Lesson video

Hello there.

My name is Mr. Tilston.

I'm a teacher.

One of my favourite things in the whole world is maths, so it's a real delight to be here with you today to teach you this maths lesson.

If you're ready for the challenge, let's begin.

The outcome of today's lesson is this.

I can scale division facts derived from multiplication facts by 100.

And we've got some keywords.

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

Dividend.

Your turn.

My turn.

Devisor.

Your turn.

My turn.

Quotient.

Your turn.

And my turn.

Scaling.

Your turn.

We've got some great maths words there, but they're not everyday words, so you might need a little reminder about what they mean.

So the dividend is a number being divided.

The divisor is a number we are dividing by.

The quotient is a result after division has taken place.

It's the whole number part.

So in this example, 10 divided by 2 is equal to 5.

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

Scaling is when a given quantity is made mm times the size.

It might be 2 times the size or 5 times the size or 1/10th times the size.

Our lesson today is split into two cycles.

The first will be deriving related division facts and scaling by 100 and the second will be multiplicative arithmagons.

Let's start by deriving related division facts and scaling by 100.

In this lesson you're going to meet Izzy and Jacob.

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

Izzy and Jacob represent a known multiplication fact using a bar model.

What could it be? Any guesses? What do you think? Something multiplied by something's equal to something.

What could it be? There are 4 equal parts making a whole.

So it's 4 multiplied by something or perhaps something multiplied by 4.

So 4 multiplied by something equals something.

Now we can choose.

Let's make the second factor 6.

It could be anything but let's go for 6.

So now we've got 4 multiplied by 6.

4 lots of 6.

Can you see that? What does that equal? 4 groups of 6 is equal to 24.

So the product is 24.

There we go.

It could also be described as 6 four times, which will be 6 multiplied by 4 is equal to 24.

Yes, multiplication is commutative.

We can swap over those factors and still arrive at the same product.

Izzy and Jacob discuss related division facts.

So something divided by something is equal to something.

What can you see there? Could you fill that in? Look at that bar model.

Which division fact is being shown? Well, 24 is being divided into equal parts.

So that's the dividend.

The dividend is 24.

24 has been divided into groups of 6.

So that's the divisor.

The divisor's 6.

And the quotient is 4 because that's how many groups there are.

4 is the quotient.

24 divided by 6 is equal to 4.

"I think there's another division fact shown here," says Jacob.

Do you think so too? Can you spot another one? I wonder if we can keep the dividend the same but have a different device still using that same bar model.

What do you think? The dividend will still be 24, because it's a number being divided, however we look at it, still the dividend.

The divisor could also be 4 'cause that's a number of parts.

Can you see 4 parts there? So that could be the devisor, instead of 6, four parts.

So 24 divided by 4.

And then what would that make the quotient? That would make the quotient 6 because that's the size of each part, each of those 4 parts.

They compare the multiplication facts with the inverse division facts.

What's the same and what's different? Have a good look at that.

What can you see? Well, says Izzy, "Both sets of related equations feature the same numbers." Did you spot that? 4, 6 and 24 appear in all of them.

"Yes," says Jacob, "But the order they appear in the equation reverses here." Multiplication and division are inverse processes.

They're opposites.

Although if the position of the factors change, we can still say they're related and you can see examples of that there.

The factors are changing position.

Here, the product of the multiplication becomes the dividend of the related division.

So you can see it's changing position in the equations from product to dividend.

And the first factor of the multiplication becomes the quotient.

Can you see that, the first factors of 4 and 6 there? There the first factors become 4 and 6, and the quotient in the division facts.

Let's scale up the dividend by 100.

To do that, we need to multiply it by 100 and I'm sure by now you're getting really confident at multiplying whole numbers by 100.

Remember to multiply a whole number by 100, you need to place two 0s after the final digit of that number.

We're not adding two 0s, we are placing two 0s.

Here we go.

So that's become 2,400.

The bar model looks wrong.

It does, doesn't it? We've made a change.

Do we need to make any more changes to it to make it correct? What do you notice? Well, we have scaled up the dividend.

It's a 100 times the size, but we haven't scaled up the divisors.

Hmm, you are right.

We need to multiply the divisors by 100 as well.

Let's do that.

So at the minute it's saying 6, but 2,400 divided into 4 equal parts, it's not 6, it's 600.

Let's write this scale up division fact as an equation and here it is.

So 2,400 divided by 600.

Is it equal to 4? Can you see that in that bar model? What do you notice? If you multiply the dividend by 100 and the divisors by 100, then the quotient remains the same.

So 24 divided by 6 is equal to 4.

2400 or 2,400 divided by 600 is equal to 4.

So the quotient hasn't changed.

True or false? If you multiply the dividend by 100 and the divisors by 100, then the quotient will be 100th times the size.

True or false? Pause the video.

True or false? True or false? It's false.

If you multiply the dividend and the divisor by 100, the quotient will remain the same.

Let's look at that example again.

24 divided by 6 is equal to 4.

2,400 divided by 600 is still equal to 4.

600 will go into 2,400 four times.

Jacob considers scaling further.

He wonders what would happen if we scaled up the dividend but not the divisor.

Okay, so that 24, scale that one up but leave the 6 the same.

What do you think what would happen? So let's make it 2,400 divided by 6.

We're scaled up that 24, it's 100 times the size.

"Why don't we turn this into a missing factors problem using the inverse," says Izzy.

Okay, so something multiplied by something is equal to something.

Something multiplied by something is equal to 2,400.

The dividend becomes the product.

The numbers reverse order in the related multiplication.

So we could say 6 is a second factor.

So something multiplied by 6 is equal to 2,400.

I know the multiplication fact 4 multiplied by 6 is equal to 24.

That's a known fact.

Hopefully it's for you.

The product has been scaled up by 100, that 24 has been made 100 times the size, which means one of the factors must have been too.

And you can see that one hasn't changed.

What's not changed? The 6 has not changed.

We know that already.

So it must be the 4.

So we're going to scale up 4 one hundred times.

Make that 100 times the size.

4 multiplied by 100 is equal to 400.

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

We can use that now.

That means that the quotient in the related division fact is also at 400.

So it can complete that equation.

What do you notice? 24 divided by 6 is equal to 24.

2,400 divided by 6 is equal to 400.

What do you notice there? If you multiply the dividend by 100 but keep the divides of the same and we've done that here, look.

So 24 has been made a 100 times the size, but the 6 has been kept the same.

You must multiply the quotient by 100.

Here we go.

So 4 multiplied by 100 is equal to 400.

We're going to read it again.

This time, will you read it with me? We're going to read what Izzy says and what Jacob says.

Are you ready? Let's go.

If you multiply the dividend by 100, but keep the divisors the same, you must multiply the quotient by 100.

Okay, very good.

Now one more time.

This time just you.

Okay, are you ready? 3, 2, 1, go.

Let's have a check.

True or false? If you multiply the dividend by 100 and keep the divisor the same, then the quotient will be one hundredth times the size.

Is that true or is that false? And can you explain it? Pause the video.

True or false? True or false? It's false.

If you multiply the dividend by 100 and keep the divisor the same, then the quotient must be 100 times the size, not one hundredth times the size.

So here's that example again.

24 divided by 6 is equal to 4.

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

It's time for some practise.

Number one.

Fill in the missing numbers in these bar models using your knowledge of times tables and scaling by 100.

And we've got some more examples here.

Number 2.

Complete the stem sentence below and 2 example equations to evidence a statement.

So if you multiply the mm by 100 and the mm by 100, then the quotient will remain the same.

What are those missing words? Think about our key words.

If you multiply the mm by 100 and keep the divisor the same, then the mm will be 100 times the size.

What are the missing words? Then can you give 2 examples? Number 3.

Find the missing numbers using your knowledge of multiplication facts and their related division facts.

Remember to consider your known multiplication facts there, your times tables facts.

You might find it helpful to work with somebody else so that you can share ideas with each other.

Pause the video and away you go.

Welcome back.

How are you getting on? Are you feeling confident about this? Are you feeling good? Let's give you some answers.

1a, the missing number is 4 in the bar model.

So therefore in the second bar model, the related bar model, the missing number is 400.

And for b, the missing number is 5.

So in the related bar model, that's 100 times the size, the 5 has become 500 is 100 times the size.

And for c, the missing number is 8 and in the related bar model 800.

And for d, the missing number is 7.

And in the related bond model that's a 100 times the size is 700, which is a 100 times the size.

We scaled it up by a 100.

Number 2.

Complete the stem sentences below and two example equations to evidence the statement.

So if you multiply the dividend by 100 and the divisor by 100, and you may have got those the other way round as well, then the quotient will remain the same.

And if you multiply the dividend by 100 and keep the divisors the same, then the quotient will be 100 times the size.

And number 3.

1,500 divided by 300 is equal to 5.

And we can derive that from the known division fact, 15 divided by 3 is equal to 5.

2,400 divided by 800 is equal to 3.

And again, we can derive that from 24 divided by 8 is equal to 3.

C is 400, d is 600, e is 9, f is 6, g is 900 and h is 1,200.

You're doing really, really well and you are ready for the next cycle.

So let's do that now.

That's multiplicative arithmagons.

Quite the mouthful.

Izzy and Jacob look at multiplicative arithmagons.

So here's an example.

What do you notice? Take some time to look at that.

What can you see? Can you see what's going on? Can you see how it works? Can you see where those numbers have come from? Izzy notices the numbers inside the square.

So look at those.

12, 28 and 21 are products of the factors in the circles adjacent.

12, that's adjacent to 4 and 3 in the circles.

4 multiplied by 3 is equal to 12.

Here we go.

So 4 multiplied by 3 is the same as, is equivalent to 3 multiplied by 4 and they both equal 12 no matter which way round you look at it.

We can also say the numbers in the squares are dividends if the numbers in the circles are divisors or quotients.

So we've got 12.

12 divided by 4 is equal to 3 and 12 divided by 3 is equal to 4.

So that's how we could get those circle values.

Okay, let's do a little check.

Write a multiplication fact and a related division fact from the numbers outlined in the arithmagons below.

Pause the video.

You might get more than one.

Maybe you got as many as four facts from that.

2 multiplication and 2 division.

So what's about 3 multiplied by 7 is equal to 21.

I can see that.

And 21 divided by 7 is equal to 3.

Or did you get 7 multiplied by 3 is equal to 21 that's using our commutative law, and 21 divided by 3 is equal to 7? Izzy and Jacob try scaling by 100.

Jacob says "Let's scale up one of the dividends by 100 and see what will change." Remember to multiply a whole number by 100, you need to place two 0s after the final digit of that number.

You knew that didn't you? You are confident with that? I can tell.

So let's change the 12 into 1200 or 1,200.

Now what else needs to change? Because at the minute that arithmagon is not correct.

We know that 12 divided by 4 is equal to 3 and the dividend of 12 has been multiplied by 100.

So we could also multiply the divisor by 100.

Well let's do that.

Here we go.

So 12 multiplied by 100 is equal to 1,200.

That divisor, that 4, we could also multiply that by 100 to create 400.

The quotient will remain as 3.

It is the 4 that will change.

Let's do that.

There we go.

Now that's better.

"Hang on, though," says Izzy.

"I've noticed something else." Okay, have you noticed something else? What do you notice? Jacob says, "There needs to be another change here.

28 divided by 400 is not equal to 7." You are right Jacob.

It's not.

Let's write it as a division fact again and look at what's been scaled up.

Okay, so we've got 28 divided by 7 is equal to 4, that's correct, but we scaled up that quotient by 100 to make 400.

So we need to scale up that divisor as well.

So it's not 28 anymore, it's 2,800.

2,800 divided by 7 is equal to 400.

And now the arithmagon is correct.

Izzy says, "I think there's more than one solution for the changes." Hmm.

Can you see another solution? Is there another way that we could alter the arithmagons? "Well," says Jacob, "I know that 12 divided by 4 is equal to 3, so I also know that 1,200 divided by 4 is equal to 300.

I know that 21 divided by 3 is equal to 7, so I also know that 2,100 divided by 300 is equal to 7." Let's have a check.

The numbers in the squares have been scaled up.

What will the 3 become? So think about what those numbers were before.

They've been scaled up by scale factor of 100.

They've been made 100 times the size.

What does the 3 need to change to? Pause the video.

Did you get it? We need to scale that up too, that needs to become 300.

Let's carry on.

Izzy and Jacob solve an arithmagon with missing numbers.

Let's start with the bottom circle.

I know that 3 multiplied by 5 is equal to 15.

So I know that 15 divided by 5 is equal to 3.

That's using our inverse.

3 multiplied by 5 is equal to 15.

That's our equation.

15 divided by 5 is equal to 3.

That's our equation.

15 multiplied by 100 is equal to 1,500.

The divisor of 5 remains the same.

That's not changing.

The 15 is multiplied by 100 to get 1,500.

If you multiply the dividend by 100 but keep the divisor the same, the quotient must be multiplied by 100.

So let's do that.

3 multiplied by 100 is equal to 300.

So we can write 300 here.

"I'll do the circle on the left," says Jacob.

"I'll write it as a missing factor problem." I like the teamwork here.

So mm multiplied by 5 is equal to 45.

Hmm, what's that? 45 divided by 5 is equal to something.

We can use our inverse.

It's 9.

There are 9 groups of 5 in 45.

So the missing number must be 9.

Nearly there.

Now to complete the arithmagons by working out the square, that's bottom left.

Can you do that? Have we got enough information? Well, we've got the circle values, haven't we? The 9 and the 300.

I'm going to use a related multiplication fact.

I know that 9 multiplied by 3 is equal to 27 and we've scaled up one of those factors by scale factor of 100 is 100 times the size, so it's 9 multiplied by 300.

So therefore we need to multiply the product by 100 as well.

So that's going to be 2,700.

The missing number is 2,700.

It is time for some practise.

Let's see if you can handle the arithmagons.

Number one.

Make the value of the selected square 100 times greater and then change other values in the arithmagon to ensure they have the correct place value.

Can you find more than one way? Number 2.

Find the missing numbers in these arithmagons using your understanding of scaling up multiplication and related division facts by 100.

And we've got a couple more examples here.

Good luck with that.

I hope you enjoy the challenge just like Jacob and Izzy did.

You might find it helpful to work as a pair.

Pause the video and I'll see you soon for some feedback.

Welcome back.

How did you find those arithmagons? They take a little bit of thinking about, a little bit of reasoning, but hopefully you got there in the end.

Let's find out.

So number one, we're making that 24, one hundred times the size.

We're scaling it up.

So that's become 2,400.

So therefore these values need to change.

So that's going to be 400 and 800.

Or we could do it this way.

2,400 and then 600 and 1,200.

We could scale up those values.

Here are 2 different ways of making that arithmagon correct.

Did you get one of those? Did you get both of those? Well done if you did.

And let's have a look at b.

We're scaling up that 72 by 100 to make 7,200.

So we need to make some changes.

So the circle could then become on the right 800 and the other value 5,600.

You might have gone the other way down the left, 7,200, and then the other two values would be not 9, but 900 and not 63, but 6300 or 6,300.

Did you get one of those? Well done.

Did you get both of those? Very well done.

And number 2, find the missing numbers in these arithmagons.

Here we've got 1400 or 1,400 and the factors of that are 700 and 2.

And for b, 4,400, the factors of that are 1,100 and 4.

And for c, we've got 4,200 and the factors are 6 and 700.

And for d, 60 would be the product of 12 and 5.

And then the product of 5 and 300 is 1,500.

So that's the missing numbers there.

We've come to the end of the lesson.

Today, we've been scaling division facts derived from multiplication facts by 100.

It's been a nice challenging lesson and I hope you've enjoyed that level of challenge.

I certainly have.

Scaling up by 100 requires multiplication by 100.

If both the dividend and the divisor are multiplied by 100, the quotient will remain the same.

If the dividend is multiplied by 100, then the quotient will be 100 times the size.

This knowledge can be used to solve multiplication puzzles like arithmagons among others.

Well done on your accomplishments, in your achievements today.

You deserve a pat on the back.

So why don't you give yourself a pat on the back now.

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

But until then, enjoy the rest of your day.

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

Take care and goodbye.