Fill in the blank: 15 x 16 = 15 x (___ x 8)

Fill in the blank: 15 x 16 = (5 x ___) x 16

Fill in the blank: 36 x 25 = ___ x 100

Fill in the blank: 55 x 20 = ___ x 100

Fill in the blank: 2.2×50 = ___ x 100

Associativity

﻿Hello, and welcome to today's lesson about the distributive property. For today's lesson, all you'll need is a pen and paper or something to write on and with. Please take a moment to clear away any distractions, including turning off any notifications. Finally, if you can, please try and find a quiet space to work where you won't be disturbed. OK, when you're ready, let's begin. OK, time for the Try This task. I'd like you to pause the video and have a go. Pause in three, two, one. OK, welcome back. Now, I'm not sure exactly how you worked it out, but I'm going to show you the way that I did it, OK. And it doesn't matter if you've done it in a different way, but this is the way that I kind of want you to get understanding from today. So can you see this? This is 24. Now, what I've done here is I have split this calculation into 20 and 4. So it's 5 times 20 plus 5 times 4. Now, why have I done that? Well, one of the reasons why is because it's easier. And how can I do that? Am I allowed to do that? And, yes, I am. And that is a thing called the distributive property, and that is what we're going to be learning about in today's lesson. OK, so 5 times 4, 20, and then that gives me a total of 120. OK, next one, well, what I'm going to do is I'm going to split this 16 and make it nice and easy for myself into a 10 and a 6. Now, you could have split it in lots of different ways. You could have done it in 11 and 5 and 13 and 3, but I put this 90 there on purpose because I wanted you to split it into 10 and 6, OK. Even though you can split it into lots of different ways, it's often easier if we split it into a multiple of 10. OK, so at 9 times 6, I have 63. Nope, not 63, 54. And add them together, 144. Split that up. OK, so I've split the 15 up into 10 and 5, and the 6 stays the same. So I get 60 plus 30, and I get an answer of 90, OK? All right, why does this work? Why can we do this? And I'm going to show you why using this, an array. OK, can you see 32 here? OK, yeah, we've got 8 and 4. OK, that gives me 32 cubes. Now, I've split it into groups, a group of 2 and a group of 6. So 8 is equal to 2 plus 6, so I can split it up into a 2 and a 6. And can you also see this, 4 times 2, here, plus 4 times 6, here? So 4 times 2 plus 6 is the same as 4 times 2 plus 4 times 6, and that is called the distributive property. OK, distributive, practise saying that word. It's very important, distributive. OK, so they both equal 32. Can we partition the array in a different way? Well, yes we can. We can do it in lots of different ways. What about if I did it like this? That one and that one over here, what calculation would that be? Have a think. Hmm, well it would be 4 times 3 plus 5. OK, we've got our 3 and our 5, which makes 8. Or we can do it as 4 times 3, this group, plus 4 times 5, this group. OK, we could have done it that way. We could have done it, we could have done 6 and 2, going the other way. We could have done 4 and 4, or we could have even done it this way, this as our group. Well, that, what's going on here? Well, I've got a 2 plus a 2 times an 8, which is the same as 2 times 8, the top group, plus 2 times 8, the bottom group. So there's loads of different ways that we can partition numbers to change our calculations, and this property of changing one multiplication into the sum of two multiplications, or more, we could do it in more ways, that is called distributivity. In fact, I'm going to show you in more ways. All right, let's do one. Go on, so now I'm just going to change this one and split that group like this. So, here, I've got 8 times 4, all right, 8 by 4. Now, I can split the 8 up into a 2, a 1, and a 5, and that is the same as 2 times 4 plus 1 times 4 plus 5 times 4. OK, so we can use distributivity to split it up into lots and lots of different ways. OK, so I'd just like you to have a think what is the same and what's different about these arrays. So pause the video and have a think. Pause in three, two, one. OK, so hopefully you've had a think. What is the same? What is different? Hmm, well, what's this length here? 15. What's this length here? 15. That length is 20, and that length is 5. So what's this length? 15. OK, these, this area here and this area here and this area here, they are all the same. They have just been partitioned in different ways. Now, I probably wouldn't partition the one like this because that would make the calculations really tricky. This one might make the calculations easier, and this one might make the calculations easier. And it's just important that you know that it can be split into so many different ways and they will still have the same area. So let's write what these calculations would represent. So, here, I've got 9 times 8 plus 7 which is the same as 9 times 8 plus. OK, so the area of 9 times that length is the same as 9 times 8 plus 9 times 7. And for this one, the next one, we've got 9 times 3.6. Don't, come on. That area plus that area is equal to 9 times the big length. And it doesn't matter that I did this one first that time and did that one first that time. It's just about understanding that there's two different ways that we can work out the area based on this picture, based on this diagram. And this one, what about this one? Hmm, well, there's going to be a take away, here, or a subtraction. What about this? Can you see this? 9 times, OK, because that length is 20 subtract 5, or we could do this. So distributivity, it works for subtraction as well. If this has a subtraction, in there, that is fine. Now, I want you to understand that these calculations are the same, and I want you to have a practise of trying to write two equivalent calculations based on arrays. OK, so it's time for the Independent Task, now, so I'd like you to pause the video and have a go. Resume once you're finished. OK, so here are my answers. You may need to pause the video to mark your work. And just on this last one, which way is your preferred method? Which way do you think was easiest? Hmm, I quite like the first one because we've got something times 10, and I think the 10 times table is quite easy. But I also like the last one because I think 5 times 20 is quite nice and then 5 times 3, I think, is easier than 5 times 7. So, for me, it'd probably be either the first or last one, but you might have preferred something else, and that doesn't matter. It doesn't matter which you prefer. They will all get the same answer. What is important is that you can understand that we can break down numbers in this way, and we can split one product into the sum of two products. OK, so now it's time for the Explore task. I'd like you to pause the video and have a go. Pause the video to complete your task. Resume once you're finished. OK, now, you may have got something slightly different to this, but this is the way that I thought they might have worked it out. So I thought Anthony might have done this. He might have done spit the 36 into 30 and 6 and worked it out like that, and the 64 like this, and then added those two products together. However, I really like Carla's way. She's saying that 36 plus 64 is equal to 100, so she's just done 100 times 8. And look how much easier that's made her calculation. Look at all this working compared to this, and it's quite powerful. And we start to use this idea as well when we move to algebra. And I just think it's really important that you go and think about it and try and understand it. And see if you can, maybe see if you can make a calculation like this. See if you can make a calculation that might trick someone. What about if you had decimals in it, something like this? 3.6 plus 6.4, why would I choose them numbers? Have a think about that, OK. And that is all for this lesson. All right, hopefully you enjoyed that, and hopefully it all made sense. And I very much look forward to seeing you next time. Thank you.

Fill in the blank. 8 x 56 = 8 x (___ + 6)

Fill in the blank. 4 x 26 = 4 x 20 + ___ x 6

Fill in the blank. 13 x 7 = 11 x 7 + ___ x 7

Which shows the distributive property

Which shows the commutative property

Which calculation is shown using the distributive property represented by the array

Distributive Property

In this lesson, we will learn about the distributive property and how it can make calculations easier.

Maths

Y7 Maths

Key Stage 3

Year 7

The distributive property

Axioms and arrays

Number

THEME

UNIT

Models of multiplication

Array models

Multiplication and division

Multiplication facts

Number pyramids

Number talks

If I know 209 ÷ 11 = 19, then I know...

Which calculation is not in the same fact family

Addition is commutative

Two numbers can be divided in any order

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

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

a x b = b x a

Lesson overview:The distributive property

Core Content

Axioms and arrays: