Lesson video

Hello there, my name is Mr Tilston, I'm a teacher.

I teach all of the different subjects, but the one that I enjoy the most is undoubtedly maths.

So I'm really excited and really happy to be here with you today to teach you this math lesson.

I hope you're ready for a challenge.

If you are, let's begin the lesson.

And this is the outcome of the lesson, I can explain how making a factor 10 times the size affects the product.

And our key words, if I say them, will you say them back please? My turn, factor.

Your turn.

My turn, product.

Your turn.

I guarantee you've heard those words before, but how about a little reminder? Because they're important words and they're coming up a lot today.

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

So for example, three is a factor of 12.

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

So for example, 12 is a product of multiplying three and four.

Our lesson today is split into two cycles.

The first will be making a factor 10 times the size and the second will be applying the concept in context.

Let's begin by looking at making a factor 10 times the size.

In this lesson you're going to meet Alex and Aisha.

Have you met them before? They're here today to give us a helping hand with the maths.

Alex and Aisha look at a multiplication.

It's a very simple one.

It is two multiplied by three is equal to six.

This is two groups of three, which is equal to six.

So can you see that? Three, twice, two groups of three.

I see it differently, says Alex.

I think this is two, three times.

Can you see that? Two, one, two, three times.

Who do you agree with? Who's correct? What do you think? Are they both right? Is Aisha right, is Alex right? Are neither of them right? Let's see.

Well actually both are correct and have the same product because of commutativity.

For today, can we use my interpretation to explore the lessons, says Aisha.

Yeah, sure.

So look at the equation and its description by Aisha.

Can you represent this using base ten one blocks? Hopefully you've got them in front of you.

If not, you can draw them, draw or arrange them.

So this is two groups of four which is equal to eight.

So can you draw or represent that? So once more, this is two groups of four, which is equal to eight.

Pause the video.

Did you represent that, maybe with blocks, maybe with drawings? Oh, that's what this looks like, that's two groups of four.

One group of four, two groups of four.

Alex and Aisha look at a multiplication.

This is two multiplied by three is equal to six.

And Aisha says let's model this with place value counters this time.

Have you got place value counters in front of you? You can draw them if not, have a go at that.

It looks like this.

It is two groups of three ones.

What would happen if we make three 10 times the size? This is what would happen.

The value of each counter needs to be made 10 times the size.

I'm sure that recently you've multiplied by 10 and exchanged one counter for a different value in doing so.

That's what we've done here.

Each one counter has changed value to become a 10 instead.

So everything is 10 times the size.

So now we haven't got two multiplied by three equals six.

We've now got two multiplied by 30 equals 60.

There are now two groups of 30 which is equal to 60.

We started with two groups of three, says Aisha, which is equal to six then says Alex, we scaled up a factor by making it 10 times the size.

And the factor that we scaled up in this case was three.

And that became 30.

That meant that the product was 60, which is also 10 times the size.

So instead of being two multiplied by three equals six, it became two multiplied by 30 equals 60.

So we scaled up one of the factors by making it 10 times the size and that meant we scaled up the product by making it 10 times the size.

Let's do a quick check.

Fill in the missing labels that show what the factor and product have been scaled up by in these related multiplications.

So the four has become 40 and the eight has become 80.

Pause the video.

They've been scaled up by the same amount and that is 10.

They've been multiplied by 10.

One of the factors has been multiplied by 10 and that's meant the product has been multiplied by 10.

Alex and Aisha are still looking at the multiplication, two multiplied by three is equal to six.

But Aisha says last time we scaled up the group size by 10.

What would happen if we scale up the number of groups by 10? So Alex says that would mean changing two to be 10 times the size.

Well what is 10 times the size of two? It's 20.

That would be 20 groups of three.

Let's pile up the counters and counts.

So here you can see two groups of three, but we're going to make some piles.

Two groups of three counters is equal six counts.

What if we piled on another two groups of three counters? That would make four groups of three counters, so that's 12 counters.

So the number of groups is going up in twos.

The number of counters is going up in sixes.

Let's see if we can continue, see if you can predict what's going to happen next.

We've got six groups of three counters, that's 18 counters.

What's next? Eight groups of three counters, that's 24 counters.

10 groups of three counters is 30 counters.

Remember going up in twos for the number of groups and sixes for the number of counters.

12 groups of three counters is 36 counters.

What's next? 14 groups of three counters is 42 counters.

And next, 16 groups of three counters is 48 counters.

Getting there, almost there.

18 groups of three counters is 54 counters.

Then one last one, 20 groups of three counters is equal to 60 counters.

Aisha says now we've got 20 groups of three piled up altogether here.

There are 60 ones here, so the product is 60.

We started with two groups of three, which is equal to six.

Then we scaled up to by making it 10 times the size, 20.

Two multiplied by 10 is equal to 20.

That meant that the product was 60, which is also 10 times the size.

So once again, one of the factors was scaled up by making it 10 times the size and then the product was scaled up by making it 10 times the size.

Let's have a quick check for understanding.

Complete the equations to match the place value counter representations, pause the video.

What did you write for those equations? Let's have a look.

Well the first one is showing one 30 times.

So one multiplied by 30 is equal to 30.

And the second one is showing ten three times.

So 10 multiplied by three is equal to 30.

Alex and Aisha compare the equations from both sets of scaled up equations.

What's the same and what's different? You might want to take a moment to have a look and have a think what's the same and what's different.

Well different factors have been made 10 times the size in each equation.

So in the first one it was the three that was made 10 times the size.

And in the second one it was a two that was made 10 times the size.

The product is 10 times the size in both.

If one factor is made 10 times the size, the product will be 10 times the size.

That's really important, that's the crux of this lesson.

So I'd like to say that again, but this time will you read it with me please? Are you ready? If one factor is made 10 times the size, the product will be 10 times the size.

Okay, now just you.

You're gonna do it in three, two, one, off you go.

Let's do a quick check.

True or false, if one factor is made 10 times the size, the product will be 10 times the size.

Is that true or is that false? And more importantly why? Okay, pause the video.

It's true, if a factor has been made 10 times the size, then the value of each unit in the product is 10 times the size.

Alex and Aisha explore further.

She says, now we can use our times tables knowledge to find related multiplication facts.

Three multiplied by five is equal to 15.

That's a known fact.

I know that she says, so I can calculate three multiplied by 50 and also 30 multiplied by five.

Remember that to multiply a whole number by 10, you place a zero after the final digit to show the value is 10 times the size.

So in this case we're going to multiply one of the factors that's five by 10 and that becomes 50.

And so the product we will also multiply by 10 and that becomes 150.

So three multiplied by 50 is equal to 150.

And we can get that from our known fact.

Three multiply by five is equal to 15.

We could multiply the other factor, the three in this case by 10.

So what would that do to the product? It would mean we would multiply the product by 10 as well.

So 15 multiplied by 10 is equal to 150.

Alex and Aisha, complete a multiplication wheel.

This is a four times tables wheel.

I'm sure you've done lots of things like this before.

We need to know our four times tables here to find the products, yes you do.

Each sector is showing four groups of a number and a product.

So here we've got four multiplied by three and the product is 12.

Let's complete the rest.

Can you complete the products? You might want to take a little bit of time to do that.

But here are the answers.

What if we took the factor four number of groups and made it 10 times the size? So instead of being four times it was 40.

What do you think? How would that change the wheel? Remember if one factor is made 10 times the size, in this case the four, the product will be 10 times the size.

So we need to multiply each product this time by 10, so those outer numbers.

To multiply a whole number by 10, we can place a zero after the final digit to show it is 10 times the size.

So there we go, each of those products has been made 10 times the size 'cause the factor has been made 10 times the size.

Shall we now make the factor showing group size 10 times the size each time, says Aisha.

That's a good idea, I like the idea.

Let's do that.

What will change this time and how will it change? If one factor remember is made 10 times the size, the product will be 10 times the size.

Either factor still makes a product 10 times the size.

So we've explored one of them.

Let's explore the other one.

That's the middle numbers in this wheel, they've been made 10 times the size and the product has been made 10 times the size.

Now you might notice it's the same product as before when we made the four 10 times the size.

What is a mistake in the jottings below? Five multiplied by eight is equal to 40.

So 50 multiplied by eight is equal to 40.

That's not right, but can you explain it? Pause the video.

What was the mistake there? Well, Aisha says the factor of five has been made 10 times greater, but the product has not been scaled up.

So the product should also have been scaled up by 10 times.

So it should be five multiplied by eight is equal to 400, not 40.

Well done.

If you spotted that and corrected that.

Let's do some practise.

Number one, complete the jottings below by completing the missing numbers using your knowledge of times tables and your understanding of making a factor 10 times the size.

And hopefully lots of these tables facts are automatic to you now.

And number two, complete the times tables facts and use this to complete the multiplications featuring a factor 10 times the size.

So in the case of A for example, three multiplied by nine is equal to something.

So three multiplied by 90 is equal to something.

They're linked.

Number three, complete the related multiplication wheels.

So we've got six times and then various times tables.

And then instead of six we've changed that factor to 60, we've made that 10 times greater.

And then we've gone back to six, but made the other factor 10 times greater.

See what you notice.

And the same for B, this time we're using eight and 80.

Well I hope you get on well with that and I'll see you soon for some answers.

Pause the video.

Welcome back, how did you find that? Did you get some well with it? Let's have a look.

Number one, two multiplied by six is equal to 12.

So we're going to scale up that factor by 10.

So that becomes 60.

So that's two multiplied by 60 equals 120.

For B, we know two times six is equal to 12.

So therefore if we scale up the other factor two that becomes 20 multiplied by six is equal to 120.

And for C, six multiplied by four is equal to 24.

So if we scale up one of the factors, the four by 10 that gives us six multiplied by 40 is equal to 240.

We scale the product by 10 as well.

And D, six multiplied by four is equal to 24.

Therefore, 60 multiplied by four is equal to 240.

5 multiplied by six for A is equal to 30, therefore five multiplied by 60 is equal to 300.

And for F five multiplied by six is equal to 30, therefore 50 multiplied by six is equal to 300.

In each case we scaled up one of the factors by a factor of 10, meaning we scaled up the product by factor of 10.

Number two, complete the times tables facts and use this to complete the multiplication featuring a factor 10 times the size.

So three multiplied by nine is equal to 27.

So three multiply by 90 is equal to 270.

Once again we scaled up one of the factors, so we scale up the product as well.

And B, six multiplied by four is equal to 24.

So 60 multiplied by four is equal to 240.

C, four multiplied by eight is equal to 32.

So, 40 multiplied by eight is equal to 320.

D, two multiplied by 10 is equal to 20, therefore 20 multiplied by 10 is equal to 200.

E, nine multiplied by six is equal to 54, therefore 90 multiplied by six is equal to 540.

And F, seven multiplied by five is equal to 35.

So, seven multiplied by 50 is equal to 350.

And these multiplication wheels.

The first one was a straightforward times tables wheel, the six times table specifically.

And for the next one, the middle one we scaled up.

So it wasn't the six times tables wheel, it was a 60 times tables wheel.

And the products became 10 times the size.

And then we went back to the original times table six, but made the other factor 10 times the size.

Meaning, we made the product 10 times the size as well.

And did you notice they're exactly the same? The second two wheels have got the same products.

And the same for B.

This is the eight times table.

This is at 80 times table.

All the products are 10 times the size.

And this is at eight times table again.

But with the other factor made 10 times the size and the products have been made 10 times the size, they've been scaled up.

You're doing very well so far.

Let's look at applying that concept in context.

Alex and Aisha look at a multiplication.

This is 60 multiplied by eight is equal to something.

Previously we've started with a times table, says Aisha, but we don't have one this time.

Can you see a times table buried within that though? So now we have to decide which times table it is best to recall to help here.

I can sort of see one hidden there, can you? What do you think? Let's look, let's look carefully.

What times tables fact, jumps out at you for that? The six multiplied by eight is what I saw, which equals 48.

6 multiplied by 10 is equal to 60.

So we scaled at one of the factors by a scale factor of 10.

Therefore we need to do the same to the product.

So 48 multiplied by 10 is equal to 480.

Circle the most relevant times tables fact and then use it to find the missing number.

So we've got 50 multiplied by six is equal to something.

Which of those is most closely related to that? And then what's the answer? Pause the video and have a go.

Well, the answer is this one, five multiplied by six is equal to 30 is most closely related to 50 multiplied by six.

One of those factors, the five in this case has been scaled up by scale factor of 10.

So five multiply by six is equal to 30.

We multiply the five by 10 to get 50.

So that means we multiply the product, the 30 by 10 as well and that gives us 300.

Well then if you got that you're on track.

Alex and Aisha look at a problem involving equivalent expressions.

So four multiplied by 80 is equal to eight multiplied by, what do you think, what could the answer be here? Neither of these expressions are a product.

So how can we solve it? What do you think? You got a good strategy? What's your thoughts? This uses commutativity, that law where we can swap over the factors and still get the same product.

I know that four multiplied by eight is equivalent to eight multiplied by four.

How about you knew that too, didn't you? So four multiplied by eight is equal to eight multiplied by four.

I also know that either factor can be made 10 times the size for the product to be 10 times the size.

So in this case, look at this factor, it's been made 10 times the size to give us 80.

The missing number is 10 times the size of four, which is 40.

I see, says Aisha, both of these expressions give the same product even though different factors have been made 10 times the size.

Let's see if you've got that.

What is a missing number in the equivalent expressions below? 40 multiplied by eight is equal to multiplied by 40.

Pause the video.

Let's see, four multiplied by eight is equal to eight multiplied by four.

That's a commutative law.

The factor four has been made 10 times the size to give us 40.

And you can see that on both parts of the equation.

In both expressions four has been made 10 times the size.

So the unknown is unchanged, it's still eight.

There are four classes.

How many children are there all together? This time we need to work out what's being asked.

I don't think we have enough information, no we don't.

What's missing, what do you need to know here? What do you think? We need to know how many children are in each class so we can calculate the total, yes.

Try this, there are four classes, there are 30 children in each class.

That's better, isn't it? How many children are there all together? Okay, we can work with that, can't we? This involves multiplication because there are four classes of 30, which is four multiplied by 30.

Alex says, let's use a known times table to help us solve the problem.

Which times tables should we use? What's jumping out at you there? What times tables is that in that four multiplied by 30, what's buried within that? It's four multiplied by three.

Four multiplied by three is equal to 12.

So four multiplied by 30, each class has 30 children, which is 10 times as many as three.

So the product should also be 10 times as many.

So we've made one of the factors 10 times as many.

Let's make the product 10 times as many, that 12, what will that be? So four multiplied by three is equal to 12.

We've multiplied that three by a scale factor of 10.

So three multiplied by 10 is equal to 30.

So we need to do the same to the product, 12 multiplied by 10 is equal to 120.

There are 120 children altogether.

Circle the times table that would be the most useful for solving the following worded problem.

Are you ready? There are seven classes, there are 20 children in each class.

How many children are there all together.

So what's a times tables fact sort of buried within that that will be helpful.

Is it two multiplied by seven is equal to 14? Is it seven multiplied by two is equal to 14? Or is it seven multiplied by five is equal to 35? Pause the video and have a think.

What did you come up with? It's this one.

Seven multiplied by two is equal to 14.

That's very similar to seven multiplied by 20, which is what we're trying to find out.

And you may even have gone one step further and worked out that it's 140 children altogether.

And we can get that from that known times tables fact, seven multiplied by two is equal to 14.

Time for some final practise.

Number one, match the related pairs of calculations by drawing lines between them and then solve them.

So I can use this calculation and we've got some known times tables facts to help me solve this calculation where one of the factors has been made 10 times the size.

Number two, fill in the missing numbers.

Number three, match the equivalent expressions by drawing a line between them.

Number four, solve the following worded problems. Party bags can hold 40 sweets.

There are seven bags.

How many sweets are there in total? Right, I can see at times tables fact there.

B, each child in a class needs four exercise books.

There are 30 children.

How many books are needed? Once again, it's not a times tables fact as it is, but I can see one buried within those numbers.

And C, three classes each raise 60 pounds for charity.

How much have they raised altogether? What's the times tables fact that's jumping out at you there.

Okay, pause the video.

Best of luck with that, but I'll see you soon for some feedback.

Welcome back, how did you get on? How are you finding that? Are you confident? Let's see, number one.

I can use two multiplied by seven to help me solve 20 multiplied by seven.

Well two multiplied by seven equals 14.

So therefore 20 multiplied by seven is equal to 140.

We made one factor 10 times the size.

So we made the product 10 times the size.

Six multiplied by eight and six multiplied by 80 are linked.

Well six multiplied by eight, you might have known that automatically, hopefully you did, that's 48 and we can use that to work out that six multiplied by 80 is equal to 480.

One of the factors in this case, eight was made 10 times the size.

So the product was made 10 times the size.

Knowing three multiplied by nine will help us to work out 30 multiplied by nine.

Three multiplied by nine is equal to 27, it's a times tables fact.

So 30 multiplied by nine, that factor three has been made 10 times the size is equal to 270.

The 27 product has been made 10 times the size.

And five multiplied by six is linked to five multiplied by 60.

5 multiplied by six, our times tables fact is equal to 30.

One of the factors that's a six has been made 10 times the size to make 60.

So the product, the 30 has been made 10 times the size, going from 30 to 300.

Number two, find the missing numbers.

Five multiplied by 70 is equal to 350.

And we could use five multiplied by seven to help us with that, couldn't we? And then we made one of the factors, the seven 10 times the size.

So we made the product the 35, 10 times the size.

B is 270, C is 720, D is 280, E is 550, F is 480, G is 240 and H is 420.

And each of those could be worked out by starting with hopefully a known times tables fact and then scaling up one of the factors by a scale factor of 10 and then doing the same to the product.

Then number three, match the equivalent expression by drawing a line between them.

So two multiplied by 70 is equivalent to seven multiplied by 20.

And the law of commutativity helps us with that.

Three multiplied by 80 is the same as 80 multiplied by three.

40 multiplied by nine is the same as 90 multiplied by four.

110 multiplied by two is the same as 20 multiplied by 11.

And finally 12 multiplied by 70 is the same as 120 multiplied by seven.

And number four, solve the worded problems. A, party bags can hold 40 sweets.

There are 70 bags.

How many sweets are there in total? Well, four multiplied by seven is a times tables fact that I could see within that.

So four multiplied by seven is equal to 28.

For me that's automatic, maybe it is for you.

And we've made one of the factors 10 times the size.

So that's seven becomes 70.

So therefore the product is made 10 times the size, 28 becomes 280.

That's 280 sweets.

And for B, each child needs four exercise books.

There are 30 children.

Could you see four multiplied by three, that times tables factor in there? I could.

And again, for me that's a known fact.

Is it for you? Hopefully it is.

It's worth knowing these times tables facts off by heart.

You're more efficient when solving this kind of problem.

So four multiplied by three is equal to 12 and we've made one of the factors 10 times the size.

So we make the product 10 times the size.

So four multiplied by three is equal to 12, helps us to work out that four multiplied by 30 is equal to 120.

And that's the answer, 120 exercise books.

And then C, three classes each raise 60 pounds for charity.

That's not a times tables fact, but I can see one buried within there that will help.

That's three multiplied by six.

So three multiplied by six pounds is equal to 18 pounds.

We can scale up the factor, the six pound by 10, so that becomes 60 pounds.

We've gotta do the same to the product.

So that becomes 180 pounds.

That's how much it raised.

We've come to the end of the lesson and you have been amazing.

Today we've been explaining how making a factor 10 times the size affects the product.

If one factor is made 10 times the size, the product will be 10 times the size.

And we've explored the fact that it doesn't matter which of the two factors you choose.

This can be applied in context by drawing upon known times table facts and making one of the factors 10 times the size.

And therefore making the product 10 times the size.

It is important to analyse the numbers to select the most appropriate times tables fact to help.

Well, you've been great today, well done.

I hope you're proud of yourself and your achievements and I'm sure your teacher is too.

I hope I get the chance to spend another math lesson with you at some point in the near future.

But until then, have a fabulous day.

Whatever you've got in store, be the best version of you that you can possibly be.

Take care and I'll see you soon, goodbye.