Lesson video

Hello, my name's Mrs. Hopper, and I'm really looking forward to working with you in this lesson from our unit on using equivalence to calculate.

So, are you ready to use all your knowledge of the different operations and think about how the numbers within them relate to each other, how we can maintain equivalence whilst changing the way an equation looks? If you're ready, let's make a start.

So, in this lesson, we are going to be explaining why the product stays the same when one factor is doubled and the other is halved.

So, we're going to be thinking about multiplication.

So, are you ready? Let's make a start.

We've got three key words in our lesson today.

We've got equivalent, factor, and product.

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

Are you ready? My turn: equivalent.

Your turn.

My turn: factor.

Your turn.

My turn: product.

Your turn.

Excellent! I'm sure you know what those words mean, but they're gonna be really important in our lesson today.

So, let's just check the definitions.

So, equivalent means having the same value, exactly the same value.

Things may look different, but the value is the same.

A factor is a number which exactly divides another whole number.

And the product is the result of two or more numbers multiplied together.

So, we're going to be using those words, factor and product, to describe the different elements of the equations that we look at.

And equivalence is going to be really key.

There are two parts to our lesson.

In the first part, we're going to be doubling and halving using known facts.

And in the second part, we're going to be thinking about same product, different factors.

So, let's make a start on part one.

And we've got Andeep and Laura working with us today.

Andeep and Laura each buy a box of eggs.

I wonder if they're doing some baking.

Andeep says, "This is my box of eggs.

There are 2 rows of 6 eggs." Laura says, "This is my box of eggs.

There are 4 rows of 3 eggs." Laura says, "My box contains the same number of eggs as Andeep's." Had you spotted that? How do you know? So, Andeep's box of eggs has 2 rows of 6.

2 multiplied by 6 is equal to 12.

And Laura's box of eggs has 4 rows of 3.

4 multiplied by 3, which is also equal to 12.

Those are well-known times table facts.

Laura shows that there's the same number in both boxes.

She says, "I can split my box into two parts and rearrange the eggs to show that they're equal." Can you sort of imagine what's gonna happen? So, she split her box, and then she's rearranged that bottom half alongside.

And can you see now that she's made 2 rows of 6? Ah, Andeep says, "Laura now has 2 rows of 6 eggs too." What do you notice about the equations? Can you see the 2 times 6 is equal to 12 and the 4 times 3 is equal to 12? Laura says, "If I double one factor, I must halve the other factor for the product to stay the same." Double 2 is 4, 2 multiplied by 2, and half of 6 is three, 6 divided by 2.

So, we've doubled one factor, and we've halved the other one, but the product has remained the same, and we can see that in the way the eggs have been arranged.

Andeep represents 6 multiplied by 4 using counters.

So, 6 times 4, and he's made a sort of area model out of it.

He says, "There are 24 counters altogether." I expect you knew that from your times table fact.

6 times 4 is equal to 24.

He says, "I can rearrange the counters." So, he's cut 'em in half, and he's put half underneath.

Can you see what's happened? Laura says, "Half of 6 is equal to 3, and double 4 is equal to 8." Have we got the same number of counters? We have, haven't we? Because none of the counters went away, they were just rearranged.

And she says again, "If I double one factor, I must halve the other factor for the product to stay the same." 6 times 4 was equal to 24.

But we halved 6 and we doubled 4, and we created the new equation of 3 multiplied by 8 is equal to 24.

The product remained the same.

And Andeep says, "6 times 4 is equivalent to 3 times 8." It may look slightly different, but it has exactly the same value.

Those two expressions are equivalent.

Andeep tries a different arrangement.

He says, "There are 24 counters altogether," and he's gone back to his 6 times 4 arrangement.

How's he gonna split them this time, do you think? Ah, he split them horizontally this time.

So, which factor has he halved, and which has he doubled? Oh, he doubled the 6 this time.

Double 6 is equal to 12, and half of 4 is equal to 2.

And Laura is reminding us, "If I double one factor, I must halve the other factor for the product to stay the same." So, let's look at the equations.

6 multiplied by 4 is equal to 24.

If we double the 6 and halve the 4, we can rewrite the equation as 12 multiplied by 2 is equal to 24.

So, "6 times 4 is equivalent to 12 times 2," says Andeep.

Andeep and Laura look at other multiplication equations.

Andeep says, "I can halve 8 and double 7." 7 times 8 is 56, is my favourite times table calculation.

So, let's see how he's gonna change it.

So, he's halving 8 and doubling 7.

4 multiplied by 14 is equal to 56 as well.

So, 8 times 7 is equivalent to 4 times 14.

The product will still be 56.

Laura says, "I can double 9 and halve 4." So, 9 multiplied by 2 is 18.

4 divided by 2 is 2.

So, 18 multiplied by 2 is equal to 36 as well, and equivalent to 9 times 4.

The product will still be 36.

And there's our generalisation, and this is worth remembering and also worth being able to picture so you can understand why it works.

If you double one factor, you must halve the other factor for the product to stay the same.

Over to you to check your understanding.

Find an expression that is equivalent to 8 multiplied by 12.

Is it A, B, or C? Andeep says, "Which expression also has a product of 96?" And Laura is reminding us, "If I double one factor, I must halve the other factor for the product to stay the same." Pause the video, find the equivalent expression, and when you're ready, we'll get together for some feedback.

How did you get on? Well, Andeep says, "You could halve 8 and double 12 and the product would still be 96." So, if we halve 8, we get 4, and if we double 12, we get 24.

So, 4 multiplied by 24 is equal to 96 and is equivalent to 8 times 12.

For B, 8 has been halved to make 4, but the 12 has not been doubled.

So, this expression will not be equivalent to 8 multiplied by 12.

It won't have a product of 96.

What about C? Well, yes, you could double 8 and halve 12 and the product would still be 96.

So, we've doubled 8 to make 16, and we've halved 12 to make 6.

So, 16 multiplied by 6 will also be equal to 96.

16 multiplied by 6 is equivalent to 8 multiplied by 12.

Well done if you've got both of those.

Laura finds the missing number in the equations.

Andeep says, "What do you notice about the equations?" What can you see about them? Oh, we can see a halving there.

Laura says, "4 is halved to equal 2." So, if we halve one of our factors, what have we got to do to the other one? Ah yes, she says, "I'll need to double 11, which is equal to 22." So, 11 multiplied by 4 is equal to 22 multiplied by 2.

What about the second equation? What do you notice? Can you see something there? Laura spotted 7 doubled is equal to 14.

So, if we've doubled one of our factors, what do we need to do to the other one? That's right.

So, she needs to halve 12, which is equal to 6.

So, 12 multiplied by 7 is equivalent to 6 multiplied by 14.

If you double one factor, you must halve the other factor for the product to stay the same.

Have we actually worked out the products this time? We haven't, have we? But we can reason to say that these are equivalent even if we haven't worked out what the products are.

Over to you to check your understanding.

Can you find the missing number in the equation? So, we've got 14 multiplied by 3 is equal to something multiplied by 6.

Andeep says, "What's the missing factor?" Pause the video, have a go, and when you're ready, we'll get together for some feedback.

What did you work it out to be then? Laura's reminding us, "If you double one factor, you must halve the other factor to keep the product the same." So, what can you see that's happened there? Aah, that's right.

Andeep says, "3 doubled is equal to 6." So, we've doubled the factor 3.

So, what must we do to the 14? That's right, we need to halve 14, which is equal to 7.

So, 14 multiplied by 3 is equal to 7 multiplied by 6.

Again, we didn't need to work out what the product was, we know by reasoning that those two expressions will be equivalent.

Time for you to do some practise.

So, question 1 says can you fill in the missing numbers in the equations? And we've given you some doubling and halving clues there.

So, "Double one factor or halve the other factor," Andeep says.

And in question 2, you're going to find the missing numbers in the equations.

And Andeep says, "Look carefully to find which factor has been halved and which has been doubled." And Laura says to notice that in F, there are two missing factors.

So, pause the video, have a go, and when you're ready, we'll get together for some answers and feedback.

How did you get on? Here are some answers.

Andeep's looking at A, and he says, "In A, half of 6 is equal to 3, and 3 doubled is equal to 10.

So, 6 multiplied by 5 is equal to 30, and 3 multiplied by 10 is equal to 30.

They're both well within our times table facts, so we'd be able to check those very easily just to check that our reasoning about doubling and halving is correct.

And the answer to B was that 7 multiplied by 4 is equal to 28, and 14 times two is equal to 28.

We've doubled one factor and halved the other.

Laura's looking at C.

She says, "Half of 12 is equal to 6, and 3 doubled is also equal to 6.

Ah, so 12 multiplied by 3 is equivalent to 6 multiplied by 6.

Depends which times table you like best, I suppose.

And for question 2, here are the answers.

Andeep's focusing on D.

He says, "In D, 11 doubled is 22, and half of 8 is 4." So, 11 times 8 is equivalent to 4 times 22.

And Laura's looked at F where we had two missing factors, and she says, "Half of 12 is equal to 6 and 12 doubled is equal to 24." She says, "Your factors may have been the other way round." You might have doubled the first 12 and halved the second one.

So, 6 multiplied by 24 or 24 multiplied by 6 is equal to 12 times 12.

We know that multiplication is commutative.

Well done if you've got those right.

Let's move on to the second part of our lesson.

So we're going to be looking at the same product, different factors.

So Andeep is trying to find the product of 5 and 28.

My 5 times table doesn't go up to 28 times 5, does yours? So he says, "What is 5 times 28?" Laura says, "Let's represent the calculation with an area model." So we've got 28 multiplied by 5, and you could imagine 5 rows of 28 counters.

Too many to put out on the table, so we'll imagine the counters this time.

She says, "We could double the 5 and halve the 28 and the product will stay the same." So let's think about how we change that rectangle.

So we divide the length of it into two lots of 14, and then we can put one underneath the other.

So we've got 5 lots of 14 and another 5 lots of 14.

So altogether we've got 10 times 14.

Oh, Andeep says, "Well, 10 times 14 is 140, so 5 times 28 must be equal to 140 as well." That made our calculation much easier to do, didn't it? Let's have a look at those expressions though.

5 multiplied by 28.

Well, if we double the 5 and halve the 28, we get 10 multiplied by 14, and we know that it's easy to multiply numbers by 10, and the product will be 140.

And it will be the same for 5 multiplied by 28.

So we can use this strategy to help to simplify some calculations perhaps.

Andeep and Laura are going to use a number line representation.

I think they need a bit more convincing about what's happening here.

So our number line goes from zero to 140.

Andeep says, "We worked out that 10 multiplied by 14 is equal to 140, and 5 multiplied by 28 was also equal to 140." How can we show that on the number line? Well, Laura says, "Each step of 28 is equal to 2 steps of 14." So there we go.

One step of 28 is equal to 2 steps of 14.

So if we carry that on along the number line, we can see that we've got 5 lots of 28 is equal to 10 lots of 14.

We can see 10 steps of 14 and 5 steps of 28, but we land on 140 at the end of our number line both times.

So another way of showing that those equations are equivalent.

Andeep is trying to find the product of 8 and 35, again beyond my times table knowledge, Andeep.

He says, "What is 8 times 35?" Laura says, "Let's represent the calculation with an area model again." So we've got 8 multiplied by 35.

What can you see here? Well, she says, "We could double 35 and halve 8 and the product will stay the same." I wonder why she hasn't halved 35.

Can you think? It's an odd number, isn't it? So we wouldn't get a whole number as one of our factors.

So doubling the odd number and halving the even number seems to be the right way round to do it.

So let's have a look and see what happens.

So this time we're going to divide the height of 8 into two 4s, and then we can make one long thin rectangle.

And this time it is 70 along.

So we've got 70 multiplied by 4.

Oh, Andeep says, "I know that 4 times 7 is equal to 28, so 4 times 70 must be equal to 280." So therefore 8 times 35 must also be equal to 280.

Again, by using this idea of halving one factor and doubling the other, we've managed to give ourselves a much simpler calculation to do.

8 times 35.

We halve the 8, we double the 35 to create 4 times 70, and then we could use our times table and place value knowledge to work out that the product was 280.

So the product of 8 and 35 must also be 280.

And they're going to check using that number line representation to make sure they can really see why those two equations are equivalent.

So we've got a number line from zero to 280.

And Laura says, "Each step of 70 is equal to 2 steps of 35." So there we can see it.

2 steps of 35 is equal to 1 step of 70.

And if we carry that along the number line, we can see that 8 lots of 35 is equivalent to 4 lots of 70.

Time to check your understanding.

Can you find the product of 5 and 42 using the strategies we've been practising ? So, "What is 5 times 42," says Andeep? And Laura says, "Double one of the factors and halve the other.

The product will stay the same." So pause the video, have a go, and when you're ready, we'll get together for some feedback.

How did you get on? Did you spot that double 5 is equal to 10 and half of 42 is equal to 21? So we can double our 5 and halve our 42, and we've got 10 multiplied by 21.

And again, multiplying by 10 is easy.

We just move the digits and put in that placeholder in the ones.

So 10 multiplied by 21 is equal to 210, and 5 multiplied by 42 must also be equal to 210.

Well done if you spotted that.

Andeep is trying to find the product of 24 and 25.

Ooh, we got 2 two-digit numbers this time.

Right, well, I'd rather not do a long multiplication, so let's see if we can use our strategy to simplify this.

He says, "How can I use doubling and halving to quickly calculate 24 multiplied by 25?" I wonder if you can think of a way.

Well, Laura says, "We could halve 24 and double 25 and the product will stay the same." So half of 24 is 12 and double 25 is 50.

Oh, she says, "We could then halve 12 and double 50." We could do it again, couldn't we? "And the product will still stay the same." So 6 multiplied by 100.

Oh, well that's easy to work out, isn't it? The product of 6 and 100 is equivalent to the product of 24 and 25.

And Andeep says, "Well, 6 times 100 is 600, so 12 times 50 must be equal to 600, and 24 times 25 must also be equal to 600." So that strategy really helped us to simplify what could've been quite a tricky calculation to do.

Laura finds the missing number in these equations.

Andeep says, "What do you notice about the equations?" Have you spotted anything? Ah, she says, "4 is halved to 2, so I need to double 27 to equal 54." And doubling numbers is quite easy, isn't it? And again, in this one, we've doubled 5 to equal 10, so I need to halve 34 to equal 17.

So again, we've managed to change two calculations that looked quite tricky into two that look much easier, one that just involves doubling and one that involves multiplying by 10.

All by knowing that if you double one factor, you must halve the other factor for the product to stay the same.

A really, really useful thing to remember and one that it's useful to understand so that you know why that works.

So can you find the missing number in this equation? Time to check your understanding.

Andeep says, "What's the missing factor?" So pause the video, have a go, and when you're ready, we'll get together for some feedback.

How did you get on? Well, Laura says, "If you double one factor, you must halve the other factor for the product to stay the same." Well done, Laura.

Keep reminding us.

So what's been doubled and what's been halved, do you think? Ah, Andeep says, "33 is doubled to equal 66, so you need to halve 8 to equal 4." So we've got 4 multiplied by 66.

I think we might need to do some more doubling and halving to make that into an easier equation, but we do know that 8 multiplied by 33 is equivalent to 4 multiplied by 66.

Time for you to do some practise.

For question 1, you're going to fill in the missing numbers in the equations.

And Andeep says, "Look carefully at the numbers in each equation." And Laura says to remember if one factor is doubled, the other factor must be halved for the product to stay the same.

And in question 2, you're going to complete these calculations.

And Andeep says, "Use doubling and halving to help you." How could you make it easier to find the answers? Remember all those ways we were looking at to find out that we could create equivalent calculations that involved doubling or multiplying by 10? I wonder if you can find that in these calculations.

And question 3 asks you to write different equations with the same product.

So Andeep says, "We can halve one factor and double the other." And Laura says, "How many different equivalent equations can you find?" We've given you 488 multiplied by 1 is equal to 488.

How many other equivalent equations can you find? Pause the video, have a go at the three questions, and when you're ready for some answers and feedback press play.

How did you get on? Here are the answers to question 1.

So you might just want to check through those, and then we'll have a look at a couple of them together.

So in A, 13 was doubled to equal 26, and 10 must have been halved to equal 5.

So 13 multiplied by 10 is equal to 26 multiplied by 5.

I know which one I'd find easy to calculate.

And Laura says, "In F, half of 12 is equal to 6 and 45 doubled is equal to 90." Well, 12 times 45 is quite tricky, but 6 times 90, I can use my times table fact of the 6 and 9 times table to work that one out if I needed to find the product.

So we have managed to find a simpler way of finding the product of 12 and 45.

I hope you were successful with all of those.

Here are the answers to question 2, but I wonder how you worked them out.

This time, we had an odd number again, so that was the one to double, wasn't it? So Andeep says, "Double 125 is equal to 250 and half of 6 is equal to 3." And Laura says, "250 times 3 is a less tricky calculation than 125 times 6." We might know our 25 times table.

25 times three is 75, so 250 times 3 must be 750.

Did you spot for B? Again, we had an odd number, so we needed to double that, but by doubling that factor, we had to halve the other.

So it gave us a times 2, a doubling fact in itself.

Two times 130 is equal to 260.

And what about C? Ah, well, we could double 5 to equal 10, and then we could halve 118, and we can halve 100 and half of 18 to give us 59.

And then we've got to multiply by 10, which is very straightforward.

So again, by doubling one factor and halving the other, we could create an easier calculation to solve.

And here are the answers for question 3.

And Andeep says by halving one factor and doubling the other, we can create all of these different equations, and they're all equivalent.

"They're all equal to 488," says Laura.

I hope you had fun creating those equivalent equations.

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

We've been explaining why the product stays the same when one factor is doubled and the other is halved.

So what have we learned about? We've learned that when one factor is doubled and the other is halved, the product remains the same.

Rearranging arrays can show the product staying the same when one factor is doubled and the other is halved.

And we also showed that on a number line as well.

And doubling one factor and halving the other maintains equivalence; it keeps the product the same.

And sometimes we can then use that to create an easier equation to solve.

Thank you for your hard work.

I hope you've enjoyed exploring that, and I hope I get to work with you again soon.

Bye-bye.