Lesson video

Hiya, my name's Ms. Lambell.

Thank you so much for popping along today to do some maths.

I hope you enjoy it.

Welcome to today's lesson.

The title of today's lesson is Checking and Securing Understanding Multiplication and Division.

This is within the unit arithmetic procedures with integers and decimals.

By the end of this lesson, you'll be able to use the mathematical structures that underpin multiplication and division of positive integers.

Most of what you see today will be a review and a recap 'cause remember, we are checking and securing into today's lesson.

Some words that we will be using in today's lesson that it would be worth having a little quick reminder of are partition or partitioning.

Partitioning is the act of splitting an object or a value down into smaller parts, and we're going to do that frequently throughout today's lesson.

The dividend is what we are dividing and the divisors is what we are dividing by.

So for example, in the calculation six divided by three, six is the dividend, and three is the divisor.

Like I said, we'll be using these words throughout today's lesson.

We're going to split today's lesson into two separate learning cycles.

The first of which we are going to make sure that we are really secure with our understanding of multiplication.

And in the second learning cycle, we are going to make sure that our, we are really secure with our understanding of division.

So let's get started, the first one, and we're going to concentrate firstly on multiplication.

So you will be familiar with partitioning numbers to make multiplication easier.

So this is something that you will have seen previously.

So for example, 20 multiplied by 300.

Now you'd probably partition that in your head, but for the purpose of this lesson, I want to partition this very carefully so we can see what's happening.

So 20, we could partition into two multiplied by 10 and 300 into three multiplied by 100.

We can then rearrange the order of those multiplication.

I can put my two and my three together, and then I can put my 10 and my a hundred together.

And then I can calculate two multiplied by three is six.

And then I can also calculate that 10 multiplied by a hundred is a thousand.

And then I can multiply those two together.

Now like I said, you probably did that type of partitioning but in your head, but like I said, moving forward, we are going to be using this method with some much harder numbers.

So I think it's really important to understand how we do each of these steps and why each step is important.

Andy and Sophia, we've got here, and they're calculating 24 multiplied by 15.

They both decide to partition the 24 and the 15.

Let's have a look and see what they've done.

So here is what Andeep's done.

He's partitioned the 24 into two multiplied by 12 and he's partitioned the 15 into three multiplied by five.

Maybe have a look now, what pairs do you think Andeep might make? What pairs of numbers are going to make this calculation a bit easier? Have a thinker moment.

Well, Andeep has decided that he's going to pair the two and the three and he's going to pair the 12 and the five.

Two multiplied by three is six and 12 multiplied by five is 60.

And then he's going to multiply those two together.

She's 360.

So there he did a partition, but in his head, did the six multiply by six and then multiplied that by 10.

Why do you think Andeep paired the 12 and the five? I think he probably paired the 12 and the five because then that gave us a nice number to work with 60.

Let's have a look and see what Sofia has done.

Sofia is doing the same calculation but decides to partition it in a different way.

So she takes the 24 and does four multiplied by six and takes the 15 and does three multiplied by five.

And then we're going to multiply those two together.

What pairing do you think Sofia might choose to do here? Have a thinker moment.

Let's see what she did.

She's decided to pair the four and the five and the six and the three and then multiply those two together.

Four multiplied by five is 20, six multiplied by three is 18, and then multiply those two together.

And then she's probably going to do a bit of partitioning in her head.

Two multiplied by 18 and then multiplied by 10.

We get exactly the same answers.

Why might Sofia have chosen to put the four and the five together? I think probably for the same reason that Andeep chose to put the 12 and the five together.

Can you think of some other ways to calculate 24 more supplied by 15? Think of other ways to partition those numbers.

Of course there are other ways.

What happens though when we start to think about multiplications with more digits, partitioning may not be useful in that situation.

For example, if we were going to do 573 multiplied by 429, I know that I would find that extremely hard to partition into something that was going to make that calculation easier.

We are now going to look at something called an area model.

Now you may be familiar with this or you may not, but we are going to go through it one step at a time.

So if it's something new to you, don't worry, you'll be fine.

We can represent the multiplication two multiplied by three using an area model.

So just as I did previously, I started off with some very easy numbers so that we could see the structure of what was happening.

We're going to do the same here.

Here's my area model.

It is two by three.

What is the area of that rectangle? The area of that rectangle is six, the area is six.

So that means that two multiplied by three is six.

That is what we mean by an area model.

How is that going to help us with a calculation like 25 multiplied by 13? We can represent this with an area model.

So here is my model and each box is worth 10 line.

What we're going to do is add some grid lines in to help us.

So here we can see I've got 10 and then three at the bottom.

And I've got two, lots of two boxes of 10 and then the five.

So I've got 13 and then I've got 25.

Now we can find the area of each of the different sections.

One of the big boxes is a hundred, one of the columns is 10 and one of the small squares is one, that's going to be useful to help us to very quickly calculate some areas in a moment.

So remember the big square is a hundred, a column or a row is 10 and one square is one.

Here's our calculation back again, then 25 multiplied by 13.

So here we can see we've got 200 and then 50 because we have five columns of 10.

And then 30 we have three rows of 10, Another three rows of 10, which is another 30.

And then if we count up those boxes, we've got 15.

So the area of that rectangle would be the sum of all of those parts, which is 325.

This means that the answer to 25 multiplied by 13 is 325.

And we've used the area model to help us find that answer.

You may be familiar with something called a grid method for multiplication, is actually just a version of the area model.

Sticking with the same calculation, so 13 multiplied by 25.

This time I've not split each of those sections into those smaller parts because we wouldn't want to be doing that particularly when we get to very big numbers.

So what we're going to do is draw a slightly simplified version of the area model we had previously.

So this is 10 and three and then 20 and five.

So notice I've split it into its tens and it's ones in each way.

Now I'm going to work out the area of each of the different rectangles.

So the area of the top left rectangle is 200, 10 multiplied by 20 is 200.

The top right is 50 because 10 multiplied by five is 50.

The bottom left rectangle is going to be 60 because three multiplied by 20 is 60.

And then the bottom right hand corner is going to be 15 because three multiplied by five is 15.

Notice here now I only have four rectangles, whereas previously I think I had six.

I'm going to get exactly the same answer.

So the area of the big rectangle, the entire rectangle is the sum of the area of the four smaller rectangles and we can see it gives us exactly the same answer.

So this is just a simplified version of the one with all the grid lines on.

It's not always possible however to draw an area model to scale, but it doesn't matter 'cause remember it's just a representation of what we are going to be calculating.

We're now going to calculate using our area model 34 multiplied by 218.

So 34.

And then notice going across this time I have three columns because I've got hundreds, tens and ones.

So I've got 200 and 10 and eight.

We're going to find the area of each of the separate sections.

So the first one is 6,000 because 30 multiplied by 200 is 6,000.

The next one is 300 because 30 multiplied by 10 is 300.

The next one is 240 because 30 multiplied by eight is 240.

The next one is 800 because four multiplied by 200 is 800.

The next one would be 40 because four multiplied by 10 is 40.

And then the last one in the bottom right corner is going to be 32 because four multiplied by eight is 32.

Now to find the area of the whole rectangle, we need to sum the area of those six separate rectangles.

So we're going to sum those, remember to make sure that all your ones are lined up, all your tens and hundreds, et cetera.

And the answer is 700, sorry, 7,412.

So that diagram there, I had tried to sort of draw it to some sort of scale, but you don't even need to go that far.

We can actually draw them and they're not to scale at all.

Now let's have a look at a three digit number multiplied by a three digit number.

So we've got a three digit by three digit, which means I need a three by three grid.

Across the top, I'm putting 245 and down the side I'm putting 324.

Would it matter if I put those the other way round? 'Cause I think in the previous questions I had my first number going down the left and my second number going across the top.

Will it matter? No, you're right, it won't.

Because we know from previous that if we've got a rectangle and we rotate it, it still has the same area.

We need to fill in all nine of these boxes then.

And remember, we are finding the area of each box even though it's not drawn to scale that that's how we're going to think of it.

So the first one is 60,000, the 300 multiplied by 200.

The next one, 12,000, 300 multiplied by 40.

And the top row last one, 1,500, 300 multiplied by five.

Onto the second row then.

So we've got 4,000, 20 multiplied by 200.

The next one is going to be 20 multiplied by 40, which is 800.

What two numbers are we going to multiply together to get the next box? Brilliant, well done.

Yes, we're going to multiply the 20 and the five, which is a hundred.

What about the next box? What are we going to multiply to find the area of the next box? The first one on the bottom row? Yeah, well done.

Four multiplied by 200 is 800.

The next one is going to be 160 and then the final one is 20.

What do we do now to find the area of the whole shape? That's right.

We sum together all of those numbers.

Looks a bit scary, doesn't it, but it isn't.

So just remember to make sure you've aligned all of your ones and tens, hundreds, et cetera, and then you'll know that you've got the answer right.

And the answer to this is 79,380.

Wow, big number.

So just to recap them, we do not need to draw these rectangles to scale.

Okay, we are finding the area of them, that's how we're thinking of it.

But we don't need to draw them to scale.

What we're going to do now is we're just going to have a go at one more question together using the area model and then you are going to have a go at a question independently.

And then when you've got, we know that you can do that, you'll be ready to start on the independent task.

So we're going to calculate 37 multiplied by 86.

So here's my grid, has to be two by two because I've got two digit number multiplied by two digit number.

And remember, it doesn't matter which way around I decide to put the numbers.

So the first box is going to be 80 multiplied by 30.

Eight multiplied by three is 24, multiplied by a hundred because remember 80 is actually eight multiplied by 10 and 30 is actually three multiplied by 10.

So here we've got 2,400.

Top right box.

We're going to do 80 multiplied by seven.

Eight multiplied by seven is 56, multiplied by that by 10 is 560.

Bottom left box we're going to do six multiplied by 30.

Six multiplied by three is 18, multiply that by 10 is 180.

And then the final box, we are going to do six multiplied by seven and that's 42.

We now need to sum all of those, making sure we've made all of our alignments.

And the answer to that is 3,182.

I think now you'll be ready to have a go at one of these by yourself.

If you are not, just rewind the video and go back through the examples and then you'll definitely be ready to go through the next question independently.

So here's the one that I'd like you to have a go at, please.

52 multiplied by 39.

So you'll need to draw yourself a two by two grid, 52 and 39.

Remember, doesn't matter if you choose to put them the other way round.

Pause the video, good luck with this.

Remember no calculators and when you've got an answer, come back and see how you got on.

Great work.

Let's have a look and see whether you have got the right answer.

I'm sure you have.

So we've got 1,500, 60, 450 and 18, and then when we add all of those together, we get an answer of 2028.

Did you get that right? Well done.

Task A.

Here what I'd like you to do is to use the area model and remember we are using the unscaled version to calculate the answers to these questions.

When you've done that, you need to find your answer in the grid and then you are going to write down the letter.

You'll have nine letters and what I'd like you to do is to rearrange those letters to reveal a word.

Don't worry if you can't reveal the word.

The most important bit is that you are able to multiply all of those numbers together.

Good luck with this and then come back when you're ready.

You can pause the video now.

Great work, well done.

Here are our answers then.

One, 1,508, which was an I.

Two, 2,146, which was a T.

Three, 1,312, which is a P.

Four, 4,922, which is an A.

Five, 1,508, which was an I.

Six, 9,310, that gave us an R.

Seven, 2,146, which was a T.

Eight, 4,482, that was an N.

And then nine was 8,975, which was an O.

And when you rearrange those letters, you get the word partition.

Remember, if you didn't get that, that doesn't matter.

The most important thing is, is that you are able to do those calculations using that area model.

Let's move on now then to looking at division.

So we're going to look at division.

Lucas knows that multiplication and division are inverses of each other.

So he wonders if he could use the area model to divide.

So if one is the opposite of the other, could we use the area model we've just done for multiplication? So we starting to have a think about that.

What do you think? Well let's take a look, shall we? Let's see.

Let's look at 90 divided by six.

Now this time we know the answer, we know the total area of the rectangle is 90.

We also know one of the dimensions is six.

What we need to do is we need to work out this missing dimension here.

I'm just going to move that 90 up 'cause I'm going to start writing some things in my rectangle, right? If I split my rectangle here, I put 60 there, 10 multiplied by six is 60.

The area of my entire rectangle was 90.

So what needs to go in that final rectangle? 60, add what makes 90? And that's 30.

So how long is that bit? It's five, because six multiplied by five is 30.

So how long is that entire side of the rectangle? Well it's 10 and five, so it's 15.

So Lucas is right, we can use the area model to do a division, 90 divided by six is 15.

Lucas has also decided that partitioning numbers could be a useful way to divide.

Here are some ways of partitioning 98.

Can you think of any more? So I'm going to show you some and then I'm going to pause for a moment to see if you can think of any more.

So 95, add three.

70 add 28.

50 add 48.

80 add 18.

And 90 add eight.

Can you think of any more ways of partitioning 98? I'm sure you can.

Wonder how many you came up with.

There are lots and lots of different ways.

What I want you to do now though is to think about which of those partitioning is most useful if we want to calculate 98 divided by seven.

So look at the partitions I've given you on the right hand side, which of those is most useful if I'm dividing by seven? And Lucas has decided that 70 add 28 is the most useful.

Why do you think Lucas might have said that that is the most useful if we are dividing by seven? Yeah, that's right.

'Cause both 70 and 28 are divisible by seven.

So I could divide each part separately and then I could then put my answers back together.

We could also do 98 divided by seven using the area model.

So we know one dimension is seven.

What's the other dimension? We know that 10 gives us the 70.

What's the last bit of the rectangle got to be to make 98? That's 28.

And then 28 divided by seven is four.

So we've ended up with an answer of 14.

Let's see how that compares to the short division method.

So the short division method, I'm sure you are very, very familiar with.

Let's have a look.

So we do seven into nine is one remainder two.

Notice we had one 10 in our rectangle and then we had a remainder of 28.

We've got exactly the same here, haven't we? We've got a remainder of 28.

Now we're going to do sevens into 28, which is 14.

So we can see that the two methods are really linked.

Wonder which one you prefer? Let's have a look at another calculation.

259 divided by seven.

Here's my rectangle.

So we're going to use the area model first.

So we know one of the dimensions is seven and we need to work out the other dimension.

So this time I've got 70.

Can I make another 70? Yeah I can.

Can I make another 70? 70 and 70 is 140.

Add 70, 210.

Yep I can fit another one in.

So this is 210.

What's left of my 259 is 49.

And 49 divided by seven is seven.

So here we can see that the answer is 37.

This is only one way to partition 259.

Could you have done it differently? You might have spotted that you could have just done 30, which you knew was 210 and then the seven.

Doesn't matter how we get there, as long as we get to the right answer.

Let's now compare that with the short division method.

Seven into two.

Well that doesn't go, so we're going to do sevens into 25, which is three.

And we can see here looking at our area model, yes we did get three tens, didn't we, out of that rectangle.

Our remainder, so sevens into 25 is three remainder four.

Yeah, that's the same isn't it? As on our area model, we were left with 49.

And here we are left with 49 and then 49 divided by seven is seven.

So like I said, you can see how closely linked these two methods are.

Which do you prefer now? Has it changed because the numbers have got slightly bigger or are you still happy with the one you were previously? Doesn't matter.

Remember any method is great.

What I'd like you to do here on this check for understanding is to decide what you think the value of each letter in the diagram is.

So we've just been through a couple of examples of splitting up using the area model to do a division.

Now it's your turn to have a go.

So you're going to work out the value of each letter.

Pause a video, and when you're done you can come back.

Great work.

Let's have a look at those answers.

So A was 294.

That was the area of the entire rectangle.

B was 60.

C was 54.

D was nine and E was 49.

Absolutely superb if you've got that right.

Well done.

We're going to go back to our partitions for 98.

Because those two methods we've just used are useful, but sometimes these partitions could be useful as well.

So let's have a look at these.

98 divided by two.

Have a think which of those partitions is most useful for that calculation.

Any of them with even numbers.

Any with even numbers would be useful because remember we know even numbers we can half easily.

So any of those would be suitable for 98 divided by two.

You might have a preference over one rather than another.

What about if we wanted to do 98 divided by four.

Which of the partitions is most useful now? The divisibility test for four is that the last two digits must be a multiple of four.

So maybe none of those.

98 is not divisible by four.

This means that the answer would be a decimal.

So maybe the best partitions would be those, if at least one of them was a multiple of four.

So 70 and 28.

'Cause 28 is a multiple of four.

50 plus 48, because 48 is a multiple of four.

80 plus 18, because 80 is a multiple of four.

And 90 plus eight, because eight is a multiple of four.

Now have a think about if we were dividing by five.

If our divisor was five, which is the most useful? 98 is not divisible by five as it doesn't end in a zero or five.

The most useful would be those with one number that is a multiple of five and this would be all of them.

All of them have a multiple of five.

95, 70, 50, 80 and 90.

Now have a think about which is most useful divided by three.

Any of the partitions containing multiples of three.

Now remember from previous lessons the divisibility test for three is the sum of the digits is a multiple of three.

So the most useful would be which ones.

50 and 48, 80 and 18, and 90 and eight.

For all of these there may be a more useful partition that we've not shown.

You may have written it down when I asked you to write down some of your own ideas earlier.

We are now ready to move on to our final task for today's lesson.

And we are going to firstly in question one, partition 52, to answer the following.

So think carefully about how you would partition 52 just like we've been through with 98 to answer those questions.

For question two, you are going to use the area model to complete those divisions.

Good luck with that.

Remember no calculators.

Pause the video and come back when you're ready.

Question three and question four.

Sam is doing a calculation involving division and has chosen to partition their dividend into 40 plus 16.

What might their divisor be? And four, Sofia is doing a calculation involving division and has chosen to partition her dividend into 50 plus six.

What might her divisor be? Pause the video, have a think about these two questions and then come back when you're ready.

Great work.

Let's check our answers then.

One A, where these are just examples.

So 40 divided by four is 10 and 12 divided by four is three.

So 52 divided by four is 13.

You may have decided to partition it in a different way.

That's okay.

Remember that's why it says e.

g.

B, 50 divided by two is 25 and then two divided by two is one.

So 52 divided by two is 26.

And C, 30 divided by five is six.

22 divided by five is 4.

4.

So 52 divided by five is 10.

4.

Use the area model to complete the following.

So for example, there's an example there, but the answers are here.

A is 46.

B, 63.

C, 76.

D, 73, and E, 347.

Question three.

I think Sam's divisors would've been four or eight and Sofia's two, five or 10.

You may have something different there.

We're now ready to summarise our learning from today's lesson.

You've done superbly well.

Let's just summarise what we've done.

So we started off by looking at the fact that we can can partition integers to make multiplication easier.

So we've got an example there.

That was the first example we went through or one of the first.

Any multiplication can be represented using the area model.

The grid method is an unscaled version of the area model.

Remember we don't have to draw it to scale because it's just a representation.

And then division can sometimes be made easier by partitioning and we can see there the partitioning by using the area model.

Remember, this is not the only way, and you may still feel that you prefer to use the short division method, which is absolutely fine.

Thank you so much for joining me today.

I've had a really good time.

I hope to see you soon.