Lesson video

Hi there.

How are you today? I hope you are having a really good day.

My name is Ms. Coe, and I'm really excited to be learning with you in this maths lesson today.

Now in this lesson, we're going to be thinking about skip counting, and throughout the unit, we're going to be focusing on skip counting and our multiplication tables for the two, five, and 10 times tables.

I'm really excited that we get to share this learning together.

If you're ready, let's get going.

In this unit, we are focusing on the two, five, and 10 times table, and by the end of this lesson, you will be able to say that you can explain that factor pairs can be written in any order.

There are three keywords in this lesson today.

I'm going to say them.

I'd like you to say them back to me.

Are you ready? My turn, commutative.

Your turn.

My turn, factor.

Your turn.

My turn, product.

Your turn.

Now some of these words might be new to you, so listen out really carefully for when they're used in the lesson.

In this lesson today, we're going to explain that factor pairs can be written in any order and we're going to have two cycles of our learning.

In the first cycle, we're going to explore commutativity.

Now commutative was one of our keywords, so we're going to really explore what that means.

And then in the second cycle, we're going to represent commutativity in different ways.

So if you're ready, let's get started with the first cycle of our learning.

In our lesson today, you're going to meet Laura and Aisha.

They're going to be asking us lots of questions and helping us with our learning.

So let's start here.

Aisha and Laura are practising their two times table.

Hopefully, you have been practising your two times table and you're becoming a little bit more familiar with some of these facts.

So we have zero times two is equal to zero, one times two is equal to two, and so on.

Look carefully at those equations.

What do the numbers represent? Well, that's right, Aisha.

The first factor represents the number of groups.

So in the example there, we have three multiplied by two, and we can say that as three groups of two.

So that means the second factor represents the group size.

So remember, the two numbers that we are multiplying together, we can call factors.

Three and two are factors.

That means the third number, six in this case, is the product.

It represents the total amount.

Aisha begins to say the two times table.

She starts by saying, "Zero groups of two is equal to zero." Let's say that together.

Zero groups of two is equal to zero.

How are we going to say the next times table fact? That's right.

One group of two is equal to two.

Let's say that together.

One group of two is equal to two.

Can you say the next one with Aisha? Are you ready? Two groups of two is equal to four.

What comes next? That's right.

Three groups of two is equal to six.

Can you say the next one on your own? Have a go.

That's right.

Four groups of two is equal to eight.

And we could carry on.

Oh.

What's changed here? Well, Laura has written the facts the other way around.

I wonder what that means.

So first, we had Aisha's list, and now we've got Laura's list.

What's the same? What's different? What do the numbers represent now? Well, we started off with zero multiplied by two is equal to zero.

And in Laura's list, it's two multiplied by zero is equal to zero.

Hmm.

There's definitely some similarities there.

Well, Aisha says that "Two still represents the number in each group in all of the equations." So we can read that first one as two, zero times is equal to zero.

Let's say that together.

Two, zero times is equal to zero.

The second one can be read as two, one time is equal to two.

Let's say that together.

Two, one time is equal to two.

I wonder if you can predict how we'd say the next one.

That's right.

Two, two times is equal to four.

Let's say that together.

Two, two times is equal to four.

Can you say the next one with Laura? Are you ready? Two, three times is equal to six.

Aisha and Laura now say the two times table both ways.

Let's take a look at what they say and see if you notice anything similar or different.

So Aisha says, "The first one is zero groups of two is equal to zero." Laura says, "Two, zero times is equal to zero." What do you notice? They say the same numbers.

Does it matter that the factors are in a different order? They both have the same product, so perhaps it doesn't.

What about the next one? One group of two is equal to two.

Two, one time is equal to two.

The product is still two in both cases.

Two groups of two is equal to four is the next one.

And Laura would say, "Two, two times is equal to four." "Three groups of two is equal to six," says Aisha.

"Two, three times is equal to six," says Laura.

So you can say these times table facts in two different ways, but they mean the same thing.

Time to check your understanding.

I would like you to say this times table fact both ways.

So look at the highlighted one and we're going to say, mm, groups of two is equal to mm.

Or two, mm times is equal to mm.

Pause the video here and have a go.

Welcome back.

So we had four multiplied by two is equal to eight.

And we can say that as four groups of two is equal to eight.

Or two, four times is equal to eight.

The factors are two and four in both equations.

We can use what we've learned to talk about groups in different ways.

Here, we have eight pairs of socks.

Now remember, socks come in pairs, which is a group of two.

Aisha and Laura are going to use what they know about the two times table to describe these socks in two different ways.

Let's take a look.

Aisha says, "We can describe these socks as eight groups of two because we have eight pairs of socks with two in a pair." She also says, "We can say two, eight times." What do you think? Do you agree with that? Laura agrees.

She says, "We can say two, eight times because we have two socks, eight times over." So we can write two different expressions.

Eight multiplied by two.

Two multiplied by eight.

Both of these represent the number of socks that we have.

So we can represent two, eight times as two multiplied by eight is equal to 16.

There are 16 socks altogether.

Time to check your understanding.

Look at the image.

This time, we have pairs of gloves.

Say it in different ways.

You have five groups of mm, or two, mm times.

Pause the video here.

Welcome back.

How did you get on? Did you notice that we had five pairs of gloves? So we can say five groups of two.

Or we can say two, five times.

And remember we can write this expression in different ways.

Five multiplied by two, or two multiplied by five.

And Laura is reminding us that "Both ways represent five pairs of gloves." Let's move on.

Aisha and Laura are comparing multiplication equations.

Aisha looks at the image and says she can see three multiplied by two is equal to six.

In her case, the three represents groups and the two represents the number in each group.

So she can see three groups of two apples.

So she's written three multiplied by two is equal to six because there are six apples altogether.

Laura says she can see two multiplied by three is equal to six because she can see two, three times.

So she can see that there are two apples, three times over.

So she has written two multiplied by three is equal to six.

There are still six apples altogether.

They look at another example and write two different equations again.

Take a look closely.

How many pairs can you see? How many pears are there altogether? What's the same? What's different? Well, Aisha has said that she can see four multiplied by two is equal to eight.

The four is the number of groups and the two is the number of pears in each group.

Laura can see two multiplied by four is equal to eight.

She can see two, four times.

They think about the factors and products in these equations.

Two and four are both factors in these equations.

The factors are written in a different order.

But that doesn't matter.

They're still representing that image.

Eight is the product in both equations.

"Did you know," asks Laura, "that multiplication is commutative?" Ah, that's one of our keywords.

This is what you've been exploring.

So commutativity is when you swap the factors around and still get the same product.

"Woah!" says Aisha.

"Well, that's awesome.

Let's test it!" I wonder what they're going to do to test that.

Time for your first practise task.

For question one, I'd like you to use the images to complete the sentences below.

So this time, you have groups of apples.

Think carefully.

How many groups are there? What is the value of each group? Think about how each child would explain what they can see.

Fill in the blanks in their sentences.

For question two, use what you know about commutativity, remember, that's factors being in either order, to fill in the blanks in these equations.

Let's look at the first one.

I can see that there are three groups and each group has two apples.

I can see that there are six apples altogether.

What is the missing factor in those two equations? You have three others to have a go at for that one.

Pause the video here, have a go at those two tasks, and I'll see you shortly for some feedback.

Welcome back.

How did you get on? For question one, let's see what each child said about the image.

For the first one, I can see that there are four groups.

Each group has two apples in it and there are eight apples altogether.

Aisha says she can see four multiplied by two is equal to eight.

The four represents the groups and the two represents the amount in each group.

Laura says that she can see two multiplied by four is equal to 8.

Two, four times.

So she can see a group of two, four times over.

For the second one, well, this time, I can see that there are five groups of two.

There are 10 apples altogether.

Laura says she can see two multiplied by five is equal to 10 because she can see two, five times.

Aisha can see five multiplied by two is equal to 10.

The five represents the groups and the two represents the amount in each group.

For question two, I asked you to fill in the blanks.

For the first one, there are three groups of two apples and there are six apples altogether.

The factors are three and two.

So we could have three multiplied by two is equal to six, or three groups of two.

Or we can say two, three times.

Two multiplied by three is equal to six.

For the second group of apples, there are seven groups of two apples.

So I can write seven multiplied by two is equal to 14 or two multiplied by seven is equal to 14.

I can say that as seven groups of two is 14, or two, seven times is 14.

For the third one, there are five groups of two pears, which is 10 pears altogether.

I can say five groups of two is equal to 10, or two, five times is equal to 10.

And for the last one, there were nine groups of two, which is 18 altogether.

I can say nine groups of two is equal to 18, or two, nine times is equal to 18.

Well done if you filled in those factors correctly.

Well done thinking carefully about commutativity in the first cycle of our learning.

Let's move on to the second cycle where we're representing that commutativity in different ways.

So let's start with an image that we've seen before.

Here are some bowls of apples.

So we've represented the bowls of apples like this.

We have three groups with two apples in each group.

How else could we represent it though? I wonder if you can think how you might represent it.

Aisha says that she can use repeated addition to represent this.

Let's see what she means.

Ah, well, we can write an equation, two plus two plus two is equal to six.

So I could add two, three times to make six.

That is one way of representing this.

But it's actually more efficient to use multiplication.

So we can write three multiplied by two is equal to six.

There are three groups of two apples.

There are six apples altogether.

We also know that we can swap the factors because multiplication is commutative.

So we can also write that we can see two apples, three times.

All of these equations represent the same thing.

This means that we can also write that three multiplied by two is equal to two multiplied by three.

They are equivalent.

They mean the same thing.

And we can use the equals sign between them to show that they are equal or the same.

And this is because multiplication is commutative.

Thanks for the reminder, Aisha.

Aisha and Laura continue to explore commutativity.

This time, they have cherries.

Cherries often come in pairs, or groups of two.

"There are seven pairs of cherries," says Aisha.

How many cherries would that be altogether? What would the product be? Laura can see two cherries, seven times.

Can you think about what the factors might be? Well, that's right.

We can write seven multiplied by two is equal to 14.

There are seven groups of two cherries and there are 14 cherries altogether.

We can also write two multiplied by seven is equal to 14 because we can say that there are two cherries, seven times.

Remember, we can write this using the equal sign in between.

We can say that two multiplied by seven is equal to seven multiplied by two.

Multiplication, remember, is commutative.

You can change the order of the factors and the product will stay the same.

Time to check your understanding.

Match the multiplication equation to the image below.

So we have an image there of cherries that come in pairs.

Which of these multiplications represent the cherries? Is it A, B, or C? Take a close look at the equations.

Pause the video here.

Welcome back.

Which one did you go for? Well, it's C and that's because there are 11 groups of two.

I can see 11 pairs of cherries, which is 11 groups of two.

I can also see two, 11 times.

So I can say that 11 multiplied by two is equal to two multiplied by 11.

Well done if you spotted that.

Aisha then moves on to thinking about coins.

Now remember, this is a two-pence coin.

It has a value of two pence.

She can represent the value of the coins in different ways.

I wonder if you can predict what multiplication equation she's going to write.

There are eight two-pence coins.

So we can write eight multiplied by two is equal to 16.

The total value of the coins is 16 or 16 pence.

What was the other equation that we can write? That's right.

We can say that there is two pence, eight times.

So we can write two multiplied by eight is equal to 16.

Eight is a factor because it represents the number of groups.

Two is a factor because it represents the value in each group.

In this case, two pence.

Remember, Aisha is reminding us that we can change the order of the factors and the product will stay the same.

Laura is thinking about bicycles.

Bicycles have two wheels.

She's representing the number of wheels in different ways.

Firstly, she can say that there are six groups of two wheels.

I know that six multiplied by two is 12, so there are 12 wheels altogether.

We can write six multiplied by two is equal to 12.

And what's the other way of writing it? That's right.

We can say two multiplied by six is equal to 12 because there are two wheels, six times.

In this case, six is a factor because it represents the number of groups.

Two is a factor because it represents the number in each group.

And 12 is the product.

And remember, if we change the order of the factors, the product stays the same.

We can also write that six multiplied by two is equal to two multiplied by six.

Time to check your understanding.

Write two multiplication equations to represent the image below.

Use the stem sentences to help you.

So remember, these are two-pence coins.

They each have a value of two pence.

Pause the video here.

Welcome back.

How did you get on? Well, I can see that there are ten two-pence coins.

So I could write ten two-pence coins, 10 groups of two, which is 10 multiplied by two, and I know that 10 multiplied by two is equal to 20.

I could also see this as two pence or two, 10 times.

So I could write two multiplied by 10 is also equal to 20.

Another quick check of your understanding.

Fill in the gaps.

Aisha says, "If I know that two multiplied by nine is equal to 18, then nine multiplied by mm is also equal to 18." Pause the video here.

Welcome back.

Well, hopefully, you remembered that in both of these equations, two and nine are factors.

18 is the product.

The factors can be written in a different order.

Multiplication is commutative.

So if I know that two multiplied by nine is equal to 18, then I know that nine multiplied by two is equal to 18.

Time for another quick check.

The product is 14.

There are seven groups of two.

Show two different ways to represent this.

Pause the video here.

Welcome back.

How did you get on? Well, here are some examples.

There are lots of different ways that you might have represented it.

You might have written some equations.

Seven groups of two is equal to 14.

Or two, seven times is equal to 14.

You may have drawn a picture or used cubes or counters to represent seven groups of two.

Remember, we can say this in two ways.

We can say seven groups of two or two, seven times.

Well done if you found lots of different ways to represent this.

Time for your second practise task.

For question one, I'd like you to write two multiplication equations to represent each picture.

Let's take a look at the first one together.

How many groups of two do we have? So we have mm groups of two.

How many apples are there altogether? Remember that's our product.

Think about how you can use commutativity to write two different equations.

For question two, you need to fill in the missing numbers.

So remember, we're using that equals sign to show that they are the same amount.

So you're told that zero multiplied by two is equal to two multiplied by zero.

Then we have one multiplied by two is equal to two multiplied by mm.

What is the same? Take a close look.

Good luck with those two tasks and I'll see you shortly for some feedback.

Welcome back.

How did you get on? Hopefully, you used the idea of commutativity a lot here to think about the factors in the equation and what the product was.

Let's take a look at question one.

For question one, you had to write two multiplication equations for each.

For the first one, we had three groups of two.

There are six apples altogether.

We could have written three multiplied by two is equal to six because three groups of two is equal to six.

Or two multiplied by three is equal to six because two, three times is equal to six.

For the second one, we had four pairs of cherries.

So we could say four multiplied by two is equal to eight because there are eight cherries altogether.

Or two multiplied by four is equal to eight.

For the next one, we had six groups of two apples.

So we could say six multiplied by two is equal to 12.

Or two multiplied by six is equal to 12.

And finally, we had seven pairs of shoes.

Remember, a pair is a group of two.

So we could say seven multiplied by two is equal to 14.

Or two multiplied by seven is equal to 14.

For question two, remember that when we have the equals sign, it means what is on one side is the same or equal to what is on the other side.

So if we had one multiplied by two, that is equal to two multiplied by one because we could say one group of two is equal to two, once, or two, one times.

If we look at the second column, we had two multiplied by 12 is equal to 12 multiplied by two because we can say that two, 12 times is equal to 12 groups of two.

Take a close look at those equations to make sure that you filled them all in correctly.

We've come to the end of the lesson and I hope that you've really thought carefully about these factor pairs and remembered that they can be written in any order.

We now know that that order is called commutativity.

Let's summarise our learning.

We understand that multiplication is commutative.

So if you have five groups of two, you can say that you have two, five times.

So we can write that five multiplied by two is equal to two multiplied by five.

Thank you so much for all of your hard work today, and I look forward to seeing you in another lesson soon.