Lesson video

Hello there.

My name is Mr. Tilstone.

I'm a teacher.

It's a real treat to be here with you today to teach you this lesson, which is all about multiplying and dividing by 10 and 100, and that's something you've perhaps got lots of recent experience with.

Perhaps that's something you are getting really confident with.

Perhaps we can take you even further down that road today.

So if you are ready, I'm ready.

Let's begin.

The outcome of today's lesson is this.

I can use knowledge of the composition of 100; so in other words, how 100 is made up, to multiply and divide by 100 in different ways.

We've got two keywords today.

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

Placeholder.

Your turn.

And my turn.

Digit.

Your turn.

I'm very confident that you've heard those words before and use them a lot, but you might lack a little reminder about what they mean.

A placeholder is where we use the digit zero to hold a place in a number and maintain place value.

And perhaps you've got the experience of using one or even two placeholder zeros and a digit is one of the symbols of a number system.

In our number system we use a digit zero, one, two, three, four, five, six, seven, eight, and nine.

They are all examples of digits.

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

The first will be connecting multiplying by 10 to multiplying by 100, and the second will be connecting dividing by 10 to dividing by 100.

Let's go.

In this lesson, you're going to meet Jacob and Laura, have you met them before? They're here today to give us a helping hand with the maths.

Laura and Jacob use a Gattegno chart.

I really like Gattegno charts.

I think they're really useful for helping to understand certain maths concepts, including this one.

So we put sort of circle or ring around five and the same with 30.

What have we got? Now a circle around 700.

Now what have we got? We got a number.

What number have I made? What number has been represented on the Gattegno chart Can you remember? I'm sure you've used one of these before.

You've made says, Jacob, 700 plus 30 plus five and that equals 735.

Okay, well let's do a little check.

Let's see if you've understood Gattegno charts.

Complete the equation below to show which number is being represented on this Gattegno chart.

So something plus something is equal to something.

Pause the video.

Did you get it? That's 500 plus three is equal to 503.

Laura and Jacob use Gattegno chart.

Moving one row up on a Gattegno chart is multiplying by 10, says Laura, and you might have done that yourself recently.

You might have learned that yourself.

So therefore two multiply by 10 is equal to 20.

We moved one place up and the same again and again and again and again and again and again.

And finally nine multiplied by 10 is equal to 90.

We've moved one row up on the Gattegno chart.

So says, Jacob has summarised it like this when multiplying by 10, the products are all multiples of 10.

Yep.

So we had 10, 20, 30, 40, 50, 60, 70, 80, and 90.

They roll the products and they're all multiples of 10.

They all have a ones digit of zero.

Yes they do.

Ten's got a zero at the end, 20 has for the ones digit, 30 has and so on.

Now says Laura, moving two rows up on a Gattegno chart is multiplying by 100.

So we've got an example here.

One multiply by 100 is equal to 100.

We've moved two places up this time and the same for two multiply by 100 is equal to 200.

And then let's look at that using some other examples.

You can see the same things happening each time.

So finally nine multiplied by 100 is equal to 900.

We've moved two places up on the Gattegno chart to find our product.

Jacob says we're multiplying by 100.

The products are all multiples of 100.

Yes they are.

So in this case it was 100, 200, 300, 400, 500, 600, 700, 800 and 900.

They're all multiples of 100.

The tens and ones digits are zero.

They all have tens and ones digits of zero says Jacob, let's do another check.

Decide whether or not the boxes and arrows represent multiplying by 10 or 100 by filling in the missing numbers.

So you've got two examples there.

Look at the first one on the left.

So we've gone from 30 to 3000.

Is that multiplying by 10 or 100? Notice how many places we've moved up on the chart and the same for six and 60.

How many places have we moved up? So is that multiplying by 10 or 100? Okay, pause the video.

Did you get it? While moving two rows up on a Gattegno chart is multiplying by 100.

So 30 multiplied by 100 is equal to 3000.

And moving one row up on a Gattegno chart is multiplying by 10.

So six multiplied by 10 is equal to 60.

Laura and Jacob use a Gattegno chart.

Laura says, I'm going to show multiplying two by 10 on the Gattegno chart.

So she's put a circle around the number two and then she's multiplied that by 10 with one place upon the chart and that's given her 20.

That's two, multiply by 10 is equal to 20.

And there's the equation.

Now she says I'm going to multiply 20 by 10, okay? She's got a circle, a ring around the 20.

She's multiplying that by 10 and that's given her 200.

So the equation is 20 multiplied by 10 is equal to 200.

Jacob says, I'm going to show multiplying two by 100 on the Gattegno chart.

So Laura is multiplying by 10.

Jacob is going to multiply by 100.

So he put a sort circle or ring around the two.

He's multiplied that by 100 and that's given a product of 200.

Hmm.

What do you notice? Two multiplied by 100 is equal to 200.

That's our equation.

What can you see there? What can you say about that? What did you notice? Hmm? Look at those three equations.

Laura says, I noticed that 200 appears as a product in two of these equations.

Did you see that as well? The final two.

Multiplying by 100 is equivalent to multiplying by 10 and then multiplying by 10 again.

That's really important.

We're going to say that again.

This is the main idea in this lesson.

So I'll say it with you and then in the end you'll say it by yourself.

Okay, so do it with me.

Are you ready? Multiplying by 100 is equivalent to multiplying by 10 and then multiplying by 10 again.

Hmm.

Okay, now you say it, off you go.

That's our big generalisation today.

That's something I really want you to focus on.

Let's do a little check.

Fit in the missing numbers as 10 or 100.

Pause the video.

How did you do? Well, Jacob says, multiplying by 100 is equivalent to multiplying by 10 and then multiplying by 10 again.

A place value chart is used to explore further.

You might have a place value chart you might be able to draw on really quickly on your whiteboards.

You might have some counters.

You can represent this along with us.

So Laura says, to multiply by 10 on a place value chart, the counters need to move one column to the left.

And I'll bet you've done that before.

I bet you knew that already.

So here we go.

We're going to multiply those two ones by 10.

We're going to move them one place to the left and they're going to become two tens.

Now to multiply them by 10 once more, which is the equivalent to multiplying by 100.

So let's do that again.

So those two tens are going to move one place to the left and they're going to become two one hundreds to multiply by 100, says Jacob on a place value chart.

The count has move two columns to the left.

And again, I'll bet you've already done that before.

I'll bet you knew that already.

So here we go.

We're going to move them two columns to the left, two places to the left.

What do you notice? What can you see? What can you say? Once again, the final product is the same.

It's 200 in both place value charts.

So Laura multiplied by 10 and then multiplied by 10 again.

And Jacob just multiplied by 100, but he gave the same product.

Let's use digits this time, says Laura.

We will start with a two in the ones column of each chart.

To multiply by 10 and then 10 again, the digit will move the same way the counters did.

So they'll move one place to the left and then one place again to the left.

Here we go.

So two multiplied by 10.

That's two tens and two tens multiplied by 10 is equal to two one hundreds or 200.

Now let's look at Jacob doing the same thing.

The digit will also move in the same way as the counters did when we multiplied by 100.

So he's going to move his two places to the left, just one move and he's given us the exact same answer, two one hundreds.

Both place value charts have the digit two in the hundreds column.

So placeholders are needed.

What can we do now? There we go.

Let's add our two placeholder zeros.

Laura and Jacob, focus on the equations.

Two, multiplied by 10 is equal to 20 20 multiplied by 10 is equal to 200, and two multiplied by 100 is equal to 200.

What do you notice? To multiply a whole number by 10 place a zero after the final digit of the number.

And we're going to use colour to help you to understand that.

We repeat that so that there are now two zeroes placed after the final digit of that number.

And again, colour hopefully helps you to see that we've got one in red and one in blue.

To multiply a whole number by 100.

Place two zeroes after the final digit of that number.

Let's show these processes as equivalent using an equation.

Two multiplied by 10 multiplied by 10 is equal to two multiplied by 100.

They both give the same product, which is 200.

So multiplying by 10 and multiplying by 10 again is the same as multiplying by 100.

This expression shows multiplying by 10 and then multiplying by 10 again.

And this expression shows multiplying by 100, which is equivalent and does give the same product.

Let's do a little check.

What's incorrect about the equation below? It's not right, but can you explain why? It says eight multiplied by 10 is equal to 18 multiplied by 10, which is equal to 800.

Hmm, that's not right.

Why pause the video.

Hmm.

Did you spot it? Laura says the equal sign is not showing equivalence here.

It's not doing its proper job.

The expression eight multiplied by 10 as a value of 80 or as the other two expressions have a value of 800.

So 80 multiplied by 10 is 800 and then the other expression is 800.

So there aren't equivalent and there shouldn't be an equal symbol.

It should look like this.

Let's correct it.

Eight multiplied by 10, multiplied by 10 equals 800.

While Laura and Jacob is still using their wonderful Gattegno chart.

Hopefully you've got one of these yourself as well.

It's really helpful.

Shall we try starting with a two digit number? Says Laura, she's ready for the next step.

Jacob says, yeah, okay, let's start with 72.

We have to move both digits.

Hmm.

So how are we going to move them? Well, Laura likes to multiply by 10 and then multiply by 10 again.

So that's what she's going to do.

So each part of that is going to move up one place in the Gattegno chart.

So 72 multiplied by 10 is equal to 720.

Now to multiply by 10 again, which is equivalent to multiplying by 100.

So we do the same thing again.

So 720 multiplied by 10 is equal to 7,200.

Now let's start again with 72 and multiply by 100.

So let's do it the other way around this time.

So 72, we're gonna do it in one move.

To multiply by 100 we move both digits up two rows.

So let's do that.

So just one, move two rows up and that's what we are left with.

So 72 multiply by 100 is equal to 7,200.

They both gave the same answer, multiplying by 10 and multiplying by 10 again, it's exactly the same as multiplying by 100.

Laura and Jacob use a place value chart to show some calculations.

We will start with a two in the ones column and a seven in the tens of each chart.

72 in each.

Can you see that? Yep.

To multiply by 10 and then 10 again, both digits will move to the left twice like this so that seven will move once, twice just like this.

So the seven tens are now seven one thousands and the same for the two once, twice.

So the two ones are now two one hundreds.

To multiply by 100 both digits will move to the left twice.

So we'll do that in one fell swoop and it looks like this.

Now we need to put in placeholders to show that the product is 7,200.

Placeholder what? Placeholder zeros, just like this.

So both approaches gave the same product multiplying by 10 and multiplying by 10 again was the same as multiplying by 100.

What's the same and what's different? Two multiplied by 10 is equal to 20, 20 multiplied by 10 is equal to 200, and two multiplied by 100 is equal to 200.

And then we've got 72 multiplied by 10 is equal to 720.

720 multiplied by a 10 is equal to 7,200 or 72 multiplied by 100 is equal to 7,200.

What do you notice? The starting number is different.

One digit for the first one and then two digit for the second.

So two for the first one, 72 for the second.

But this doesn't affect how the digits change place value.

Both sets of equations show that multiplying by 100 is equivalent to multiplying by 10 and then multiplying by 10 again.

They're the same thing.

They do the same job.

It is time for some practise and I think you are ready.

In fact, you definitely already.

So number one, roll a 10 sided dice or perhaps you've got a spinner if you haven't got one of them to get a one digit number.

If you haven't got a spinner, maybe your partner can give you a random number.

Represent this on a Gattegno chart and multiply it by 10 twice.

Labelling your Gattegno chart as you do this, then restart with the original number and this time multiply it by 100, drawing on a label as well.

Now that sounds complicated, don't worry.

An example is shown below.

So here we've got two.

So you roll the dice, two came out, this is what you do.

Put a circle around the two.

You're multiplying it by 10 to give you 20 and then multiplying by 10 again to give you 200.

Or you're doing it in one move, you're multiplying by 100 to go from two to 200.

So just like that one.

Can you do that with some different numbers, three times please.

Number two, complete the chart below by filling in the missing numbers.

Number three, complete each table below by filling in the missing numbers.

And Jacob says, what does this show? Something to think about there.

And number four, Laura started a drawing club on Monday, she was the only member.

on Tuesday there were 10 times as many people as they were on Monday, on Wednesday there were 10 times as many people in the drawing club as they were on Tuesday.

How many people were in the drawing club on Wednesday? You might need to read that a couple of times.

Do take your time on that, make sure you fully understood the question and then have a go.

And if you can work with somebody else, I always recommend that.

Then you can make sure you've understood it and bounce ideas off each other.

Okay, well good luck with that.

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

Welcome back.

How did you get on? Are you feeling confident? Well, let's give you some answers.

Laura rolled her dice three times.

She rolled a two, a five and a nine and you can see what she came up with there.

So with the two she multiplied it by 10 and 10 again and that eventually gave her 200.

And then she also multiplied by 100 to go straight from two to 200.

And the same thing with five to get to 500.

And the same thing with nine, giving a product of 900.

Two different ways to do that.

Multiply by 10, multiply by 10 or just in one step, multiply by 100 and then complete this chart.

This is the complete row.

What do you notice? And this is a complete bottom row.

What do you notice? There's a very clear pattern.

The middle row is multiples of 10 and the bottom row is multiples of 100.

What does this show? Let's have a look.

So we've got two multiplied by 10 equals 20 multiplied by 10 equals 200.

We've got some other examples.

Here are the other answers.

Okay, now let's use the same starting point.

So 2, 6, 12, 17, 24 and 99 and multiply them by 100.

So two multiply by 100 is equal to 200.

What do you notice? It was the same as multiplying by 10 and 10 again.

And here are the other products you may notice, the final product is exactly the same as when we multiply by 10 and 10 again.

Multiplying by 100 is equivalent to multiplying by 10 and then multiplying by 10 again.

And then Laura started drawing club Monday, she was the only member.

Tuesday, there were 10 times as many people as there were on Monday, on Wednesday there were 10 times as many people in the drawing club as there were on Tuesday.

So did you pick up that multiplying by 10 multiplying by 10 again? How many people were in the drawing club on Wednesday? Well she says I was the only member on Monday.

So that was one person to start with.

That's Monday.

That was one.

Tuesday there were 10 times as many.

So that's one multiplied by 10.

That equals 10.

And then on Wednesday there were 10 multiplied by 10 again, 10 times as many people again.

So that gave you 100.

So there were 100 people in the drawing club on Wednesday.

Popular club.

I think you are ready for the next cycle.

You're doing ever so well, let's move on.

Now we're going to look at connecting dividing by 10 to dividing by 100.

Laura and Jacob decide to explore division.

These equations show what we learned about the connection between multiplying by 10 and multiplying by 100.

What about division? Let's explore the connection between dividing by 10 and dividing by 100.

Maybe you could have a little prediction here.

What do you think you're going to find out? Hmm, then we can prove it.

Division is the inverse operation to multiplication.

It's the opposite.

I wonder, says Jacob, whether dividing by 10 and dividing by 10 again is equivalent to dividing by 100 in the same way multiplication is.

What do you think? Hmm and how could you prove it? Do you agree with Jacob? What would you do? Well, we could use this Gattegno chart again, couldn't we? So moving one road down on a Gattegno chart is the same as dividing by 10.

So in this case, look, we've got 10 divided by 10 is equal to one, 20 divided by 10 is equal to two, and so on and so on.

So in each of these cases we're moving one place down on the Gattegno chart.

One row down.

So 90 divided by 10 is equal to nine.

When dividing a multiple of 10 by 10, you remove the zero in the one's place, says Jacob.

Yes you do.

Now moving two rows down on a Gattegno chart is like dividing by 100.

So this is 100 divided by 100 is equal to one.

So that's moved two places down, two rows down.

And let's see that with some other examples.

The same thing is happening each time.

We're dividing a multiple of 100 by 100 to give us a single digit answer.

And in this case load 900 divided by 100 is equal to nine.

When dividing a multiple of 100 by 100, you remove the zeros in the ones and tens place.

So that nine in the one hundreds column in that last example has moved two places to the right.

Okay, let's do a check.

Decide whether or not the boxes and arrows represent dividing by 10 or 100 by filling in the missing number.

Pause the video.

What do you think? Well Laura says, moving two rows down on a Gattegno chart is dividing by 100.

So we can write 100 there and moving one row down on a Gattegno chart is dividing by 10.

So we can write 10 there, we've moved one place down.

Shall we try dividing a four digit multiple of 100 to start with? I like your ambition, Laura.

She's ready for the next step.

Let's go for it.

So we've got a four digit multiple of 100, that's 7,200.

Now we've got to move both digits down.

Let's divide by 10 and then by 10 again, that's method one.

So that's move one place down.

So 7,200 divided by 10 is equal to 720.

And then again, 720 divided by a 10 is equal to 72.

So that was divided by a 10, divided by a 10 again or moving down one place.

Moving down one more place.

Now let's start with 7,200 again.

And this time divided by 100 in one step.

So the 7,200, we're going to move two rows down.

Each of those parts of that number is going to move two places down, just like that.

So 7,200 divided by 100 is equal to 72.

What do you notice? Is it what you expected? Laura says, I notice that 72 appears as a quotient in two of these equations.

Yes it does.

Can you see them? The last two.

Dividing by 100 is equivalent to dividing by 10 and then dividing by 10 again.

So it is similar to multiplying by 100, which is multiplying by 10 and multiplying by 10 again, it's just the inverse.

A place value chart is used to show the same calculations.

So we're gonna place the digits from 7,200 into their correct place value positions just like so.

Let's use both of those methods.

So to divide by 10 and then 10 again, both digits will move to the right twice.

Let's do that.

So the two one hundreds will move once, twice to become two ones.

And the seven one thousands will move once twice to become seven tens.

Now to divide by 100, both digits will move to the right twice.

So let's do that.

So that 200, let's do it in one jump, two places to the right.

That's now become two ones.

And this seven one thousands, two places to the right now become seven tens.

Well you can see very clearly that's given as the same quotient.

The quotient are the same.

The digits have moved and replaced the placeholders.

We don't need those anymore.

Laura and Jacob focus on the equations.

What do you notice? What can you see? What can you say? What can you deduce? What can you generalise? To divide a multiple of 10 by 10 remove the zero after from the ones column.

Here we go.

Then we'll repeat that again.

Removing the zero in the ones place again.

To divide a multiple of 100 by 100, remove the two zeros from the ones and tens column.

So 7,200 divided by 10 divided by 10 again is equal to 7,200 divided by 100.

They will give the exact same quotient.

This expression shows divided by 10 and then divided by 10 again.

And this expression shows dividing by 100, which is equivalent and does give the same quotient.

What's incorrect about the equation below? Can you explain? So 800 divided by 10 is equal to 80 divided by 10, which is equal to eight.

(Mr. Tilstone disagrees) It isn't.

So can you explain why not? Pause the video.

What did you say? Well, Laura says the equal sign is not showing equivalence here, just like it didn't before in that previous check.

The expression 800 divided by 10 has a value of 80, whereas the other two expressions have a value of eight.

So they aren't equivalent and they shouldn't be an equals symbol.

That's not the correct use of the equals symbol, but this is correct.

800 divided by 10 divided by 10 is equal to eight.

It is time for some final practise.

Complete the ratio chart below by filling in the missing numbers.

And then what do you notice compared to chart in task A? Asks Laura.

And then complete each table below by fitting in the missing numbers.

What does this show? What can we learn from this? You may notice that once again, the numbers are the same in both tables.

Number three, Lucas collects Battle Robot cards.

He has 400 in his collection now.

This is 10 times as many as he had last year, which is 10 times as many as he had two years ago when he started his collection.

How many Battle Robot cards did Lucas have in his collection at the start? Take your time on that again, think about that carefully.

Make sure you've understood.

And then number four, Lucas has read 10 times as many books as Laura this year.

Laura has read 10 times as many books as Sam.

Altogether they've read more than 300 books, but fewer than 500.

How many books could each of them have read? Hmm, that's a real thinker.

Do take your time on that.

And once again, if you can work with somebody else, I always recommend it.

Pause the video and away you go.

Welcome back.

How did you get on? Let's give you some answers and you can see.

So number one, here are the missing numbers in that row, the middle row.

What do you notice? All multiples of 10.

And in this row all multiples of one.

Jacob says, I notice the numbers are the same, but the top and bottom row have swapped.

Compared to that first chart in the first cycle.

That's because division and multiplication are inverse processes.

They're the opposite.

And then let's have a look at some answers here.

So 400 divided by 10 is equal to 40 and then 40 divided by 10 is equal to four.

And then here are the other answers in that first table.

Okay, that was dividing by 10 and dividing by 10 again.

This is just dividing by 100.

So 400 divided by 100 is equal to four.

What do you notice? The quotient is exactly the same as when we divided 400 by 10 and 10 again.

And here are the other quotients.

And you might notice they're exactly the same as in the other table because the original starting numbers were the same.

And from that we can say that dividing by 100 is equivalent to dividing by 10 and dividing by 10 again.

Lucas collects Battle Robot cards.

He's got 400 in his collection now.

That's 10 times as many as he had last year, which is 10 times as many as he had two years ago when he started his collection.

How many Battle Robot cars did Lucas have in his collection at the start? Well, Jacob says the question uses 10 times as many.

Did you pick that up? But we're going back in time.

So we need to use the inverse to undo, which means division.

So now it's 400.

Last year, 400 divided by 10, which is equal to 40.

Two years ago, 40 divided by 10, which is equal to four.

So the answer is he started with four cards.

And then Lucas has read 10 times as many books as Laura this year.

Laura has read 10 times as many books as John.

Altogether they've read more than 300 books, but fewer than 500.

How many books could each of them have read? Well, we're going to multiply by 10 to go from John to Laura and then multiply by 10 to go from Laura to Jacob.

Our combined total must be a three digit number between 300 and 500.

I must have read hundreds, Laura tens and John ones, but with the same digit.

Hmm.

I could have read 300 or 400 for the total to be 300 to 500.

So let's have a look at this.

So John, read four books, multiplied by 10.

That means Laura's read 40 books and multiply it by 10.

Jacob's read 400 books.

What a book worm.

That's fantastic.

And that equals, that's equivalent to 444 books.

That's a lot of books between them.

What about if it was three books that John had read? Multiplied by 10, that gives us 30 books for Laura.

Multiplied by 10 gives us 300 books for Jacob, and that equals 333 books.

That's our two possibilities.

That was a very challenging question.

So if you got that fantastic, brilliant work.

We've come to the end of the lesson.

Today we've been using knowledge of the composition of 100 to multiply and divide by 100 in different ways.

Multiplying by 100 is equivalent to multiplying by 10 and then multiplying by 10 again.

And likewise, dividing by 100 is equivalent to dividing by 10 and then dividing by 10 again.

They're inverse.

I think you deserve a pat on the back for your achievements and your accomplishments in this lesson.

You've been wonderful.

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

But until then, I hope you have a very productive and very wonderful day.

Whatever you've got in store, take care and goodbye.