Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson on representing the 5 times table and linking it to the 10 times table.

So are you ready to do some counting in 5s and 10s, and explore how the 5 and 10 times table are related to each other? Are you? Excellent, let's make a start.

And in this lesson we're going to be representing multiplication equations in different ways.

In this lesson we've got two key words.

We've got commutative and we've got factor pairs.

So let's practise saying those words together.

I'll take my turn and then it'll be your turn.

Are you ready? My turn, commutative, your turn.

My turn, factor pairs, your turn.

Some important words to be looking at today.

So commutative we're going to explore is all about the order of our factors in a multiplication and factor pairs are the 2 factors that we have in a multiplication.

So as you go through, we'll talk about those words and then we'll use them to help us describe what we are learning about today and how we're working.

There are 2 parts to our lesson today.

In the first part of our lesson we're going to be looking at the commutative law and we're going to be looking at it with the 2, 5 and 10 times tables.

And then in the second part we're going to be representing equations in different ways.

So if you're ready, let's make a start on part 1.

And we've got Aisha and Laura helping us with our learning today.

So what do we know about factors so far? 2 and 3 are a factor pair here.

What can you see? Well, we can say there are 2 groups of 3, which is equal to 6, and we can record that with our multiplication.

2 times 3 is equal to 6.

2 represents how many groups we've got and 3 represents that there are 3 in each group, and you can see 2 groups of 3 in the counters.

But we can also say 3, 2 times, which is also equal to 6.

We've got 3 counters and we've got those 3 counters 2 times.

So we can write our factor pair in the other order.

Our factors can be the other way round, but the product stays the same.

2 times 3 is equal to 6 and 3 times 2 is equal to 6, the product in both is 6.

So you can record the representation in 2 ways and this is known as commutativity, and that means that the factors can be in any order and the product will remain the same as we can see there with our 2 times 3 counters and our 3, 2 times.

But there's also another way of looking at this arrangement of counters.

Aisha says, "I can see 2 groups of 3 counters." 2 times 3 is equal to 6, and there are the 2 groups of 3 counters highlighted in some purple boxes.

Laura says, "Well I can see 3 groups of 2 counters." Ooh, so can you see in green she's highlighted 3 groups of 2.

So we were saying with our 2 times 3 that that was 2 groups of 3 counters.

So we can read our multiplication a slightly different way, 3 times 2 rather than being 3, 2 times, we can say it's 3 groups of 2 counters and we can see those highlighted with green.

But we can also see that there are still 6 altogether, our product is still 6.

So what do you notice? That's right, the factor pair and the product remains the same.

Our factors are 2 and 3 or 3 and 2 and our product is 6.

The order of the factors though is different, isn't it? 2 times 3, when we looked at the 2 groups of 3 counters, and now we can think of 3 times 2 as 3 groups of 2 counters as we can see in the representation now.

Let's have a look at another one then.

How many groups of 2 are there now and how can we represent this as a multiplication equation? Do you want to have a look before Laura and Aisha share their thinking? Aisha says, "We can say that there are 5 groups of 2 counters." So we've got 5 representing the number of groups and 2 representing the size of the group.

5 times 2 is equal to 10.

And we can see the 5 groups of 2 highlighted with the purple boxes.

5 groups of 2 is equal to 10, 5 times 2 is equal to 10.

Laura says, "I can group it a different way." Ah, I wonder what Laura's going to do.

So how many groups of 5 are there and how can we represent this as a multiplication equation? We've still got the same array of counters there.

Laura says, "We can say there are 2 groups of 5 counters.

2 groups of 5 is equal to 10, 2 times 5 is equal to 10." So we can record that way of thinking about our counters as a multiplication as well.

This time 2 represents the number of groups which is 2 and 5 represents the size of the group, which is 5.

So 2 groups of 5 is equal to 10.

What's the same and what's different? Well the factor pair and the product remains the same, our factors are 5 and 2 or 2 and 5 and the product is 10, but the order of the factors is different.

In the first one we had 5 lots of 2, and in the second one we had 2 times 5 representing 2 groups of 5.

But we can put them on top of each other and we can see that they're exactly the same.

5 times 2 is equal to 2 times 5, and that's because multiplication is commutative.

We can change the order of the factors and the product stays the same.

Time to check your understanding.

There are 2 groups of 6, but how many groups of 2 are there, and can you represent that with a multiplication? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Did you spot that there were 6 groups of 2? We still had 12 counters.

So we had 2 groups of 6 is equal to 12 and now we've got 6 times 2 is equal to 12, 6 groups of 2.

And Laura says, "Remember that multiplication is commutative." The factor pair and the product remain the same, but the order of the factors is different and commutative means that we can change the order of the factors and the product remains the same.

How many groups of 3 are there and how can we represent this as a multiplication equation? Well, Aisha says, "There are 10 groups of 3 counters." Can you see all the purple boxes around 10 groups of 3 counters? 10 groups of 3 is equal to 30, 10 times 3 is equal to 30.

The factor pair is 10 and 3, the product is 30.

Laura says, "I can group it a different way!" I wonder if you can think how Laura's going to group the counters.

She says, "How many groups of 10 are there and how can we represent this as a multiplication equation?" She says, "There are 3 groups of 10 counters." 3 groups of 10 is equal to 30, 3 times 10 is equal to 30.

The factor pair is still 3 and 10, but this time we've got 3 groups of 10 counters.

The factor pairs are the same and the product is the same because multiplication is commutative.

We can change the order of the factors and the product remains the same.

But we might want to draw a different picture to show that we were thinking about 10 groups of 3 and now 3 groups of 10.

Time to check your understanding.

How many groups of 10 are there and how many groups of 2 are there? And can you represent those with multiplication equations? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Did you see the 2 groups of 10, 2 times 10, and that's equal to 20.

What about the groups of 2? Ah, did you see that there were 10 groups of 2? 10 times 2 is also equal to 20, so there are 2 groups of 10 and 10 groups of 2.

The factor pair and the product remains the same, but the order of the factors is different.

And in 2 times 10 we were thinking about 2 groups of 10.

And in 10 times 2 we are thinking about 10 groups of 2.

Time for you to do some practise now.

You are going to use the commutative law to circle the groups and then write down the multiplication equations for each.

So think of the different ways that you could use the factors to make those groups and group sizes different.

So in the first one we've got 6 counters and we know that 2 times 3 is equal to 6.

How else could you group the counters you've got a times 2 there, and then you're going to do the same for 8 counters and 10 counters and create the 2 different multiplication equations for each.

And in question 2, you're going to gather 12 counters.

How many groups of 2 can you make and how many groups of 6 can you make? And can you represent each way that you've grouped them as equations? And then for questions 3 and 4, you've got some equations to complete, you've got some gaps there.

Can you use your new knowledge of the commutative law to work out what the numbers are that go into those missing gaps? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? So for question one you had to think about the different ways you could group the counters.

So we'd done 2 times 3 is equal to 6, 2 groups of 3, but you could also think about 3 groups of 2 is equal to 6, 3 times 2 is equal to 6.

For the second set of counters, you had 8 counters, you could have had 2 groups of 4, 2 times 4 is equal to 8, or 4 groups of 2, 4 times 2 is equal to 8.

And for the last set you had 10 counters.

So you could have thought about 5 times 2, 5 groups of 2 counters, or 2 times 5, 2 groups of 5 counters.

And you could see that it didn't matter about the order of our factors, the product stayed the same.

For question 2, you gathered 12 counters.

How many groups of 2 could you make and how many groups of 6 could you make? Well we could make 6 groups of 2 and 2 groups of 6.

So we could represent the 2 groups of 6 as 2 times 6 is equal to 12, and the 6 groups of 2 as 6 times 2 is equal to 12.

The order of the factors can change, but the product stays the same.

And here are the answers to question 3.

If 5 times 3 is equal to 15, then 3 times 5 is equal to 15.

If 5 times 7 is equal to 35, then 7 times 5 is equal to 35.

And if 5 times 8 is equal to 40, then 8 times 5 is equal to 40.

We can change the order of the factors, but the product stays the same.

And for question 4, if 10 times 5 is equal to 50 then 5 times 10 is equal to 50.

If 7 times 10 is equal to 70 then 10 times 7 is equal to 70.

And if 9 times 10 is equal to 90 then 10 times 9 is equal to 90.

So for 3 and 4 we didn't have counters to look at, we were just using our knowledge that if we change the order of the factors, the product stays the same.

And time to move on to the second part of our lesson where we're going to be representing equations in different ways.

Aisha and Laura went to a school party.

Aisha represents the number of sweets she saw using counters.

Can you see them on the plates there? And she's got counters to represent the sweets.

What could she see? She could see 6 groups of 2 sweets, couldn't she? 6 times 2 can represent 6 groups of 2 and 6 times 2 is equal to 12.

Aisha now represents this in a different way.

Oh look, she's collected all the sweets and put them onto 2 plates this time.

How many are on each plate? Well there are 6 aren't there? So now we've got 2 groups of 6.

2 times 6 can also represent 2 groups of 6 or 6, 2 times.

So 2 times 6 is equal to 12, 2 groups, with this time, 6 in each group is equal to 12.

So the factors have changed but the product has stayed the same.

We've still got 12 sweets.

So over to you to check your understanding, 6 times 2 is equal to 2 times 6, do you agree? We've got the pictures there that Aisha drew.

Do you agree with her and can you explain your thinking to your partner? So pause the video, have a discussion and when you're ready for some feedback, press play.

What did you think? Did you agree with Aisha? Well 6 groups of 2 is 6 times 2 and that's 12.

And 2 groups of 6 is 2 times 6 and that's 12 as well.

So 6 times 2 can also represent 2 groups of 6 or 6, 2 times.

So we can think about our multiplication, we can think about what each factor represents and each factor could represent the number of groups or the group size depending on how we think about it.

But 6 times 2 is equal to 2 times 6 because multiplication is commutative, we can change the order of the factors and the products stays the same.

Aisha has some coins.

She wants to represent 2 times 5 equals 10 using her coins.

Oh, how's she decided to represent it this time, do you think? 2 times 5 she says can represent 2 groups of 5 or 5, 2 times, and she's got 2 5p pieces there to represent her 2 groups of 5 coins.

2 times 5 can also represent 5 groups of 2 or 2, 5 times.

So she could represent the same value with 5 2p coins.

She says, "The number of coins represents the number of groups," and, "The value of the coins represents the value of the group." So we could have 2 groups of 5p coins or we could have 5 groups of 2p coins.

So depending on whether we think of the 2 as the size of the group or of the number of the groups depends on how we might read our multiplication.

Laura has some coins.

She wants to represent 10 equals 10 times 1 using her coins, hmm, how's she going to do that? Well 10 times 1 can represent 10 groups of 1 or 1 10 times.

So she's got 10 groups of 1 there and she's got 10 groups of 1p or 1p 10 times, but 10 times 1 can also represent 1 group of 10 or 10, 1 time.

So she's got 10 in one group represented with 1 10p coin, 1, 10 times, 1 times 10 is equal to 10.

So can you prove that 5 times 1 is equal to 1 times 5, and you could use coins or counters.

Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? So you may have shown 1p 5 times, so 5 groups of 1p, or you could have said 1 times 5, 1p 5 times, or you might have shown 1, 5p coin, 5 times 1 can also represent 5p 1 time, or 1 times 5 could be 1 times 5p.

So the way we think about the multiplication depends on the way we might read it or think about it, but we know that 5 times 1 is equal to 5 and 1 times 5 is equal to 5 and we can represent that as 5 1s or 1 5.

Time for you to do some practise now.

For question 1, gather some 1, 2, 5 and 10p coins or some counters and represent the equations given in different ways and write the different equations.

So you've got 2 times 5 is equal to 10, can you represent that a different way? And 20 is equal to 5 times 4.

For question 2 there are some sweets in a box.

What situations could the equation represent? Remember Aisha was looking at the sweets at the school party.

What could this look like? Could you draw 2 pictures and write down the equations to match them? And then there are 3 things to do in question 3.

Can you answer the following questions and represent each with a multiplication equation? A says, Aisha and Laura have some coins.

Aisha has 20p altogether.

If Aisha has 2, 10p coins and Laura has the same value as Aisha but in different coins, what set of coins could Laura have? So she's got 20p as well, but she doesn't have 2, 10p coins, what could she have? And in B, Laura has 50p altogether.

If Laura has 10, 5p coins and Aisha has the same value as Laura but in different coins, what coins could Aisha have? So she's got 50p, but she doesn't have 10, 5p coins.

And C says, 4 times 5 is equal to 5 times 4.

Can you prove that's correct? Pause the video, have a go at the questions and when you're ready for some feedback, press play.

How did you get on? So for question 1 we were looking at 2 times 5 is equal to 10 and we can represent that as 2 groups of 5 represented by those 2, 5p coins or 5 groups of 2 represented by the 5, 2p coins.

And we can represent that as 2 times 5 or 5 times 2, 2 times 5 could be 2 lots of 5p coins or it could be 2p coins 5 times and 5 times 2 could be 5 lots of 2p coins or it could be 5p, 2 times, but whichever way round we have the factors and how we describe those factors, the product will always be 10 when the factors are 2 and 5.

And for the other one, 20 is equal to 4 times 5.

We've used counters this time, or we could have represented 20 as 4 lots of 5p, but could we have represented it as 5 lots of 4p? We haven't got a 4p coin have we? So this time we've represented it with the counters, 4 groups of 5 and 5 groups of 4.

So 20 is equal to 4 times 5 and 5 times 4.

In question 2 there were some sweets in a box.

What situations can this equation represent? So 5 times 3 is equal to 15.

So we could have had 3 groups of 5 or 5, 3 times.

So we could have said 3 times 5 is 3 groups of 5, or we could have said 5 times 3 is 5, 3 times.

But we can also think of our 15 sweets as 5 groups of 3 or 3, 5 times.

So those equations could represent 5 groups of 3 or 3, 5 times.

But again, 3 and 5 are our factors, so 15 is our product, so different ways we could think about those sweets in the box.

The question 3, you were answering these questions.

So the first question, A, had coins that had a value of 20p.

We knew that Aisha had 2, 10p coins, but what coins could Laura have to have the same value? Well, she could have 10, 2p coins, 10 times 2p is equal to 20p, or she could have 5, 4p coins.

5 times 4 is also equal to 20.

So 2 different ways of writing it down.

If we just wanted to use the same factors that Aisha had, then she would've had 10, 2p coins.

And Laura has 50p altogether.

And if she has 10, 5p coins and Aisha has the same value as Laura but in different coins, what coins could Aisha have had? Well, she could have had 10p coins, couldn't she? She could have had 5, 10p coins, so 10, 5p coins, and 5, 10p coins would've been represented by 5 times 10 or 10 times 5.

The factors are the same, and so whichever order they are, the product will be the same, there will be 50p in total.

And finally, C asked you to prove that 4 times 5 is equal to 5 times 4, and we could use counters, couldn't we? And we could show that our counters could be seen as 5 groups of 4 or 4 groups of 5, and we could write those 2 equations and we know that they're both equal to 20.

Well done for working hard on those tasks.

And we've come to the end of our lesson.

We've been representing multiplication equations in different ways.

So at the end of our lesson we now understand that the same multiplication can be represented in 2 different ways.

So for 5 times 2 or 2 times 5 we can think of having 5 groups of 2 or 2 groups of 5.

And we used arrays so that we could think of 10 as 5 groups of 2 or 2 groups of 5.

And the expressions are equivalent.

And we also understand that factor pairs can be written in any order and the product will be the same.

And this is because multiplication is commutative.

Thank you for all your hard work and your mathematical thinking today.

I hope you've enjoyed the lesson and I hope I get to work with you again soon.

Bye-bye.