Lesson video

Hello there.

My name is Mr. Tilstone and I'm a teacher.

How are you? Are you having a good day today? I hope we can make that day even better by having a successful maths lesson.

So if you are ready for that challenge, let's begin the lesson.

The outcome of today's lesson is this.

I can explain how making the dividend 100 times the size affects the quotient.

You might have some recent experience of making a factor 100 times the size and how that's affected the product.

I wonder what's going to happen this time.

Can you make a prediction? We've got three key words.

So if I say them, will you say them back please? Ready? My turn, dividend, your turn.

My turn, divisor, your turn And my turn, quotient, your turn.

Those words can be easily confused.

So let's have a little look at what they mean.

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 example, we've got 10 divided by 2 is equal to 5.

10 is the dividend, 2 is the divisor, and 5 is the quotient.

We've got two cycles in today's lesson.

The first will be making the dividend 100 times the size and the second applying the concept in context.

So let's begin by making the dividend 100 times the size.

And in this lesson, you'll meet Alex and Sam.

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

Alex and Sam label a division equation.

Let's see if you can do it too.

So it's 12 divided by 3 is equal to 4.

What can we say about each part of that? Well, 12 is a dividend because that's a number we are dividing.

And today we're going to be looking at making that dividend 100 times the size.

3 is a divisor because that's the size of the groups we are creating, or the number of groups we are dividing the dividend into.

And 4 is the quotient.

Alex and Sam show the division using a bar model.

So here's a dividend, 12.

That's the number we are dividing.

We split it into 3 equal parts because that's the divisor.

The quotient is 4 because that's the result of the division.

But Alex sees it a different way.

He says, I think there's another way of showing this division on a bar model.

Here's the dividend again, just like before.

This is the divisor.

This time we count up in the divisors of 3 to see how many fit into 12.

So 3, 6, 9, 12, it's 4.

The quotient is 4 just like it was before because that's how many groups of 3 fit into 12.

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

So in both the dividend is 12.

And in both the divisor is 3.

And in both the quotient is 4.

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

Read the context and fill in the missing numbers to create a division equation.

Label and explain your choices.

So Sam has 20 erasers, which she puts into piles of 5.

How many piles does she have? So think very carefully.

It's going to be hmm divided by hmm is equal to hmm.

Pause the video.

Let's see.

The dividend, that's the number of erasers, do we know that? Yes, we do.

And the divisor is the number of erasers in each pile.

And the quotient is the number of piles.

Do we know that? No, we don't.

So the dividend is 20.

We knew that the divisor is 5.

20 divided by 5 is equal to 4.

We didn't know that, that's what we've worked out.

And 4 is the quotient.

Sam and Alex look at a division.

It's 6 divided by 2 is equal to 3.

Nice and straightforward.

Let's model this with place value counters.

Have you got place value counters? If so, would you like to have a go before we show you? 6 divided by 2 is equal to 3.

You might have done something like this.

It's 6 divided into 2 groups.

Can you see those 2 groups of 3 ones? So you can see 6 divided by 2 is equal to 3.

Now what would happen if we make that dividend 100 times the size? And that's what we are really focusing on in this cycle.

So that dividend at the moment is 6.

What if we made it 100 times the size? What could we do? How could we represent that here? We could do it with counters.

The value of each counter needs to be made 100 times the size.

So instead of being 6 ones, it's 6 100s.

Each one counter has changed value to become a 100 instead.

So now we've got a different equation.

We've got 600 divided by 2.

So divided into two equal groups is equal to 300.

The quotient is now 300.

So hmm, what happened to that quotient there? We made the dividend 100 times the size.

What happened to the quotient? Sam and Alex, look at the equation.

6 divided by 2 is equal to 3.

We started with 6 divided into 2 groups of 3.

Then we scaled up the dividend by making it 100 times the size.

So it wasn't 6 anymore, it was 600.

That meant that the quotient was also 100 times the size.

It became 300 instead of 3.

Filling the missing labels that show what the dividend and quotient have been scaled up by in these related divisions.

So we've got 8 divided by 4 is equal to 2 and 800 divided by 4 is equal to 200.

So they're related.

What have we scaled that 8 by and what have we scaled the 2 by? Pause the video.

Well, 8 multiplied by 100 is equal to 800, and 2 multiplied by 100 is equal to 200.

So we've scaled them by the same amount.

They're both 100 times the size.

What's the same and what's different here? Have a look at these two sets of related equations.

6 divided by 2 is equal to 3.

600 divided by 2 is equal to 300.

Look what we've done to them.

Look how we've changed them.

Look what has changed.

And then 8 divided by 4 is equal to 2 and 800 divided by 4 is equal to 200.

They're related.

And again, see how they've changed.

See what part of them has changed and how.

Both equations, say Alex, featured different numbers.

Yes, they do.

In both, when the dividend was made 100 times the size, the quotient was 100 times the size too.

Did you notice that? So let's look again.

So 6 was made 100 times the size to become 600, in the same way that 8 was made 100 times the size to become 800.

They're the dividends.

Then the quotient 3 has been made 100 times the size to become 300 and 2 has been made 100 times the size to become 200.

So when the dividend was made 100 times the size, the quotient was 100 times the size too.

That's how the equation was affected.

But that's confusing, says Alex.

Doesn't division make whole numbers smaller? Hmm.

Yes, says Sam.

But we have scaled up the dividend not the divisor.

So we're starting with more to divide.

That makes sense.

So let's have a little check.

If the dividend is made 100 times the size, then the quotient becomes 1/10 times the size.

Is that true or is that false? Hmm, have a think about that and can you explain why? Pause the video.

True or false? True or false? That's false.

The amount being divided, the dividend, is 100 times the size, so the group size or number of groups is also 100 times the size, not 1/10.

Sam and Alex explore further.

Now we can use related division facts from times tables knowledge.

So here's a known fact.

Hopefully this is a known fact for you as well, you know, off by heart.

3 multiply by 5 equal to 15.

So by using the inverse, Alex knows that 15 divided by 5 is equal 3.

There we go, that's the equation.

Let's make the dividend 100 times the size.

That's what we're doing all through this cycle by multiplying by 100, says Sam.

Can you do that? What's 15 multiplied by 100? To multiply a whole number by 100, you place two zeroes after the final digit to make the value 100 times the size.

So now we've got 1,500 divided by 5.

That dividend is 100 times the size.

How do you think that's going to affect the quotient? It's also going to become 100 times the size.

So 3 multiplied by 100 is equal to 300.

15 divided by 5 is equal to 3, so 1,500 divided by 5 is equal to 300.

Let's do a check.

What's the mistake in the jottings below? 3 multiplied by 6 is equal to 18, so 18 divided by 6 is equal to 3, so 1,800 divided by 6 is equal to 3.

Hmm.

Part of that was not correct.

See if you can spot it.

Pause the video.

Did you spot it? It's this part, 1,800 divided by 6 is not 3.

The dividend of 18 has been made 100 times greater.

So that's been multiplied by 100.

So therefore, we should have also multiplied the quotient by 100.

So that 3 should have been multiplied by 100.

Can you correct it? What should it be? It should be this.

1,800 divided by 6 is equal to 300.

And you might think of it this way, 18 divided by 6 is equal to 3, 1,800 divided by 6 is equal to 300.

It's time for some practise.

Number one, complete the jottings below by completing the missing numbers using your understanding of making the dividend 100 times the size.

And number two, use the times table fact to complete the related division fact.

Then use your understanding of making the dividend 100 times the size to find the final quotient.

Number three, who do you agree with and why? Let's see.

Alex says, if the dividend is made 100 times the size, the quotient will be 100 times the size.

Is that true? Or is what Sam says true? She says if the dividend is made 100 times the size, the quotient will be 1/100 times the size.

Hmm, who's correct? See if you can explain that.

See if you can explain that clearly by using mathematical language.

Good luck with that and pause the video.

Welcome back.

How did you get on? Let's have a look.

Number one, let's give you some answers.

So 24 divided by 6 is equal to 4.

So therefore 2,400 divided by 6 is equal to 400.

We multiply the dividend by 100, so we have to multiply the quotient by 100.

And B, 32 divided by 8 is equal to 4, so therefore 3,200 or 3,200 divided by 8 is equal to 400.

And again, we've multiplied both the dividend and the quotient by 100 in each case.

This is C.

Here's D.

Here's E.

And here's F.

And again, in all of those examples, we multiplied both the dividend and the quotient by 100 to get our final answers.

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

So A, 3 multiplied by 9 is equal to 27.

That's a known fact hopefully.

So 27 divided by 9 is equal to 3.

So therefore, 2,700 divided by 9 is equal to 300.

We made both the dividend and the quotient 100 times the size.

This is B.

And here's C.

The same rule applied.

And number three, who do you agree with and why? In this case, Alex was correct.

If the dividend is made 100 times the size, then the number being divided is greater.

So it will be split into parts that are 100 times greater or there will be 100 times a number of groups.

You probably didn't explain it quite like that, but well done if you explained it nice and clearly.

Okay, now we're going to look at applying the concept in context.

Sam and Alex, look at a division.

72 divided by 8 is equal to something.

We can rearrange this division equation to create a missing factors problem.

Something multiplied by 8 is equal to 72.

And if you know your times tables off by heart that might have come to you straight away like that.

Let's see.

The dividend has become the product of a related multiplication fact or times table.

The divisors has become the second factor.

The first factor is unknown.

But this does make it easier.

Alex knows that 9 multiplied by 8 is equal to 72.

So that's our missing number here.

So I know that 72 divided by 8 is equal to 9.

Yes it is.

That's using the inverse.

Let's look at a different one.

400 divided by 2 is equal to something.

Previously we've started with a times table and related division fact We don't have one here.

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

Can you see a times tables fact in there or related fact? What do you think? Did you see this? Let's use 4 divided by 2 is equal to 2.

'Cause that has 4 as a dividend, which is 1/100 of 400.

So they're related.

So we can say that 4 multiplied by 100 is equal to 400.

How can we use that to find the quotient? If the dividend is made 100 times the size, the quotient will also be 100 times the size.

So let's do that.

2 multiplied by 100 is equal to? The missing number is 200.

2 multiplied by 100 is equal to 200.

So once again, the dividend has been made 100 times the size, the quotient has been made 100 times the size.

Let's do a little check.

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

So we are looking to find out 1,600 divided by 4 equals something.

What's helpful? Is it 16 divided by 4 is equal to 4? Is it 16 divided by 2 is equal to 8? Or is it 8 divided by 4 is equal to 1? Which one resembles it the most closely? Pause the video.

Is this one, 16 divided by 4 is equal to 4 can be used to help us with 1,600 divided by 4? The dividend has been made 100 times the size.

So therefore, the quotient needs to be made 100 times the size.

And 4 multiplied by 100 is equal to 400.

So well done if you've got that, you're on track.

Let's look at some problems in context.

A school buys 1,200 exercise books.

How many books does each year group get? This time, we need to work out what's being asked.

I don't think we have enough information.

No, we don't.

What do we need to know do you think here? What's missing? We need to know how many year groups there are because that will be the divisor.

Yes.

We've got our dividend, the 1,200, we just don't have the divisor.

What about this? These are shared equally between 6 year groups.

Okay, now we've got our divisor.

This involves division because there are 1,200 books being divided across 6 year groups equally.

So here we go.

1,200 divided by 6.

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

Which division fact should we use? Hmm.

Can you see a division fact within that? What do you think? I think we should use 12 divided by 6 is equal to 2.

Good call, Alex, so do I.

Because 12 is 1/100 times the size of 1,200.

So 12 divided by 6 is equal to 2.

And we've made that dividend 100 times the size to make it 1,200.

So we need to make the quotient 100 times the size as well.

So instead of being 2, it's 200.

Each year group gets 200 books.

Let's do a little check.

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

A school buys 900 exercise books.

They are shared equally between 3 year groups.

How many books does each year group get? So what's the known division fact that would help here? Is it 27 divided by 3 is equal to 9? Is it 10 divided by 2 is equal to 5? Or is it 9 divided by 3 is equal to 3? Which one resembles it the most closely? Pause the video.

Well, 900 divided by 3 is like 9 divided by 3.

They're related.

So that's the one that will be helpful.

9 divided by 3 is equal to 3.

And then well done if you took that a step further and solved it.

So we are making our dividend 100 times the size.

We're scaling up.

So we need to make our quotient 100 times the size.

We're scaling that up too.

So the three will become 300, 300 books.

It is time for some final practise and you are ready.

I can tell you are.

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

Number two, find the missing numbers.

And again, top tip, look for that known division fact buried within there.

And number three, solve the following worded problems. A, a factory makes 5,400, or if you like 5,400 squash balls every hour.

They are put into packs of 6.

How many packs are made every hour? Can you see a division fact? Hopefully a known division fact within there.

And B, the school council raises 2,400 pounds or 2,400 if you like, for 8 classes to spend on their class library.

If the money is shared equally between classes, how much do they each get? And I can see a related division fact within there that's known.

Well good luck with that and I will see you soon for some feedback.

Pause the video.

Welcome back.

How did you get on? Let's have a look.

So we can use 35 divided by 7 to help us work out 3,500 divided by 7 or 3,500 divided by 7.

They're related.

35 divided by 7 is equal to 5, so 3,500 divided by 7 is equal to 500.

48 divided by 8 is related to 4,800 divided by 8, or 4,800 divided by 8.

So 48 divided by 8 is equal to 6, so 4,800 divided by 8 is equal to 600.

81 divided by 9 is related to 8,100 divided by 9.

So the answer to the first is 9 and the second 900.

And 40 divided by 8 is related to 4,000 divided by 8.

40 divided by 8 is equal to 5.

We've scaled up the dividend is 100 times the size.

We're scaling up the quotient that's 100 times the size.

That's 500.

And the missing numbers.

So for the first one.

Did you start by thinking of 16 divided by 4 is equal to 4, a known division fact then work from there? Making the dividend 100 times the size and the quotient 100 times the size.

So 16 divided by 4 is equal to 4.

1,600 divided by 4 is equal to 400.

And here are the rest of the answers.

And in each of these cases, starting with a known division fact and then scaling up was helpful.

Well done if you've got those.

And number three, solve the following word problems. A factory makes 5,400, or if you like 5,400, squash balls every hour.

They are put into packs of 6.

How many packs are made every hour? Well did you start with this? 54 divided by 6 is equal to 9.

So 5,400 divided by 6 would've made it 100 times the size.

The dividend, we make the quotient 100 times the size.

So it's 900.

900 packs every hour.

And then the school council raises 2,400 pounds, or 2,400 pounds, for 8 classes to spend on their class library.

If the money is shared equally between classes, how much do they each get? Did you start with this? That known division fact, 24 divided by 8 is equal to 3.

And then did you scale up, making it 100 times the size? So the quotient is 100 times the size.

It's 300 pounds.

Well done if you got that.

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

Very well done.

I think you deserve a pat on the back.

We've been explaining how making the dividend 100 times the size affects the quotient.

If the dividend is made 100 times the size, the quotient will be 100 times the size as well.

This can be applied in context by drawing upon known times table facts and their related division facts before making the dividend 100 times the size.

So if you know your times tables really well and you know them automatically, that makes this even easier.

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.

I hope you're very proud of yourself.

I hope you have a wonderful day, whatever's left in store.

And that you are the best version of you that you could possibly be.

You can't ask for more than that.

I would love to spend another math lesson with you.

But in the meantime, take care and goodbye.