# Lesson video

In progress...

Hello, my name is Mrs. Behan.

And for this lesson, I will be your teacher.

Together, we are going to explore the law of commutativity.

So basically, if we change the order of factors we're multiplying, the product will stay the same, but does it happen when we divide? Let's find out.

Let's begin by going through our lesson agenda.

We're going to do a little bit of a recap on what commutativity actually is.

Then we're going to explore commutativity and division.

We then have a practise activity, and following that, there is an independent task for you to do.

And I know you'll be keen to find out how you got on.

So I will make sure that I go through the answers with you.

There are just two things that you will need for this lesson.

Something to write with, so a pencil or a pen and something to write on.

If you don't have those things to hand, just pause the video whilst you go and get them.

Remember to try and work in a quiet place where you won't be disturbed for the lesson.

Can you see the array? You're going to start by making some number facts based around the array.

I'll give you some clues.

commutative law, partitioning and recombining.

How can you use the law of commutativity, partitioning and recombining to come up with loads and loads of number facts? I'm going to leave that one with you for just a couple of minutes.

Pause the video here whilst you have a go.

Okay then, let's see how you got on.

If it was me, the first one I would come up with would be five multiplied by four, and that's because I can see five equal groups with four in each.

I'm then going to use the law of commutativity and change the order of the factors.

My story changes, because I've got this picture.

I now have four groups of five.

What I did then, was I put this line down the centre, so I have partitioned the array into two groups.

I now have a group of two multiplied by five.

I've got two columns of five.

And I've got the same on this side, but I'm going to write that in a slightly different way.

This time I've got five rows of two.

So, altogether, I can say five multiplied two plus five multiplied by two or two multiply by five is equal to 20.

I then thought, right, I'm sure I can define that set somewhere else, I can partition it.

So I drew the line here.

So you can see, I now have two groups of four on this side of the line and I have three equal groups of four on the bottom side of the line.

So, two multiplied by four plus three multiplied by four is equal to 20.

What happened here? That's right, I've applied the law of commutativity and changed the order of the factors.

So instead of looking at equal groups this way on rows I'm now going for columns.

So I have four equal groups of two, and you can see underneath I have four equal groups of three.

So, if I recombine them, four multiplied by two plus four multiplied by three is equal to 20.

So we're going to change our story now.

Read the words on the screen with me.

15 flowers, five in each vase, how many vases? So we know the whole, we have 15 flowers.

There needs to be five in each vase, but the unknown is how many vases.

The unknown is the number of vases.

So, we can think there's something missing.

The missing information is something multiplied by five equals 15.

We don't know how many groups, our vases group the flowers together, don't they? So we're missing that information.

So something multiplied by five is equal to 15.

Five should be in each group, but we don't know how many groups.

So multiplied by something is equal to 15.

Both of them give us the same product, because they are commutative.

So to work out that missing number we need to use the inverse operation.

Tell me what the inverse of multiplication is? That's right, it is division, but which division calculation is it that we need? 15 divided by five or five divided by 15? Which one is it do you think? That's right, 15 divided by five is our calculation.

Five divided by 15 would not give us our whole number.

When we divide, the dividend is usually, well, it's the largest number, and then we split it up into smaller groups, smaller parts, okay? This one is not going to work for this number story.

So 15 divided by five, what's the quotient? Well, the quotient is three, because there are three vases.

We've worked out our unknown.

So 15 flowers, five in each vase, there are three vases altogether.

This time we have a slightly different number story.

Say the words on screen with me, 15 flowers, three in each vase, how many vases? 15 flowers, three in each vase, how many vases? So, we can see here we have our 15 flowers and they have been grouped or shared out between the vases.

There are three in each vase.

So our missing number is going to be telling us the number of vases.

That's what we've been asked to find out.

So again, you can see we have a missing box here to represent our unknown.

We don't know how many vases.

We can see them on the picture, I know we can.

But in our calculation, we have the unknown here.

So box multiplied by three is equal to 15 or three, because that's our group size, multiplied by the number of groups, which we don't know, is equal to 15.

So we know it's five, don't we? We can see that in the picture and we know are fives and three times table.

But if we're relating this now to our calculation 15 divided by three equals five and 15 divided by five equals three.

Both of these calculations have been calculated correctly, but which one do we use for this story? Have a think, which one do we use for this story? We need to use 15 divided by three equals five, because we have 15 flowers, we know there are three in each vase, and we didn't know the number of vases, but now I've calculated it, we know there are five vases.

15 divided by five equals three is the correct fact but that doesn't match with this story.

We can't change them 'round like we can with the multiplication.

It's a little bit more complicated.

So let's just take a look at the two stories that we have looked at so far.

We started with 15 flowers in both stories.

We started with five in vase, how many vases? The missing number was three.

We didn't know the number of vases here.

So three times five equals 15, five times three equals 15.

15 divided by five is equal to three.

Okay, we knew that we had five in a group.

So we needed three groups.

Up here three in each vase, how many vases? We didn't know the number of groups.

With them, we found it out that it was five multiplied by three is equal to 15, because we know our number facts.

The division that we use here was 15 divided by three equals five, because 15 was the whole, there were three in each vase, so we need five vases.

So look here, factor multiplied by a missing number will give us a product.

So whenever we're multiplying, it will give us a product.

Think about what that product has been.

Product when we multiplied was 15.

When we had the missing number at the start, so box multiplied by factor equals product, the product again was at the end, the product was 15.

So, three multiplied by five equals 15, five multiplied by three equals 15.

So the product was still the same at the end.

When we looked at division just have a look at the number fact here, 15 divided by five equals three, 15 divided by three equals five.

What do you notice about this number here, the dividend? Did you notice that the dividend, which is at the start of the division equation is the same number as we got for the product? The dividend divided by the divisor is equal to the quotient.

But we notice that the product and dividend are the same number.

So here's a summary of what we have learned.

The product in the multiplication equation has the same value as the dividend in the matching division equation.

Say those words with me, the product in the multiplication equation has the same value as the dividend in the matching division equation.

So we're starting to realise that division isn't commutative like multiplication is.

We've already looked at a grouping example where we had to group flowers into vases.

Let's have a look at a division by sharing example.

So, on your screen you can see 12 conkers.

These 12 conkers are shared between four children.

Think about yourself and three friends that you would share these conkers with.

How many conkers would they have each? So, if we looked at a multiplication example here, box multiplied by four is equal to 12, or four multiplied by box is equal to 12.

Let's just think about what each of those numbers and boxes represent.

So the box here represents the number of conkers that the children have.

So it's the number of conkers multiplied by four children is the total or the product of 12.

This time we've used the law of commutativity to change it 'round.

Four children multiplied by how many conkers they have each is equal to 12.

So to work out that missing number we're going to use the inverse.

The inverse of multiplication is division.

So 12 divided by four, when we've worked it out 12 shared between four children means they each have three.

When we multiply, we know that we can change the order of the factors and the product will still remained the same.

So what happens when we divide, if we change the order of the numbers? So we know that each child would have three conkers.

We saw that on our previous screen.

So let's look at 12 divided by three, what's that equal? Well, 12 divided by three is equal to four, but that would tell us that one child has four conkers.

Four is our missing number, because we have four children, and we know they each have three conkers.

So when we look at the multiplication, this works for us.

But 12 divided by three is equal to four.

This in this story is telling us how many it's been shared between.

Well, we're not sharing between three children, we're sharing between four children.

So you can see that if we change the order of the numbers, our story becomes different.

Look at each example, side by side.

You might want to pause here, just whilst you have a look.

This example here, 12 conkers shared between four children, how many conkers each? You can see we have four children and they each have three.

So this example is correct to match this story.

12 conkers shared between four children, how many conkers each? So if we have 12 and we put the three here as the divisor, we only share between three children and they would each have four.

So, even though we're using the correct numbers four, three and 12, have a relationship, for this story it doesn't do what we want it to do.

The calculation is not correct for the story that we need to look at.

So by looking at the flowers and conkers and multiplication and division, we can now conclude that multiplication is commutative but division is not commutative.

So we're coming towards the end of our lesson now.

But just before we go I just want you to have a look at these calculations.

We've got some multiplication calculations down here and some division calculations down here.

Just have a look, what's the same, what's different, and what do you notice? You might want to pause the video here whilst you have a little think.

Did you notice anything? Did you notice anything about the number zero? Let's have a look.

Zero multiplied by one is equal to zero.

Zero multiplied by two is equal to zero.

So, this factor here is zero.

What impact has that had on the product? The product is zero too.

And we know that because of the commutative law that if it was one multiplied by zero the product would also be zero.

Let's have a look on this side.

The dividend is zero, divided by one which is the divisor gives us the quotient of zero.

Zero divided by two is equal to zero.

Zero divided by three is equal to zero.

So if the divisor, no, that's not the divisor.

That is the, correct me, the dividend.

The dividend is zero, the quotient is zero.

But we can't change the order of these numbers can we? They have different purposes.

We can't change the order, division is not commutative.

But we can summarise that when zero is a factor, the product is zero.

When the dividend is zero, the quotient is zero.

Say those words with me, when zero is a factor, the product is zero.

When the dividend is zero, the quotient is zero.

That's a very easy fact for us to remember.

We're nearly at the end of our lesson and we're just going to look at how multiplication and division equations can match up.

So I'm going to start you off.

Have a look at the multiplication down the left-hand side.

15 is equal to three multiplied by five.

We need to match it to one of the division calculations over here.

I know that 15 divided by three is equal to five, so those two join up.

24 is equal to eight multiplied by three, that one connects to 24 divided by eight is equal to three.

Pause the video here and carry on matching up the equations.

Ready to go through the answers? I'm going to show them on the screen and you can check your own work.

Well done.

I'd like you to write a matching multiplication or division question to the one you have been given.

So I'm giving you eight times two is equal to 16.

So I'd like you to change that into a division question oh, sorry, a division calculation and write it in this box here.

20 divided by two equals 10, write the multiplication in there.

So we're looking at the inverse.

On this side I'd like you to think whether you need to multiply or divide.

So, to find the answer, would you divide or multiply? Write the correct symbol in the box.

Lucy had 20 stickers, she shared them between her five friends.

Does she need to multiply or divide? You can write the symbol in the box.

Lola bought three toys for each of her four dogs.

Does she need to multiply or divide? Write the symbol in the box.

Toby made bunches of five flowers, he used 20 flowers altogether.

Does he need to multiply or divide? Write the answer in the box.

When you're finished, come back to me and we'll go through the answers together.

Okay then, let's see how we got on.

So, we were asked to write a matching multiplication or division question to the one we've been given.

Eight multiplied by two is equal to 16.

We could write 16 divided by two is equal to eight.

And remember, the dividend must go at the start.

The product and the dividend are the same number.

These two could change.

But the dividend when we're rearranging the equation is the same number as the product.

20 divided by two equals 10.

We could write 10 multiplied by two equals 20.

What was the other option you could have had? Yes, you could have had two multiplied by 10 is equal to 20.

24 divided by two equals 12.

That can be rearranged to 12 multiplied by two is equal to 24.

We could change the order of the factors and have two multiply by 12 equals 24.

12 multiplied by three is equal to 36.

We can rearrange that as 36 divided by 12 is equal to three.

Now remember, our dividend has to be the same number as our product.

22 multiplied by two is equal to 44.

We can rearrange that to 44 divided by two is equal to 22.

All right, let's see how our characters over here got on.

She shared them between her five friends.

Did Lucy need to multiply or divide? Well, we know that she had 20 stickers to begin with and she shared them out between five friends.

So this is a division question.

20 divided by five is equal to four.

So they each got four stickers.

Lola bought three toys for each of her four dogs.

Does she need to multiply or divide? Well, she needs to multiply because three toys for each of her four dogs would be equal to 12 toys altogether.

Toby made bunches of three flowers.

He used 21 flowers altogether.

Does he need to multiply or divide? Well, here he needs to divide, because he started with 21 flowers or used them altogether.

And he made bunches, the group size was three.

There were three flowers in each bunch.

21 divided by three is equal to seven.