So not the way I do it. But if you want to do it that way, feel free. Jack, let's start. Look at that. Sorry. Yeah, I want take a Green shot. Sure. This is, this is the not the method that I would suggest stelllar, just to be clear, this is, this is one way you can do it and one way some people like to do it. But I think it's much, much easier to just add the whole numbers together. Go three plus two is five. And a quarter plus a half, we've already done it. Another going to do again is three quarters. And then just add those two together and get 53 quarters. It's way easier, way easier. So Jack, see if you can kick us off with two and an. Plus 33 quarters hmm hmm hmm. Where? Did I simplify? I don't think there is anything to simplify here. Yet, at least. Birds, very, very nice. And as you can see, Stella, what Jack did just there, he broke the question up. He did two plus three is five. And one eighth plus three quarters is really the same thing. If you think about it, the smallest number in the eight and the four times table is eight. So we can keep the first fraction as it is, and that becomes six over eight, which gives us seven over eight. And then you just put those two answers together, you add them together, and you get an answer of 57, eight. Okay. Oh, Oh, Oh, Oh, okay, I know. All right, see if you can do one now. First time you've ever done one of these. So don't worry if it goes wrong. Completely understandable. 22 thirds plus one and a quarter. Oh, wait, it's not going to be like that. This one is six. Eight. The common mondenomanswer here is what's the smallest number in the three and the four times table? Oh, this is five. So here is five. I see what you're doing. Okay, okay. Sure, sure, sure. If you want to do it that way, that's fine. Yep, because. Let me see three so this gonna be three us. So this will be eight. And Hebe, think about what the smallest number in the three in the four times table is. It's not three. What's the lowest common mons denominator of 34? I'm not. It's one. Let's just think about three times table. We've got three, six, nine, twelve, 15, 18, blah, blah, blah, blah, blah. Four, eight, twelve, 16, 20. You see the first number, the features in both twelve, twelve. So that's the number that we want on the bottom. Remember, we've got to have the so this twelve is like it's like eight can be Oh, like three can be times four equals twelve and four times three equals twelve, right? Exactly. So whatever you do to the bottom, you've got to do to the top. Oh, Oh conow. I know. Like Yeah okay, now it's like cure twelve yeabsolutely, right. Then it's going and up here is same number or we need to change. You absolutely need to change it. Otherwise you've changed the value of the fraction as long as you you if the bottom goes up and the top goes up at the same between, you haven't changed. It's gonna to be times or this one times this one. So it's gonna be it's times four times eight, eight equals 32 and then it's gonna be five times three. Twelve plus and 40. Well done. 47 over twelve. Absolutely. A notice that you've ended up with an improper fraction there. So depending on the question, you either want to keep that answer as it is or change it to a mixed number. Sometimes they might say, give your answer as a mixed number, in which case you go, okay, well, what is 47 over twelve? The closest I can get to 47 without going over 47. And the three times table is three times twelve, which is 36. So do three and eleven over twelve. But again, they might say, give your answer as an improper fraction, in which case 47 over twelve would be correct. So just read the question. Improper fraction. Either it is an improper fraction. Next number, lead. The next number. I just wanted to show you, Stella, because you did it. You did it the way the alternate methds, which is fine if you prefer to do it that way, do it that way. Everyone has their own preferences. But do I do think it's easier, kind of objectively easier, if you just split the question up into two parts? So first of all, you go, okay, what's two plus one? Two plus one is three. And what's two thirds plus quarter? Well, common denominator of twelve. So that's going to be three times four is twelve. So two times four is eight and this is going to be three. So I get eight over twelve plus three over twelve is eleven over twelve and I stick those two answers together and I get a final answer of three and eleven over twelve and I think it's just a little bit easier. There are there's there's that little bit less work to do. But that said, Stella, if you prefer this method, stick to that method. Do Yeah because if Yeah if if I like do two method, I will like mixed it up so you mix it up. Okay. All right. In that case, perfect, perfect stick to that method. So exactly the same thing is. Exactly the same thing is with minor things. So you can use exactly that method that you just use stellar to subtract mixed numbers as well. So Jack, so if you can demonstrate, do it whichever way you want to do it. Jack, seven and a quarter. Minus two and a fifth. What would that be, Jack? Yeah, just notice that this is a minus question now, Jack, not an addition. Yeah good. Five and a 20 if is absolutely White. And again, Stella, if you're interested in the way that Jack just did it there, it's not the way that you're doing it, but that's absolutely fine for you to do it. Two different ways. He just split the question up and he did. Seven minus two is five and a quarter minus a fifth is the same thing as five over 20 minus four over 20, which is one over 20. And he just added those two answers together to get five and mid twth really, really good workshop. Stella, over to you. What would and a third. Minus two and a sixth. Okay, so tobe here and then 13, always 13 here is nothing, and here -62 thirof same thing. And then equals, let me sing three times. Three times six equals just write out the three times table. Okay, okay. I know four for 44, four wait three k times four equals twelve and six can times two equals twelve two. Yeah but there's one even smaller than that. Look look at what's the first number to feature in both if you look at them. Oh, six, six. Absolutely. Yeah. Which is happy day is it? Six? Because we're lazy, aren't we? We we want to be lazy in maths if there's any excuse to not do any work at all. Perfect. If it's six I means we don't even need so here we've got a time action. So here we've got a times one. Yeah we don't even need to do any work for that, which is fantastic. That's too easy. It's going to be equal six. Six good easy, easy Yeah so just just be careful about the you had 13 here. So this was the one that you were times zing by two this one you were times zing by okay I saw I'm times two by six always makes data so then it's going be 26 and then here it's gonna be so so here is gonna be 13 that's 1314 here's 14Oh it's 13 okay so then that's gonna be 1313 over six is absolutely correct. And again, just to show me that you can do it, what would 13 over six as a mix number be? Well, okay, wait, equals a mix number. Okay, so two times six plus one. Good. Two and a six is absolutely right. Fantastic check. Ack. Let's do a harder subtraction one this time. This is where actually Stella's method can be a little more intuitive than the way that we like to do it. Jack, let's see if you can remember how to do this. Anyway, do it, do it however way you want to do it. Three and a fifth. -23 quarters. However you want to do it, check, you can just find a way to the final article. Yeah so that's not not actually right as it is. I mean, look, if we split up the work, if we do it, this is this is using the way that we like to do it, this is when it gets a little bit trickier. The reason why this question is trickier is because three quarters is bigger than a fifth. So we actually end up with a minus result when we're doing a fifth minus three quarters. So you can see that three minus two is one and a fifth minus three quarters. Common s denominator of 20. There's going to be four over 20, minus deaover 20, which can you see gives us a negative answer here. It's actually going to be minus eleven over 20. So when we're adding our two answers, kwe're combining one and minus eleven over 20. So we've got to think about what that is. What is one minus eleven over 20? Well, it's actually nine over 20, one minus eleven over 20, nine over 20. The nice way of thinking about that is, you know, if you have one minus eleven over 20, if you start at eleven over 20 and think about what you need to add to that to get up to one, it's obviously nine over 20. So that's doing it all way. But this is where stellla's way of doing it actually is really, really useful, because you don't run into that same awkward problem stelllar. So you can do exactly the same question and do it your way, and we'll see if you get to the same result. Have a. Go stokay. So three times five equals 16 and then plus one equals 16 and it's over five and minus two times four equals eight, eight plus three equals eleven. So eleven over four equals, let me say, five times two equals times. Okay, so it's all time like 20 minus one and it's over like five times. Here is four. So four, 50, 24 plus four t equals 64, 64. And here it's five, so 55. Then it's equal 26, 20 over Oh, it's gonna be nine here, nine over 20. So you can see we get to exactly the same answer and it is a little bit less awkward that way of doing it for these questions. So Jack, again, if you if that method is making sense to you, if it's making more sense to you than the way that we've done it before, feel free to use that method. Otherwise, what you have to do is account for the fact that a fifth minus three quarters gives you a negative result. So rather than doing one plus that, you're actually doing one minus that. Is that making sense? Jack? Me, let me know if it's confusing. Or not. Yeah, this makes sense. It's making sense. Good. Seeing as we've got two minutes and 40 odd seconds left, let's do a question together now. So if you can work, I don't know if you've got paper in front of you that you can use, or another whiteboard, maybe let's see if you can both work privately on this one again, whatever way you wanna use to get there, that's fine with me. I don't care about the method as long as you get the right answer. So let's do five and a sixth. -23 quarters. See if you can both get to an answer in the next to three odd minutes or so. Jacket, you can work on a piece of paper or something like that. That would be great. Just so you're not influencing each other's answers. We'll go through on the whiteboard in a second. Getting getting to an answer by the review. How we doing, guys, you got an answer, Stella. Your microphones turned off, Stella. I'm just asking if you've got an answer yet for this one. Yeah, Jack, has Stella your microphones turned off just in case you're speaking. I don't know if you're trying to. Have you, have you have you got an answer for this? Wait, I saw it. No, no, no. I was asking you both to do it, but that's all right. Jack, what did you get for this one? But let me check about everything. Have you on the computer? If you want to. Yeah. Okay. I to do this. Where can? Roy, you can write now, Jack, if you've got an answer yet, right it on the left, maybe. Thank you. Very good. Absolutely right. Both got the correct answer. Excellent. 29 over twelve or as a mix number that's going to be 25 over twelve. Good, good, good. Well done both of you. Again, some really, really great work done today. Have a wonderful rest of your Monday and I'll see you guys next week. Take care of yourselves. Bye bye.