Lesson video

Hello there, my name is Mr. Tilston.

Give me a thumbs up if you are having a good day.

Give me two thumbs up if you are having a great day.

For me, it's two thumbs up.

And would you like to know why? It's because I love maths and I get to spend this math lesson with you.

So if you are ready to begin, let's begin.

The outcome of today's lesson is this.

I can explain how making the dividend 10 times the size affects the quotient.

You might have had some very recent experience at making a factor 10 times the size and you might remember that meant the product was also 10 times the size.

I wonder what's going to be true today.

Have you got a prediction? We've got some key words.

Three of them.

If I say them, will you say them back please? My turn, dividend.

Your turn.

My turn, divisor, your turn.

And my turn, quotient.

Your turn.

Those are not common everyday words and you may well need a reminder about what they mean.

So let's do that now.

The dividend is the number being divided.

The divisors is the number we are dividing by and the quotient is the result after division has taken place, it is a whole number.

So in this case we've got an equation.

10 divided by two is equal to five, 10 is the dividend, two is the divisor, and five is the quotient.

You might want to write that down or maybe your teacher will write it down so you've got something to remind you.

Our lesson today is split into two parts, two cycles.

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

So let's begin by making the dividend 10 times the size.

In this lesson you will meet Lucas and John.

Have you met Lucas and John before? They're here today to give us a helping hand with the maths and very good they are too.

Lucas and John label a division equation.

See if you can remember which parts are which.

12 is a dividend because that's the number we are dividing.

Okay, got that.

So 12 is a dividend.

Three is the divisors 'cause that's the size of the groups we are creating or the number of groups we are dividing the dividend into.

And finally four is the quotient because that's the result of the division.

Okay, think you've got that? Lucas and John show the division using a bar model.

Let's have a look at that.

What would that look like? Can you picture it before we show it? So here's a dividend.

12 is a dividend because that's a number we are dividing.

That's going to be the big part of the bar model, the large part on the top.

And then we split it into three equal parts because that is the divisor.

So you can see 12 divided by three.

12 has been divided into three equal parts.

Three is the divisor and then four is the quotient, the quotient's four, because that's the result of the division.

I think there's another way of showing this division on a bar model says Lucas, can you picture a different way too? Well we've got 12 as a dividend, again, just like before.

Just like before that's going to be the top part of our bar model.

And the divisor is three.

This time we count up in the divisors of three to see how many fit into 12.

Three, six, nine, 12, just like that.

So that's a quotient.

The quotient is four because that's how many groups of three fit into 12.

So there's two different ways that we can represent that equation.

What's the same and what's different? The values for the divisor, dividend and quotient are the same in both.

What's the dividend in both? It's 12.

What's the divisor in both? It's three.

What's the quotient in both? It's four.

The quotient is the group size in the first model and the number of groups in the other.

So it all depends on the context.

So read the context and fit in the missing numbers to create a division equation, label and explain your choices.

So we want those words, dividend, divisor, and quotient.

John has 20 erasers, which he puts into piles of five.

How many piles does he have? Okay, so you've got something divided by something is equal to something.

Pause the video.

Let's see.

Well he's got 20 erasers.

That's going to be our dividend.

The divisor is the number of erasers in each pile.

And the quotient is the number of piles.

And we know two of those numbers.

What do we know? We know the dividend.

That's 20 erasers.

We know the number of erasers in each pile.

That's five.

So it's 20 erasers divided into piles of five.

And then the quotient, we didn't know, but we can work it out.

That's a number of piles.

So that's the answer.

John and Lucas look at a division, six divided by two is equal to three.

Let's model this with place value counts as he says.

Have you got place value counters because you could use them.

Here we go.

Six divided into two groups of three ones.

So you can see there hopefully, six divided by two is equal to three.

What would happen if we make the dividend 10 times the size? Hmm.

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

So we can exchange each one counter for a 10 counter.

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

Now it's 60 divided into two groups of 30.

So the quotient is 30.

And you can see that in each of those two groups there's 30 in each of those two groups.

Three tens.

So six divided by two is equal to three.

60 divided by two is equal to 30.

Hmm.

John and Lucas look at the equations.

Here we go.

Six divided by two is equal to three.

We start with six divided into two groups of three.

Then we scaled up the dividend by making it 10 times the size.

So 10 times the size of six or six multiplied by 10 is equal to 60.

That meant that the quotient was 30, which is also 10 times the size.

So what do you notice here? What happened to the dividend and what happened to the quotient? Fill in the missing labels that show what the dividend and quotient have been scaled up by in these related divisions.

So eight divided by four is equal to two, 80 divided by four is equal to 20.

What's happened there to the dividend and the quotient? Hmm.

Pause the video.

Well done if you spotted they were scaled up by the same amount and that amount was 10.

They've been scaled up by 10.

They're 10 times the size, both of them.

Look at these two sets of equations.

Six divided by two is equal to three.

60 divided by two is equal to 30 and then eight divided by four is equal to two.

And 80 divided by four is equal to 20.

What can we say is the same? And what is different about those equations? Hmm.

Have a think.

Well both equations feature different numbers, says Lucas.

Yes they do.

But in both says John, when the dividend was made 10 times the size, so in this case six multiplied by 10 was 60 and eight multiplied by 10 was 80, the quotient was 10 times the size too.

So three multiplied by 10 became 30 and two multiplied by 10 became 20.

"But that's confusing" says Lucas, "doesn't division make whole numbers smaller?" "Yes," says John.

But we've scaled up the dividend, not the divisor." So we're starting with more to divide, a bigger number to divide so we're going to end up with a bigger number.

Well Lucas is still a bit unsure about that and maybe you are too, or maybe you're starting to get this.

He says can you represent it for me? Good idea.

So John uses a bar model to show what he means.

So this is showing six divided by two is equal to three.

So six is our dividend, two is our divisor, and three is our quotient.

This is the original equation as a bar model with a dividend of six.

But we multiplied that dividend by 10 and this is what happened.

We didn't have six anymore.

Got six multiplied by 10 is equal to 60.

Now we split this large dividend to two equal parts because the divisor hasn't changed.

It's still two.

So we still need two equal parts.

So it's not going to be three anymore, it's going to be 30 because 30 plus 30 equals 60.

So 60 divided by two is equal to 30.

"I see now," says Lucas.

"The quotient is also 10 times the size." Yes it is.

So the dividend was 10 times the size.

The quotient was 10 times the size.

What if the quotient is the number of groups not the group size? Hmm, good question.

Well this is the original equation as a bar model, but with the quotient as number of groups.

So this time we've got three groups.

We're still multiplying that dividend by 10.

And just like before, that's going to give us 60 is our new dividend.

Now we count up in twos because the devisor is the same this time and that's what would happen.

So there will be 30 groups.

So 60 divided by two is equal to 30.

So two different ways of representing that.

But in both cases you might have noticed that the dividend was multiplied by 10 and the quotient was multiplied by 10.

"I see now," says Lucas.

"There are 30 groups of two instead of three groups, which is 10 times as many." True or false little checkpoint.

If the dividend is made 10 times the size, then the quotient becomes one 10th the size.

True or false and why? Pause the video.

True or false? True or false? False.

Why? The amount being divided the dividend is 10 times the size.

So the group size or the number of groups is also 10 times the size.

So when the dividend is made 10 times the size, the quotient is made 10 times the size, okay? John and Lucas explore further.

Lucas says now we can use related division facts from times table's knowledge.

I know that John and Lucas are very good at their times tables.

They know a lot of them.

Hopefully you do too.

"I know," says Lucas, "that three multiplied by five is equal to 15.

So by using the inverse, I know that 15 divided by five is equal to three." Very good.

15 divided by five is equal to three.

So for Lucas, that's an automatic fact.

He knows it like that.

That's good.

"Let's make the dividend 10 times the size by multiplying by 10," says John.

Just like we did before, so let's help him out.

What's 15 multiplied by 10? To multiply by 10, you place a zero after the final digit to make the value 10 times the size.

Remember that? You're not adding a zero, you're placing a zero.

So that's 150 that's been made 10 times the size.

What are we going to do now do you think? We haven't changed the divisor.

What's going to change? If the dividend is made 10 times the size and it was the quotient will also be 10 times the size.

So we've seen that lots of times now.

So three multiplied by 10.

Hopefully that's a real quick one for you.

The quotient will be 30 because three multiplied by 10 is equal to 30.

So 15 divided by five is equal to three.

150 divided by five it's equal to 30.

Let's have a check.

What's the mistake in the jottings below? Three multiplied by six is equal to eight.

So 18 divided by six is equal to three.

So 180 divided by six is equal to three.

Hmm.

I think part of that was right and part of that was wrong.

See if you can spot it.

Pause the video.

Did you spot it? The dividend of 18 has been made 10 times greater but the quotient has not been scaled up.

The quotient should have been made 10 times greater too.

That three should have been changed.

Did you realise that? Did you correct it? This is what the correct answer is.

180 divided by six is equal to 30.

It is time for some practise and you are definitely ready for this.

Complete the jottings below by completing the missing numbers, showing your understanding of making the dividend 10 times the size.

Number two, use the times tables fact to complete the related division fact.

Then use your understanding of making the dividend 10 times the size to find the final quotient.

And remember all the way through this, when we've made the dividend 10 times the size, we've made the quotient 10 times the size as well.

See if you can remember that.

Number three, who do you agree with and why? Well, Lucas says, "If the dividend is made 10 times the size, the quotient will be 10 times the size." And John says, "If the dividend is made 10 times the size, the quotient will be one 10th times the size." Who's correct? Well best of luck with that.

I don't think you're going to need luck though.

I think you're going to smash this.

I'll see you soon for some feedback.

Welcome back.

How did you get on? Number one, 24 divided by six is equal to four.

So let's scale the 24 up 10 times.

That's 240 now.

So we've done the same to the quotient.

We've multiplied that by 10 as well.

The divisor hasn't changed.

So B, 32 multiplied by 10 is equal to 320.

So four multiplied by 10 is equal to 40.

And for C, these are the answers, is D, is E, and here's F.

And in all of those cases we multiplied the dividend by 10.

So we multiplied the quotient by 10.

Number two, use the times tables fact to complete the related division fact, then use your understanding of making the dividend 10 times the size to find the final quotient.

So three multiplied by nine is equal to 27.

That's a known fact.

So 27 divided by nine is equal to three.

So therefore 270 divided by nine is equal to 30.

Because we've made the dividend 10 times the size so we need to make the quotient 10 times the size.

And the same for B.

Here's B and here's C.

Number three, who do you agree with and why? Well in this case it was Lucas.

If the dividend is made 10 times the size, the number being divided is greater.

So it will be split up into parts that are 10 times greater or there will be 10 times a number of groups.

You probably didn't say exactly that, but hopefully you said something a bit like that.

You're doing ever so well and you're ready for the next cycle, which is applying the concept in context.

John and Lucas look at a division.

This is 63 divided by nine is equal to something.

Do you know the answer? We can rearrange this division equation to create a missing factors problem.

Something multiplied by nine is equal to 63.

Do you know the answer? This is a times tables fact now, isn't it? So the dividend has become the product of a related multiplication factor or times table.

The divisor has become the second factor.

The first factor is unknown.

That makes the division easier because we can apply our knowledge of known times tables facts.

I know that seven multiplied by nine is equal to 63, so it's gotta be seven.

Well done if you said that.

So 63 divided by nine is equal to seven.

John and Lucas look at a division that's 80 divided by four is equal to something.

Previously we started with a times table and related division fact, is this one of those? No, we don't have one here.

So let's look at the digits and use our understanding of times tables and related divisions.

I can see a times tables fact sort of hidden within that.

Can you? What do you think? Eight divided by four is equal to two because that has eight as a dividend, which is one 10th of 80.

So we've made 8 one 10th the size.

We could look at it the other way.

Eight multiplied by 10 is equal to 80.

And that should help us to find that missing value.

If the dividend is made 10 times the size, the quotient will be 10 times the size too.

So let's do that.

Here's the quotient.

Let's make it 10 times the size.

The missing number then must be 20 because two multiplied by 10 is equal to 20.

So 80 divided by four is equal to 20.

Let's have a little check.

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

So 27 divided by three is equal to nine or nine divided by three is equal to three.

Or 18 by three is equal to six.

Which one is most relevant when finding the answer to 90 divided by three is equal to hmm? Pause the video.

Which one do you think it was? Which one most resembled that? It was this one.

Nine divided by three is equal to three will help us to find out 90 divided by three is equal to something.

Well if nine divided by three is equal to three, we multiply nine by 10 to get 90.

So we do the same to the three to that quotient.

Three multiplied by 10 is equal to 30.

90 divided by three is equal to 30.

Very well done if you got that, you are on track and you are ready for the next part of the learning.

Let's have a look at some contexts.

60 children are going on a trip travelling by coach.

It sounds exciting.

If the children are divided equally between the coaches, how many children will there be on each coach? Hmm.

I feel like that's a bit difficult to answer at the minute.

Would you agree something's missing? This time we need to work out what's being asked.

"I don't think we have enough information," says Lucas.

I agree.

What information is missing? What do we need to know? We've got the number of children.

Hmm.

We need to know how many coaches are being used because that will be the divisor.

How about this then, 60 children are going on a trip travelling by coach.

There are three coaches.

That's better, isn't it? If the children are divided equally between the coaches, how many children will they be on each coach? I think we've got enough information now.

This involves division because there are 60 children being divided into three coaches equally.

So that's 60 divided by three is equal to something.

Let's use a related division fact to help solve this problem.

Which division fact should we use? Can you think of a division fact that's related to a known multiplication fact here? A division fact that sort of resembles this division fact? What do you think? "I think we should use six divided by three," says Lucas.

Yeah, I do.

That equals two.

Because six is one 10th times the size of 60.

So six divided by three is equal to two.

Now we can use that, can't we? 60 children is a dividend, which is 10 times as many as six.

So we've multiplied that dividend by 10 to get 60.

What else do we need to do then? We need to do something to that quotient.

We need to multiply that by 10 as well.

If the dividend is made 10 times the size, the quotient will be 10 times the size.

There are 20 children on each coach.

Let's have a check.

Circle the division fact that will be most useful for solving the following worded problem.

120 children are going on a trip travelling by coach.

There are three coaches.

If the children are divided equally between the coaches, how many children will there be on each coach? Hmm? Is it 24 divided by 12 is equal to two? Is that the helpful fact? Is it 12 divided by three is equal to four? Is that the helpful fact? Or is it 21 divided by three is equal to seven? Is that the helpful fact? And little hint, which one resembles it the most? Pause the video.

Well, it's this one.

12 divided by three is equal to four.

That's hopefully a known fact that we can get from three multiplied by four is equal to 12.

And a special well done if you went one further there and because you multiplied the dividend by 10 to get 120, you multiplied the quotient by 10.

So instead of four it's 40.

So the answer is there'll be 40 children on each coach.

Right, time for some final practise.

You've got this.

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

Number two, find the missing numbers.

And number three, solve the following worded problems. And I'll leave you to read those.

If you can work with a partner though, I always recommend that, then you can bounce ideas off each other and help each other out.

Okay, pause the video, good luck and I'll see you soon for some feedback.

How did you get on with that? Let's see, are you feeling confident? Let's give you some answers and you can compare.

So number one, 14 divided by seven is equal to something.

Well, I can use that to help me solve this.

140 divided by seven.

They resemble each other, don't they? They're linked.

So 14 divided by seven is equal to two.

140 divided by seven is equal to 20.

And then these two are linked.

So the answers are six and 60.

These two are linked.

The answers are three and 30 and these two are linked.

The answers are five and 50.

And find the missing numbers.

So for A, it's 50.

350 divided by seven is equal to 50.

And I used 35 divided by seven is equal to five to help me with that.

And here are the other answers.

Very well done if you've got these.

And number three, there are 350 sweets, there are seven jars.

If they are divided equally, how many sweets are there in each jar? Well, we could use our known fact, 35 divided by seven is equal to five as a starting point.

Then we make the dividend 10 times the size.

So we make the quotient 10 times the size as well.

So that's 50, 50 sweets in each jar.

And the next one, 30 divided by six is equal to five is our known fact that we could start with.

And once again, we made the dividend 10 times the size.

So we make the quotient 10 times the size, so that's 50, 50 packs.

And C is our known fact based on a times table 48 divided by 8 is equal to six.

We made the dividend 10 times the size, so we make the quotient 10 times the size and the answer is £60.

That's how much they get each.

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

Today, we've been explaining how making the dividend 10 times the size affects the quotient.

If the dividend is made 10 times the size, the quotient will also be 10 times the size.

And you've explored that time and time again today.

This can be applied in context by drawing up on known times tables facts, and their related division facts before making the dividend 10 times the size.

And it's so helpful if you know your times tables up by heart and it's automatic.

It's important to analyse the numbers, to select the most appropriate times table and related division fact to use.

Well done on your accomplishments and your achievements today, you deserve a pat on the back.

So why don't you give yourself one now.

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

But until then, have a fantastic day and be the best version of you that you can possibly be.

Take care and goodbye.