Lesson video

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Hello, my name is Mrs. Beham.

And for this lesson, I get the privilege of being your teacher.

We are going to explore the law of commutativity in a little bit more detail or the commutative law.

You might have heard it called that.

Basically, we're going to look at multiplication and see if we can change the order of the factors and see what happens.

Okay when you're ready, let's make a start.

Let's begin by taking a look at the lesson agenda.

We are going to start by recapping some multiplication facts.

Then we will move on to exploring commutativity.

After that, we will have some practise time.

And then at the end of the session, there will be an independent task.

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.

You will just need two things for this lesson.

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

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

And remember try to work somewhere quiet where you won't be disturbed for the lesson.

This is called an arithmogon.

And an arithmogon is just basically a triangle and it's got some numbers in each corner of the triangle.

But then it's got some pink circles.

Those pink circles are needing to be filled in.

So the way we do it is we multiply two numbers.

So we take two numbers from the corners of the triangle, we multiply them together, and then we write the product of those numbers in one of the pink circles.

I'll show you.

So let's take the three.

If I get my laser pointer.

Let's take three and we'll multiply it by 10.

We're going to write the product here.

So tell me, three multiply by 10 is? That's right, it is 30.

Okay now if you got it right, then we're looking at this pink circle next we'll see.

So can you multiply three by five for me? We'll put the product here.

Three multiply by five is 15.

Well whew! We got it right.

And so our last circle is down here.

So 10 multiplied by five is equal to? Tell me.

I can hear you from here shouting 50.

Does it matter that we did 10 multiplied by five and not five multiplied by 10? Something we will explore later in the lesson.

So there are three arithmogons for you to have a go at completing.

Pause the video here whilst you have a go.

When you're ready, come back to me and I'll share the answers with you.

So five multiplied by four, the product is 20.

Five multiplied by three is equal to 15.

And four multiplied by three is equal to 12.

Let's look at the second arithmogon.

Six multiplied by nine is 54.

Six multiplied by four is 24.

And nine multiplied by four is 36.

I'm sure you've done a great job.

Let's look at the last one.

10 multiply by seven is equal to 70, 10 multiplied by six is equal to 60, and seven multiplied by six is 42.

Give yourselves a big smiley face if you got them all right.

Can you say the word on the screen? It says commutativity.

And this is because addition and multiplication adhere to the commutative law.

A law is a set of rules.

So the commutative law is a rule in maths, where numbers follow the rules.

The commutative law means that the order of addends or factors can be changed, but the result will always stay the same.

Let's have a look at an addition equation and the multiplication equation, and let's remind ourselves of the vocabulary to use.

So we're going to look at an array to put our equation into context.

So you can see here there are six spots or six counters.

And there is a divide after two of the spots.

So we've got a group of two and we've got a group of four.

So in an addition equation even, we can see we've got two plus four is equal to six.

This is an addend, the number two.

The number four is an addend.

You'll notice they have the same name.

So that basically is the name of a part that we're going to add together to create the sum of a whole.

So what happens if we add them in a different order? Does the sum change? Let's have a look.

So no, it doesn't.

So I've switched the order of the addend.

We've now changed it.

So four is first added to two, and then we still have a sum or a total of six.

So as you can see, if you change the order of the addends, the sum remains the same.

Let's see if it's the same for multiplication.

So it's the same array which will be coming up in a moment.

Two multiplied by three is equal to six and three multiplied by two is equal to six.

There is the array.

So we can see it as two equal groups of three, or we can see it as three, two times.

So let's look at the vocabulary.

Number two is a factor, number three is a factor, and six is the product.

So when we've changed the order, we know that the product has stayed the same.

And again you'll notice just like the addends have the same name, these both are called factors.

So these two sets that we multiplied together are called factors and they create a product.

So what have we learnt? Well, if we change the order of the factors, the product remains the same as well.

Okay look at this picture.

We can see some flowers in a vase.

You might want to count the vases.

You might want to count the flowers.

Can you write two multiplication equations? I'll give you a moment just to do that.

We managed? Okay then.

So you might've written three multiplied by five is equal to 15 because three groups of five are equal to 15.

Where can we see the three? Well, we can find three because we have three vases.

Those are our groups.

And the five is found in the group.

So we have five flowers in each vase.

So three groups of five are equal to 15.

You may have also written five multiplied by three is equal to 15.

What's the same about these multiplication equations? Well, the factors are the same, the product is the same.

What's different about the two equations? That's right, the order has been changed.

So we started with three multiplied by five.

In the second equation, we've got five multiplied by three.

But it depends on how you are telling the story.

This one, three groups of five equal to 15.

This one over here, Five, three times is equal to 15.

Five, three times is equal to 15.

So we can see here that we've put the equal sign between our two expressions.

Five times three is equal, sorry, three times five is equal to five times three.

So we know that they are, what's that word beginning with C? Commutative.

Well done.

Can you say the multiplication that's on your screen? Three multiplied by five is equal to 15.

Okay, let's have a look at this statement.

Three groups of five are equal to 15.

I just want you to think about that for a moment.

Maybe get a picture in your head of what three groups of five look like.

Now on the screen you can see it says five groups of three are equal to 15.

Five groups of three are equal to 15.

What does that look like? If you're struggling, I've got some pictures coming up in a moment for you.

Another way of thinking of five groups of three are equal to 15 is three, five times is equal to 15.

So three groups of five is equal to five groups of three.

Okay so have a look at those pictures.

Is that something like what you imagined? We can see three groups of five flowers is 15 in total.

But over here, five groups of three are equal to 15, and three, five times are equal to 15.

So it's important that in this one, we know that the number in each group has changed.

We had five in each group over here, and in this example, we've got three in each group.

So even know we might be able to use the commutative law and change the order of the factors, we've got to make sure that we think carefully about the number story or the examples that we look at.

Here we've got five multiplied by three which equals 15.

So let's have another look.

Three groups of five equal to 15.

Five, three times is equal to 15.

So each group size, what is the group size? That's right, the group size is five here.

The size of each group is five.

So the group size is five and there are three of them.

But if we had five groups of three are equal to 15, does that tell us a different story? What's the group size here? Well, the group size would be three, and there's five groups of them.

So there would be three in each vase.

Let's see if we're right.

So three groups of five, can we see the three groups of five? Three vases each with five flowers in them.

Now on this side, five groups of three, the three is inside the vase.

So three groups of five is equal to five groups of three.

So even though the story is different, when we use pictures, it tells us something different.

Actually the product stayed the same.

So even both ways we did it, the product still stayed the same.

Have a look at these two expressions.

And I want you to just think what's the same about them and what's different? Perhaps you want to pause the video here and talk to somebody in your house maybe, or just have a little think to yourself.

So again I want to know what is the same and what is different.

When you're ready, come back to me.

Okay then, so you've had time to have a little think.

What did you notice was the same? Well, I noticed that both of them need multiplying.

We don't have any division or addition or subtraction.

The operation we're going to use is multiplication.

Did you notice anything else? I noticed that we have a two-digit number in each example.

14 is a two-digit number and 12 is a two-digit number, and they're both being multiplied by a one-digit number.

So we've got a two-digit number multiplied by a one-digit number in each example.

But what's different? The value of the two-digit number is different and the value of the one-digit number is different.

So here the value is 14.

We've got one 10 and four ones.

But in our two-digit number over here, we've got one 10 and just two ones.

So I wonder whether both of these expressions will give us the same product.

Should we work through them and see? Let's go.

Okay first of all, we will partition number 14 into 10 and four.

We can now do 10 multiplied by two, and we can do four multiplied by two.

So 10 multiplied by two is 20, four multiplied by two is eight.

Can you remember what we've got to do now with our products? We need to recombine them.

So we recombine 20 and eight, which gives us a total of 28.

So we know that 14 multiplied by two is equal to 28.

Okay let's go on to the other one now.

So four multiplied by 12.

Can you cut in your 12? I'm sure you really going to do it.

Let's go, 12.

Oh, hold on a minute.

Cancelling 12 is a little bit tricky.

Why don't we use the law of commutativity and change the order of the factors? So now we will look at 12 multiplied by four.

Let's give it a bit of a switch.

Here we go.

Okay then, let's partition 12 into tens and ones.

We now have 10 and two.

Let's multiply 10 by four which gives us a product of 40.

Two multiplied by four is equal to eight.

So now what do we do? Let's recombine.

Let's recombine 40 and eight.

Oh, there's a mistake on my screen, can you spot it? 40 plus eight is not 28, 40 plus eight equals 48.

But I've got it right there.

So 12 multiplied by four is equal to 48.

Did they have the same product? No, they didn't.

I wonder why that is.

Perhaps it's because the values were different.

I think the law of commutativity will only work if we change the order of two factors that have the same value, not simply because they've got the same digits.

On your screen, you should be able to see a table.

And the table down the side has got multiplication expressions.

And on this side it says products equal question mark.

So all we're going to do is we're going to calculate these and we're going to decide whether the products are equal to each other or not.

So we'll pop a tick if they are equal or a cross if they are not equal.

Remember what we said about the factors having the same value, and then whether the product will be the same or not, so equal.

That's just little bit of a clear from me.

But pause the video here and have a go at the challenge.

Okay then, let's go through.

So two multiplied by three and three multiplied by two.

Did you find that the products were equal? The products are in fact equal.

Thumbs up there.

And that's because three and two have the same value in each of the factors.

So we can change the order of the factors and the product will remain the same.

It won't be unchanged.

It will remain unchanged even.

Five multiplied by four and four multiplied by five.

I bet you know that number factor already.

If you can do five times four, you'll know it's 20, and four times five is 20.

So we do know that the products are equal.

And again, because the values are the same.

Five multiplied by five and six multiplied by five.

Oh yeah, I don't know why you can see a red box.

We'll pretend that the red box on your screen shows a cross.

It's supposed to be a cross.

So five multiplied by five and six multiplied by five are not equal.

And that's because? That's right, the values are different.

The products will only be the same if the factors have the same value.

12 multiplied by three and 13 multiplied by two.

So here we've got the same digits; a one, a two, and a three, and a one, a two, and a three in each expression.

But what did we learn earlier? If the value is different, the product will be different.

So the factors value is different, the product will be different.

They are not equal.

So big cross there, you should be seeing a cross.

I wish you can show me a tick or a cross as we go through this last one.

Three multiplied by 22 and 23 multiplied by two.

Are the products equal or not equal? That's right, the products are not equal.

I've got another cross there, because why? The values are different.

You're a wizard at this now.

Well done.

So now that we've explored the law of commutativity a little bit further, I'm sure you can have a go at this independent task.

What I would like you to do is look at the two sets of expressions, and it says match the calculations that have the same product.

So you're going to look down here, down the left-hand side of the products, and they will all match to something else on this side.

Over here you can see some bananas and some sweets, and I'd like you to write two multiplication equations for the following sets of pictures.

So look at the groups of the bananas.

Look at how many are inside the groups and write down two multiplication equations for each picture.

Pause the video here to have a go at the task.

And when you have finished, come back to me and I will go through the answers with you.

Okay then, let's have a look at the independent task.

I'm sure you got on brilliantly.

So did you match 26 multiplied by two to two multiplied by 26? I'm sure you did.

Did you multiply, or sorry, did you match 22 multiplied by three to three multiplied by 22? Two multiplied by 34 matches to 34 multiplied by two.

36 multiplied by five matches to five multiplied by 36.

Three multiplied by 41 matches to 41 multiplied by three.

It's just important to note that on these, I haven't given you any pictures or any number stories to go of.

So it's easy just to change the order of the factors if we're working out a calculation to make it easier for ourselves.

It would be much easier to count in twos than it would be to count in 26's if we needed to.

I'm sure we would use our partitioning and recombining method to multiply a number that big though.

Okay then, we're going to write two multiplication equations for these pictures.

So did you manage to get four multiplied by three because we have four equal groups of three, or three multiplied by four because there are three in each group and we have four equal groups? Let's have a look at the sweets.

Three multiplied by six, we have three equal groups of six, or we have six, three times.

There are six in each group and we have three of them.

I'm sure you've done fantastically.