Which of the calculations below is equivalent to 8 x (9 + 10) ?

Which of the calculations below is equivalent to 9 x 8 + 9 x 10 ?

Which of the calculations below is equivalent to (8 + 9) x 10 ?

3 x ( 11 + 4) = 3 x 11 + 3 x 4 shows what property?

3 x ( 11 + 4) = (11 + 4) x 3 shows what property?

Number Pyramids

﻿Hello. I'm Mr Coward and welcome to the final lesson of the unit on Axioms and Arrays. Today's lesson is on number talk. All you'll need for today's lesson is a pen and paper or something to write on and with. If you can please take a moment to clear away any distractions including turning off any notifications that would be brilliant. And if you can, try to find a quite space to work where you won't be disturbed. Okay? When you're ready, let's begin. Okay, time for the Try This task. So what you need to do, is you need to try and fill in these gaps such that you can make calculations that you can see with the dice. Now some of them you might not be able to see straight away. You might almost have to turn your head on its side to see some of the calculations and imagine moving the dice around. Or just have a go, okay? It doesn't matter if you can't get them all but just see what you can do. All right, so I'd like you to pause the video in three, two one. Okay, welcome back. Hopefully you've managed to get a few of them but don't worry if you didn't get them all because some of them are quite tricky. Okay, the first one. Six times five. Six times, brackets, five plus something. Now, if you group this and this together you've got six times five plus two. You've got five plus two six times. Okay one. That goes with that. That goes with that. That goes with that. So we have six lots of five plus two. Now the next one. Six times ten, subtract something. Well just imagine for a second that this one had five dots on it. Then it would be six times ten but we don't have five dots. We have three less than five dots. So we'd have to subtract six lots of three. Okay, what about this one? Well, we've got six fives and six twos. What about this one? Three lots of ten, plus something. Well if we group these together and those two with those two, and those two with those two. We'd have three lots of ten plus four. Can you see this one? Two times fifteen plus something. What if we group this way going like that? Fifteen add six. We've got two lots of fifteen add six. Okay what about this one? Something times five subtract something times three. Where's that one at? Well what if we said each of them was five. That was five, that was five that was five. If we said they all had five dots we could say it was twelve times five because there's twelve dice, so twelve times five. Well we've added on three dots to six of the dice so we need to take away those dots so we do take away six times three. Now there's a lot going on there so hopefully this picture will help. Okay, can you see both of them calculations? Can you see that they're the same? What about over here? Can you see both of those calculations? Can you see how they're the same? Or how are they different? What about this one, can you see that? What about this one can you see that? Okay so here. Yep, there we are. There's our fourteen, and we've got three lots. We've got six lots of five plus two. Or six times five plus six times two. Here we've got our six times ten subtract the threes. Or we've got our twelve times five subtract the threes. Okay, so we're going to use lots of different calculation strategies Like we did with the dice. But we're going to do it numerically this time. So, I'm going to work out eighteen times five and I'm going try to do it six ways, all right? Six ways, let's have a go. Let's see if I can do it. Okay, so number one. Well, I'm going to use this property. I'm going to split eight up - eighteen up into ten plus eight and then do that times five. What's that property called, can you remember? It's called the distributive property. Okay, so that's the distributive property says that one multiplication is equal to the sum of two multiplications. Or two or more. Well it just means that we can split it up. And fifty plus forty. Okay, next method. Can I use the distributive method again? What about this? The subtraction. A hundred. Okay. Can you think of a method using the associativity principle? Have a think. What about this? Like that? Oh, sorry. Not eighteen again. I want to do a two and five first. This is why putting on the brackets definitely helps. And so that's equal to ten. Ninety. Okay, can I do it a different way using the associativity principle? Yes I could. What if I do this? Split eighteen up into three times six and then do the six times five first. Well six times five is thirty. So I've got the same answer again. Okay, same answer, different answer. And you know, you could have split that six up you could have split that up into a three and a two. So you could have done three times ten which would have gotten you your thirty. Okay, another way. Starting to struggle now. What about this? Eighteen times ten times a half because ten times a half is five and I'll do my eighteen times ten first. Eighteen times ten is a hundred and eighty times a half equals ninety. And what I could have done here is I could have used also commutativity, and done I'm writing too fast for the pen to keep up. So eighteen times a half. Done that first because you could swap those around because multiplication is commutative so it doesn't matter. So we've got eighteen times a half. Nine, and then times ten which equals that's an awful ninety. Ninety. Finally, last way. Because you know, let's get - let's let's beat our goal of six. So, can you work out what's going on here? Eighteen times four plus. What property have I used there? The distributive property. And then I can say that this, and I'll use the associative property, eighteen times two times two plus eighteen times one, which is just eighteen. So what I'm doing here is I double eighteen. thirty six. Double it again. Seventy two. And then add on eighteen. That's if you really like doubling, you can do this way. So you've got four lots of eighteen, double it twice. Plus another lot of eighteen which gives us five lots of eighteen. And that is seven different ways that we can do this calculation, which is a load. And I'm just going to show you two of them visually, in a really nice visual way. Okay, so have a look at this. So here, Ben's Calculation. Eighteen times five. She's used, what property has she used? Associativity. And done nine times two times five and then doing the two times five first. And how does this diagram relate that calculation. Have a look. Have a think. Can you see it? What's this length? Eighteen. So what's this length? Nine. So she's cut that eighteen in half and moved it down. And the area has not changed, we've just moved it so that becomes a ten by nine rectangle which is an easier calculation. Okay, what's Zachy done? What property has he used? The distributive property. So we've got ten plus eight. And what length would that be? Five. So we've got five times ten. Fifty. Five times eight. Forty. So he's split his calculation up into the sum of two products. The sum of two multiplications. Okay, so independent task now. Now I want you to use the axioms. Commutativity. Associativity. Distributivity. To try and find an efficient way to calculate each of these. And don't worry if you got, like if you've got a different way to me. There's so many ways to do this. So you might even want to try even two or three different ways and see which is the most efficient and have that kind of conversation with yourself. Which one is better? Why? Okay? So pause the video to complete your task. Resume once you've finished. Okay, so here are my possible answers. And they are just possible answers because this you know, you might have found another way that you find easier than this but I just wanted to give you some suggestions. So pause the video and compare your answers to my answers. Pause in three, two, one. Okay, so now it's time for the explore task. Now some of these I don't think there's that nice of a nice efficient way to do them. So if you don't think there's a nice efficient way to do them you might put them in here and use written multiplication whether it be long multiplication whether it be the grid method. How ever you'd normally do written multiplication you would use that method. But some of them you think - you might think Oh, I've got a really cool strategy that will make this calculation much simpler and you'd put that here. I would use a different strategy. So you decide where you'd place these five. Now, I might put these five in a different place to you and that is fine. Everyone's got their different preferences so don't stress. There's no right or wrong answer here. It's just about thinking about it. My goal of this task is for you, when you get a multiplication calculation to think Do I have to use a written method or is there a cool strategy that I can use and make it into a different calculation that makes it easier. So I just want you thinking about it. So pause the video to complete your task. Resume once you've finished. Okay, and here is a suggestion. I didn't see a nice way to break down these but I thought these could be broken down or changed, manipulated, into other calculations. And again, I just want to stress that these are not the correct answers these are just possible ways that I would do it, okay? So if you've got something different that is completely fine as long as you had a good think about what different strategies you could use for each of the questions. Okay, and that is it for the end of the unit. Thank you very much for all your hard work. I hope you enjoyed it and I hope you've learned lots. If you'd like to, please ask your parent or carer to share your work on twitter tagging @OakNational and #LearnwithOak. Thank you very much.

This model represents a way to calculate 28×5. What has the model changed the calculation to?

a x b = b x a

a x (b + c) = a x b + a x c

Number talks

In this lesson, we will use the axioms to find efficient methods to solve multiplication calculations by using distributive, associative and commutative properties.

Lesson overview:Number talks

Core Content

Axioms and arrays: