Lesson video

Hello there.

My name is Mr. Tilstone.

It's really lovely to see you today, and to be working with you on this math lesson, which is all about area.

It's actually the final lesson in the area unit.

So today is a chance to really showcase all of the skills that you've been developing.

Some of the questions might seem quite tough, but with perseverance, resilience, a positive attitude, and maybe some teamwork, you're gonna smash it.

So if you are ready, let's begin.

The outcome of today's lesson is "I can use knowledge of area to solve problems in a range of contexts." Our keywords or keyword today is my turn, "Dimension", your turn.

You may very well have encountered that word dimension before.

Can you remember what it means? Would you like a recap? "A dimension is a measure of length in one direction." So a length is a dimension.

A width is a dimension.

Our lesson today is split into two cycles.

The first will be, "Area problems involving rectangles." And the second will be, "Area problems involving compound shapes." So if you're ready to begin, let's start with "Area problems involving rectangles." In this lesson, you're going to meet Lucas, Jun, and Sam, and they may very well be familiar to you.

May well have met them before, they're going to be here today to lend a helping hand.

So, "Lucas and Sam are playing a guessing game where they have to imagine and visualise rectangles." Lucas says, "I'm thinking of a rectangle with an area of 36 centimetres squared.

What are the dimensions of my rectangle?" Hmm.

Sam says, "Hmm.

Are the dimensions a known times tables fact?" Good question.

"Yes." "Sam considers all of the times tables she knows that give a product of 36." Can you think of any, something x something = 36? "She uses a systematic approach, starting with the smallest possibility for her first number." "Okay?" She says, "I'm visualising a rectangle with the dimensions three centimetres and 12 centimetres." So that'll be 3 x 12.

That would give an area of 36 centimetres squared.

"Is that right?" She asks him.

"No." He says.

Hmm, okay, so it must be something else.

So if you thought 3 x 12, that's not right.

Think again.

Maybe you've got a different possibility.

She says, "Okay, I'm visualising a rectangle with the dimensions four centimetres and nine centimetres.

Is that it?" Well, let's have a think.

4 x 9 is 36.

So the area of that rectangle would be 36 centimetres squared.

It could be, couldn't it? Let's see.

"No." He says it's not.

Hmm, have you got another one in mind at times tables fact? One thing x something = 36.

"Only one possibility remains!" She says.

"I am visualising a rectangle with a dimension six centimetres, and six centimetres.

Is that it?" "Yes! Well done, Sam." He says.

So that's a square.

A square is a special kind of rectangle.

6 x 6 = 36.

So that square or rectangle has got an area of 36 centimetres squared.

"They play again." "I'm thinking of a tall thin rectangle, with an area of 48 centimetres squared." Says Lucas, "What could the dimensions of my rectangle be?" So maybe you want to play along with this as well.

So again, something x something = 48.

Have you got any times tables facts in mind? She says, "Hmm.

If it's tall and thin, then one of the dimensions must be a lot smaller than the other." So the dimensions couldn't be say six centimetres and eight centimetres, even though that does equal 48 centimetres squared, doesn't it? But it couldn't be that 'cause he said "It's a tall, thin rectangle." So he says, "No." No it's not.

"It could be one centimetres and 48 centimetres!" She says.

It could be, couldn't it? It's not a times tables fact, but 1 x 48, does equal for 48.

That would be very thin long.

But it could be, let's see.

"It's not." He says, "It's not as tall and thin as that." Okay, any more possibilities? "Right." She says, "Next try.

Are the dimensions two centimetres and 24 centimetres?" Let's also do a check first.

2 x 24 is 48.

So that rectangle would've an area of 48 centimetres squared.

Could it be that? Let's see.

"Could be." He says, "But no.

Try again." Hmm.

Have you got a different possibility in your mind? What else could it be? "Are the dimensions", she asks, "three centimetres and 16 centimetres?" And once again, that's not a times tables fact, but when we multiply those numbers together, it does equal 48.

So it could be, let's find out.

"Yes!" He says, "Well done, Sam.

You got there in the end." And well done if you got that, and that is what that rectangle would look like, it's not to scale.

"They play again." They like this game.

Lucas says, "I'm thinking of a rectangle with an area of 42 centimetres squared.

This one is close to being a square, but it isn't.

What could the dimensions be?" Okay, we'll think about that then.

So not a square, but almost a square.

Any idea? Something x something = 42.

Hmm.

(tongue clicks) Should we investigate? Sam says, "Hmm.

If it's close to being a square, then the dimensions must be close to being the same." That's true, isn't it? If the dimensions were six centimetres and six centimetres, it will be a square with an area of 36 centimetres squared." Which actually is pretty close to 42, isn't it? So I think she's on the right track here.

"And if the dimensions were seven centimetres and seven centimetres, it would be a square with an area of 49 centimetres squared." Hmm.

That's close too, isn't it on the other side? "So I think it's in between those.

Are the dimensions six centimetres and seven centimetres?" Sounds plausible, doesn't it? 6 x 7 is 42.

Those numbers are close together.

Let's find out.

"Yes! Well done, Sam.

You're getting really good at this!" And well done to you if you thought it was seven and six as well.

Here we go.

Seven and six or six and seven.

Let's do a check.

You are going to have a go at that game, now.

"Visualise or you could even draw", maybe you've got a whiteboard, something like that, a rectangle with a specific area.

"You may wish to use a stem sentence to help you." So let's have a look at that stem sentence.

"The area of the rectangle is hmm centimetres squared." "One dimension is hmm centimetres." "The other dimension is hmm centimetres.

"Hmm x hmm = hmm." And you might notice they're colour coded to help you out.

"Tell your partner the area.

They must work out the dimensions of the rectangle.

You may wish to give clues as Lucas did in the last two rounds." (tongue clicks) Pause the video, have fun with that.

Have a go, have a few go's even.

Welcome back.

So lots of things you might have said.

"For example, you may have given your partner the area 24 centimetres squared, and a clue that one dimension is more than double the other one." So you could be really kind of creative with that.

Well done if you gave a really tricky example, and well done if you managed to solve a tricky one, too.

Let's have another check.

"A rectangle has an area of 24 centimetres squared.

How many possible pairs of dimensions could it have that are in the times tables up to 12 x 12?" So something x something = 24 centimetres squared.

You could use that stem sentence again.

"The area of the rectangle is 24 centimetres squared.

One dimension is hmm.

The other dimension is hmm, hmm x hmm = hmm." Pause the video and off you go.

Did you manage to find a possibility? Did you manage to find more than one possibility? Let's have a look.

Well, 2 x 12 is in our times tables facts, our instant recall facts, 2 x 12 is 24.

And that fits.

So 24 centimetres squared.

(tongue clicks) And you might have had those the other way around.

It doesn't matter which one is the length and which one is the width.

They're both dimensions, so 12 centimetres x 2 centimetres = 24 centimetres squared.

Maybe you had 3 centimetres x 8 centimetres giving 24 centimetres squared.

And it might be the other way around as well.

Maybe you had 4 centimetres x 6 centimetres = 24 centimetres squared.

And maybe you had that the other way around.

So, once again, well done if you got one of those, well done if you got two of those, and very well done if you got all three of those.

Time for some independent practise.

Number one, "Sam wants to make a rectangular bookmark with an area of 30 centimetres squared for Lucas's birthday.

What could the possible dimensions be? Give as many probabilities as you can." So hmm x hmm = 30 centimetres squared.

You could use examples that are times tables and maybe some that are outside the times tables.

Two A, "Jun wants to give his best friend Lucas a small frame photo of the two of them for his birthday." Lucas is popular isn't he? Isn't he? "He's choosing between two photos.

He wants to use the one with the larger area." Now looking at those two photos, I couldn't really see which one's got the larger area 'cause the dimensions are so different.

I think it's gonna be quite close.

"One has dimensions of 6 centimetres x 6 centimetres.

The other is two centimetres wider, but the length is two centimetres less." Hmm, "Which one has the larger area and by how much?" And B, "The entire frame has an area of 96 centimetres squared.

What dimensions could the frame have? What dimensions an area could the photo then have, so that it will fit inside the frame?" So the frame might not necessarily be that particular rectangular shape.

It might be different, it might be longer, thinner, etcetera, etcetera, but something x something, is going to = 96 centimetres squared.

Okay, lots of possibilities there.

Pause the video.

Good luck.

Off you go.

And welcome back.

Let's see how you got on.

Let's give you some feedback for that.

So number one, some possibilities for the bookmark with an area of 30 centimetres squared, you could have 3 x 10, or 10 x 3, 5 x 6, or 6 x 5.

And they're the times tables possibilities, but there are possibilities outside the times tables, such as 1 centimetres x 30 centimetres, = 30 centimetres.

That will be a very thin long bookmark, wouldn't it? I'm not sure that'd be a good one, but it's a possibility.

And 2 centimetres x 15 centimetres, for example.

And you might even have gone into decimals, something like 4 centimetres x 7.

5 centimetres.

That's a possibility.

And number two, let's have a look at that.

So this is the photo problem.

"Which one has the larger area and by how much?" Let's have a look.

Well the first one, 6 x 6 is 36.

So 36 centimetres squared.

8 x 4 = 32, so 32 centimetres squared.

So there wasn't much in it, it was the slight difference.

And then if we do 36 centimetres, take away 32 centimetres squared, that gives us four centimetres squared.

And I would count on from 32 to 36.

So the first photo has a larger area by four centimetres squared.

And B, "The entire frame's got an area of 96 centimetres squared.

What could the dimensions be?" So there's lots and lots of possibilities here depending on what your dimensions were for the frame.

So, for example, if the dimensions were 8 centimetres x 12 centimetres", which equals 96 centimetres square, that's a possibility.

The photos of the area, they would have to be smaller, wouldn't they? So you could have 7 x 11, 6 x 10, and 5 x 9.

So different sizes for the photos, lots of possibilities.

It's quite a tricky one there.

"For the dimensions of the photo frame, accept any pair of numbers with a product of 96, which is 6 centimetres x 16 centimetres", that's one that's outside the times tables.

And another one that's outside, "4 centimetres x 24 centimetres." Would also give an area of 96 centimetres squared.

Are you ready for cycle B? "Area problems involving compound shapes." So before we focused on rectangles, let's think about compound shapes.

Compound rectilinear shapes.

Okay.

"A square measures 40 centimetres x 40 centimetres.

A square is then removed from it", and you can see that.

So try and picture it was a square and then it's had a square removed from it.

And the square that's removed was "half the length and half the width of the original square." And we know what the length and width of the original square were.

That one was half of it.

"What's the area of the remaining shape?" Hmm.

So have a look at that.

Have a think about that.

We've got lots of information there.

We need to do something with the numbers.

Let's see.

So, says Lucas, "The original area was 40 centimetres x 40 centimetres = 1,600 centimetres squared.

Quite a big one.

"Half of that is 20 centimetres x 20 centimetres, which = 400 centimetres squared." So that would be the area of the square that's been taken out.

"1,600 centimetre squared, take away 400 centimetres squared = 1200 centimetres squared.

So that's the area of the remaining shape.

I think there are some other ways to do that as well.

And I think some might have one.

Let's have a look.

Yes, I thought so.

So she's started in the same way.

So "20 centimetres x 20 centimetres = 400 centimetres squared.

So just like Lucas, she's worked out the area of the square that's been taken away.

And she says, "There are three of those small squares left." Ah right.

That's clever, isn't it? So each of those three remaining small squares has got an area of 400 centimetres squared.

400 centimetre squared x 3 is 1200 centimetres squared.

So you can say they've both got the same answer, but they both went a very different way about it.

And that is the case with any of these compound rectilinear shape area problems. There's more than one approach.

It might be worth if you solve it trying a different way and seeing which is more efficient.

Jun saw it a different way altogether.

Let's have a look how he saw it.

"I saw the bottom rectangle and the square at the top." Oh okay.

So he split it into two different rectangles.

I see, one of which is a square.

Hmm.

And then you could work out the area of each.

Well done, Jun.

Let's have a look.

So the area of the bottom rectangle is 40 centimetres x 20 centimetres.

Yep, can you see that? 'Cause it's half of 40.

And that = 800 centimetres squared.

So it's partly the way there.

The area of the square is 20 centimetres x 20 centimetres.

All three of them were that out, didn't they? And that = 400 centimetres squared.

He's still not there yet though.

He's got to add those two together.

So the rectangle + the square.

800 + 400 = 1200 centimetres squared.

That was different.

I like that.

Good thinking, Jun.

So let's compare the different strategies and which strategy do you think was the most efficient.

And there's no right or wrong answer to that by the way.

So it's whatever you think.

Let's have a look.

So Lucas says, "I saw the whole square, and subtracted the small square." Yeah.

Jun says, "I split the shape into the bottom rectangle and the square at the top." Yep, sure.

And then Sam says, "I knew the area of the small square and I multiplied it by three." So what do you think was the most efficient? Well, Sam did the fewest calculations this time.

However, it really does depend on the question.

Let's do a check.

It's another one about squares or shapes that were squares.

So, "An identical square shape has been removed from each corner of a larger square shape.

What's the minimum information that you will need to know to work out the area of the remaining square?" 'Cause you might notice there's no numbers on that.

So what do you need to know and, "How would you solve it?" Okay, pause the video, discuss it with a partner.

If you've got a partner next to you, bounce ideas off each other.

If you come up with one way to do it, maybe find another and maybe even another.

Okay, pause the video and off you go.

Welcome back.

Did you manage to find one or more strategies, for working either area of the remaining shape? Well, "You need one of the dimensions of the large square, and one of the dimensions of one of the smaller squares." So you do the large one first, and then you need one of those dimensions of the smaller squares.

"Work out the area of a small square, multiply it by four." 'Cause there's four of those.

And then, "Subtract the area from the largest square." So a few different steps to working that one out.

You might have said something else altogether.

So remember, it's Lucas' birthday, happy birthday Lucas.

And he's a video games fanatic.

Maybe you are, too.

"His favourite game is 'Shape Shifters', where you have to make blocks fit together to make them disappear." Hmm, I know some games like that do you? "To celebrate his birthday, his friends have each baked him a small cake in the shape of one of the blocks from the game." Oh, what a lovely idea that is, isn't it? He's going to be really happy to see all those different cakes.

That's really thoughtful.

Okay, let's have a look.

So this is one of the blocks from the game.

And "This is a top view of Sam's cake." The one she's baking.

"Before the dimensions are added, see what can be observed about the shape." What could you say about that shape? What's it look like? What does it remind you of? What's there? What's missing? Etcetera, etcetera.

Let's have a look.

Well, says Sam.

"It's made up of what looks like five small squares." Yeah, we don't know for sure that they're squares, but they do look like it, don't they? "It looks like a rectangle missing, a smaller rectangle." Yes it does.

I can see that.

Can you see that, too? Big rectangle with a small one missing, yep.

And, "It looks like a rectangle on the left with a square attached on the right." Can you see that? Rectangle on the left.

Square on the top right.

Yep, I can see that.

All of these are useful observations which could help with any potential calculation strategies.

So let's have a look.

So we're going to have a look at a couple of different ways to solve that.

We could do 8 ÷ 2 = 4.

And the reason we're doing that is that means that one of the dimensions of the smaller rectangle is four centimetres.

So we're halving that eight to find out one of the dimensions, of one of the shapes.

So, 8 ÷ 2 = 4.

And then 16 ÷ 4, for the same reason, = 4.

So, both of the dimensions are four, meaning it is a square, it looked like one and it is one.

So it's a 4 x 4 square, each of those shapes.

4 x 4 = 16.

So the area of each square is 16 centimetres squared.

We're not there yet, but we're getting getting close, aren't we? We've got lots of good information there.

What do you think we need to do now with that information? There's five squares.

Yeah, so what shall we do? Do 16 centimetre squared x 5, and that gives us 80 centimetres squared.

So that's one way to work out the area.

But there's other ways, too.

Let's have a look at a different way.

Now you might have it started by drawing a rectangle around the shape.

And you can do that when you're problem solving.

Feel free to draw, annotate, do what you need to do.

Okay, so you might have seen a larger rectangle missing a smaller rectangle.

Now 16 x 8 would give us 128.

And that would be the area of what would've been the larger rectangle.

But we've got this missing gap.

And we can work out the dimensions of that gap, too.

The 12 x 4.

So 12 x 4 = 48.

Now that's missing, that rectangle's missing.

So what do you think we need to do? What operation do you think we need if it's missing? Subtraction.

128 - 48 = 80.

So the area of that shape is 80 centimetres squared.

So we found the area in two very different ways there, wouldn't you think? But which strategy do you think is the most efficient? And again, there's no right or wrong answer to that.

Which do you like? Which one do you think would've got you there the quickest? And is there an even more efficient way to work out the area? 'Cause that's what great mathematicians do.

They don't settle on the answer.

They'd see if they can do it more efficiently.

Find better ways.

Okay, so we've got another cake.

This is Jun.

"This is a top view of Sam's cake.

He's chosen a block from level two of the game where the shapes do not have squares inside." Okay, that's going to make it a little harder to work out the area, isn't it? But okay, I think we can do it.

See what can be observed about the shape again.

What could you say about that shape? Hmm.

What do you notice? What's there? What's missing? What does it remind you of, etcetera.

Well, says Jun, "It looks like a rectangle missing a smaller square." And it might not be a square, but it looks like it, doesn't it? It's certainly a rectangle.

"It looks like a rectangle on the bottom with two small squares on the top." Can you see that? It's a bit harder to see when they're not divided up, but I can picture that with my mind's eye.

I can see that.

And again, we don't know for sure that they're squares, by the way.

We would need the dimensions for that.

But it certainly looks that way.

And Jun says, "It could be split into three rectangles." Yes.

Can you do that with your mind's eye? I can picture using a horizontal line on it and splitting it into three rectangles.

And then I can picture using a vertical line on it and splitting it into three different rectangles.

So yeah, definitely.

So here are two of the methods that you could use to work out the area.

See which one you think is the most efficient.

Okay, so we've got some dimensions added this time.

Let's have a look.

Well, you could draw a rectangle around the shape.

We've done that a few times now, haven't we? You're very welcome to do that when you're problem solving.

Okay, there we go.

Can you see how that might help? And then we can work out the area of that larger rectangle.

So 9 x 6 = 54 centimetres squared.

And then 3 x 3 will give us the area of the missing square.

So that's nine centimetres squared.

What do you think we need to do if it's missing? Subtract.

54 - 9 = 45.

So that must mean the area of this cake is 45 centimetres squared.

There's another way to do it.

There always is.

You could draw inside the shape to partition it into rectangles.

Feel free to do that when you're problem solving.

Draw lines.

Here we go.

And that's not the only way, by the way, to split it into rectangle.

This is using vertical lines.

You could use horizontal lines.

That's just one way.

Can you see now that we've got three rectangles and we can work out the area of each.

So we've got 6 x 3 rectangle on the left.

That's 18.

We've got another one on the right.

That's 18.

And then we've got that small square looking shape.

I think it is a square.

It is 3 x 3 = 9.

What do you think we need to do with those three areas? We add them together, we combine them.

18 + 18, or you might have gone straight for the 36 there.

So 36 + 9 = 45.

Meaning the area of that compound rectilinear shape, that cake, is 45 centimetres squared.

Again, two very different ways of working out that area, but both effective.

Which one do you think was the most efficient there? Let's have a check.

Here's another block from "Shape Shifters", that Aisha will be turning into a cake.

See what you notice about the shape? Can you find at least two strategies for working out this area.

And which do you think will be the most efficient? So I'm looking at that and I can see that there's lots of different things.

See what you can notice.

Okay, pause the video, have a go.

If you find a solution, find another one, and maybe another one, off you go.

Welcome back.

As you get on, did you manage to compare your methods with a partner? It's good to do that.

You can bounce ideas off each other.

Often that will open up new avenues in your mind.

"We might have said, find the area of one of the smaller rectangles," which look like squares, definitely they might be squares, it might not.

They're definitely rectangles, "and multiply it by four." Yes, that would work.

It could, "Draw around the shape to create a large rectangle.

Then subtract the area of two smaller rectangles." Yes, can you see that's a draw around it.

Make a big rectangle, take away the two small rectangles.

Or, and I like this one, this is my favourite, "Move the top right rectangle into the bottom left gap," or alternatively top left into the bottom right, but either way, "to make one rectangle rather than a compound rectilinear shape." Then you just need to look at two dimensions.

It just becomes one calculation.

So that's a great method.

I like that a lot.

"You could calculate the area of the large rectangle on the top, and add it to the area of the smaller rectangle on the bottom." And I seem to remember from earlier, Jun doing something very similar to that.

So yes, that's that could work, too.

So look at that.

Four different ways to calculate the area of that one shape.

It's time for some final practise.

Number one, "A small square has been removed from a larger square.

What is the area of the remaining shape?" So remember it's a square.

It looks like we're missing some information, but we're not because it's a square.

Think what you know about squares.

It's got a dimension of 30 centimetres.

And the missing square has got a dimension of 10 centimetres.

Can you work out the area? Can you think of an efficient way to do it? And if you do it one way and you get time, maybe try another way.

Number two, "In honour of his favourite video game, 'Shape Shifters', remember Lucas' friends have each baked him a small cake in the shape of one of the blocks from the game.

Use the dimensions given to work out the area of each of the cakes." So A, that's got dimensions of 10 and 20.

What can you do with that information? B's got dimensions of 20 and 30, what can you do with that? And then C, that looks like it's from level two of the game.

We've got some information.

They are four centimetres, 16 centimetres, eight centimetres, and four centimetres.

What are you going to do with the that information? Gonna need to do some thinking here by the way.

And D, we've got another compound rectilinear shape.

Again it looks like from level two.

In fact, the rest all do.

That's the information given.

You might not need all of it.

You might need to do something with the numbers you've got as well.

The same for E and the same for F.

I'm just gonna give you a little clue about F.

There's something that I notice about F, something special about that shape that I notice.

I wonder if you are seeing it too.

If not, do take a little bit of time to see what you can notice, because I guarantee that it will save you time when you see it.

Okay? Pause the video, have fun with that.

It's going to feel a little challenging.

Persevere with it and be positive and you will get there.

I'll see you shortly for some feedback.

Welcome back.

Are you ready for some answers? How did you find that by the way? So, "A small square's been removed from a larger square.

What's the area of the remaining shape?" So if we do 30 x 30, that gives us 900 centimetres squared.

That's a large one.

10 x 10 is 100.

And then subtract that 100 from the 900, it gives us 800 centimetre squared.

Well done if you got that.

And then the dimensions of the cake and there's lots of ways you could do it, we're going to show you a suggested strategy.

It's not the only one there.

So for A, that's 125 centimetres squared.

That's the answer, that's the important part.

So we could do 10 centimetres ÷ 2 = 5 centimetres, 5 x 5 = 25 centimetres squared.

And then 25 centimetres squared x 5, = 125 centimetres squared.

Just one way to do it.

For B, it's 400 centimetres squared and there's one suggested strategy.

And then for C, it's 80 centimetres squared.

And once again, this is one of many suggested strategies.

And for D, you've got a couple of options here.

So you could take the rectangle, the 12 x 8 rectangle, that's a 96 centimetre squared rectangle, and then work out the area of the gaps and subtract them.

So 4 x 4, that's a square, that's 16 centimetres squared, and that comes up twice.

So if you do 96 centimetres squared, take away 16 centimetres squared, take away 16 centimetres squared, and you could have done that as just taking away 32 centimetres squared.

It gives you 64 centimetres squared.

And alternatively, and perhaps a little bit easier this way, the square on the top left could be moved into the gap on the top right to create an 8 x 8 square.

And I think that's more efficient.

That took fewer steps and fewer calculations.

E is 125 centimetres squared and there's a strategy for you.

And F is 96 centimetres squared.

And again, there's a strategy.

Now that's the one that I noticed something about.

I noticed if you move the bottom square into the gap, that is square shaped, it gives a 12 centimetres x 8 centimetres rectangle.

And that's times tables fact.

So that was as simple as knowing that 12 x 8 is 96 centimetres squared.

So when you saw it, it was maybe one of the easiest questions of the day.

We've come to the end of the lesson.

And in fact we've come to the end of the unit.

Our lesson today has been used knowledge of area to solve problems in a range of contexts.

"If the area of a rectangle is known, but the dimensions are not, many possibilities exist.

You need to consider which numbers can be multiplied to give the product that is the same as the area.

The area of a compound rectilinear shape can be calculated in more than one way, and often there's an approach that is most efficient.

Take some time to consider this before working out the area." And that is often time very well spent.

I hope you're very proud of yourself.

You certainly should be.

It's been a great pleasure working with you.

Hopefully I'll see you again soon.

But until then, take care.

Enjoy the rest of your day and goodbye.